Motivic integration and Milnor fiber

We put forward in this paper a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called"nonarchimedean Milnor fiber"that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski-Loeser construction. We also establish, in a much more intuitive manner, a new Thom-Sebastiani formula, which can be specialized to the one given by Guibert, Loeser, and Merle. Finally, applying $T$-convex integration after descent, matching the Euler Characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski-Loeser proof, but is also free of other sophisticated algebro-geometric machineries.


Introduction
Recent years have seen significant development in applying Hrushovski-Kazhdan's integration theory to the study of Denef-Loeser's motivic Milnor fiber and related topics. The main goal of this paper is to articulate a uniform narrative on such interactions, and thereby not only recover several fundamental results regarding motivic Milnor fiber but also subjugate them to the same principles afforded by the new perspective, and hopefully open up new fronts of inquiry in the process. This narrative is summarized in the diagram (1.2) below.
More concretely, we shall reconstruct motivic Milnor fibers as motivic integrals, establish a general type of Thom-Sebastiani formula, and retrieve invariants of the corresponding topological Milnor fibers, all without using resolution of singularities. In fact, there are several variants of the Hrushovski-Kazhdan style integration at play here and their synergy is the driving force of our telling. Among these variants, the central one is of course the original construction as developed in [22]. It works for any algebraically closed valued fields of equal characteristic 0 and is flexible enough to allow arbitrary choice of parameter spaces that satisfy certain mild conditions. Varying the parameter space enables one to study different categories of definable sets that are equipped with suitable Galois actions, which is highly desirable in the applications we are interested in. Such a perspective is first put forward in [23] for the purpose of finding a resolution-free construction of the complex motivic Milnor fiber, among other things (see also [29,25] for further developments).
To begin with, by an (algebraic) variety over a field k, we mean a reduced separated k-scheme of finite type. We denote by Var k the category of varieties over k.
The Grothendieck semiring K + C of a category C is the free semiring generated by the isomorphism classes of C, subject to the usual scissor relation [ [B] denote the isomorphism classes of the objects A, B and " " is certain binary operation, usually just set subtraction; additional relation may be imposed, to be determined in context. Sometimes C is also equipped with a binary operation -for example, cartesian product of sets or (reduced) fiber product of varieties -that induces multiplication in K + C, in which case K + C becomes a commutative semiring. The formal groupification K C of K + C is then a commutative ring. If a group G acts on the objects of C and the morphisms of C are G-equivariant, that is, they commute with G-actions, then the corresponding G-equivariant Grothendieck ring is denoted by K G C. If G = lim n G n is profinite then we shall always impose the condition that a G-action factor through some G n -action. The archetype of this kind of Grothendieck rings is Kμ Var C , whereμ is the procyclic group of roots of unity (the limit of the inverse system of groups µ n of nth roots of unity).
In this introduction, for simplicity, we shall just consider a nonconstant polynomial function f : (C d , 0) −→ (C, 0) such that 0 is a singular point, that is, ∇f (0) = 0. For 0 < η ≪ δ ≪ 1, the topological type (or even the diffeomorphism type) of the set F a =B(0, δ) ∩ f −1 (a), whereB(0, δ) is the closed ball of radius δ centered at 0, is independent of the choice of η, δ, and a ∈ (0, η]. This topological type, referred to as the (closed) Milnor fiber of f , is denoted by F f . The open Milnor fiber, where the open ball B(0, δ) is used, is also of interest, but more so in the real environment than in the complex one. We will come back to this later.
Let L be the space of formal arcs on C d at 0. So each element in L is of the form γ(t) = (γ 1 (t), . . . , γ d (t)), where γ i (t) is a complex formal power series and γ i (0) = 0. Let L m be the space of such arcs modulo t m+1 (also referred to as "truncated arcs"). Consider the following locally closed subset of L m : It may be viewed in a natural way as the set of closed points of an algebraic variety over C and carries a natural µ m -action. The motivic zeta function attached to f is then the generating series whose coefficients are in effect the "μ-equivariant motivic volumes" of the sets of truncated arcs above: ].
It is shown in [9,10] that Z f (T ) is rational and the motivic Milnor fiber S f := − lim T →∞ Z f (T ) is then extracted from this rational expression via a formal process of sending the variable T to infinity (this process is also summarized in [23, § 8.4]). Of course, to justify calling S f a "Milnor so-and-so" one needs to show, at the very least, that invariants of the topological Milnor fiber F f can be recovered from it. This is indeed the case for, say, the Euler characteristic and the Hodge characteristic.
Originally, both the proof that Z f (T ) is rational and the proof that the Euler (or Hodge) characteristics coincide rely on resolution of singularities. More recently, in [23], these results are established by way of a more conceptual construction, namely the Hrushovski-Kazhdan integration. To briefly outline the methodology, we work in the field C((t ∞ )) = m∈Z + C((t 1/m )) of complex Puiseux series, also simply denoted byC. This field is the algebraic closure of the field C((t)) of complex Laurent series. A typical element takes the form x = n∈Z a n t n/m for some m ∈ Z + such that its support supp(x) = {n/m ∈ Q | a n = 0} is well-ordered, in other words, there is a q ∈ Q such that a n = 0 for all n/m < q. We think of k := C as a subfield ofC via the embedding a −→ at 0 . The map val :C × −→ Q given by x −→ min supp(x) is indeed a valuation, and its valuation ring ] consists of those series x with min supp(x) ≥ 0 and its maximal ideal M of those series x with min supp(x) > 0. Its residue field k admits a section onto k and hence is isomorphic to C. It is well-known that (C, O) is an algebraically closed valued field.
For a series x = n∈Z a n t n/m ∈C with val(x) = p/m, let rv(x) = a p t p/m , which is called the leading term of x. Then the motivic zeta function attached to f may be expressed as where the coefficients H m (X f ) are Hrushovski-Kazhdan integrals of definable sets that take values in Kμ Var C [[A] −1 ] and the so-called nonarchimedean Milnor fiber of f is a definable set over the parameter space (the "ground field") S = C((t)). Formulated in this way, the rationality of Z f (T ) essentially follows from certain computation rules of (convergent) geometric series. That the Euler characteristics of S f and F f coincide follows from the fact that we can express both the Euler characteristic of each coefficient of Z f (T ) and the Euler characteristic of F f in terms of traces of the monodromy action on the cohomological groups of F f , where the twisted constructible sets in the residue field k and Γ[ * ] is the category of definable sets in the value group Γ (as an o-minimal group), both are graded by ambient dimensions. The objects of RES[ * ] are twisted because the short exact sequence at the bottom of (2.1) does not admit a natural splitting, and K Γ[ * ] is not the Grothendieck ring of o-minimal groups because not all definable bijections are admitted as morphisms. Anyway, we have two retractions from K RV[ * ] onto a quotient !K RES of K RES (the gradation is forgotten), reflecting the fact that there are two Euler characteristics in the Γ-sort; these are labeled E b , E g in (1.2). The isomorphism Θ is constructed as in [23, § 4.3].
The motivic zeta function Z f (T ) now resides in Kδ ]. However, the coefficients of Z f (T ) requires a kind of crude volume forms and the integral (or other variants in [22]) is not adequate for the task. Significant modifications are in order. This work has been carried out in [18] in order to correct an oversight in [23], resulting in the canonical isomorphism ⋄ in (1.2).
The category µVF ⋄ [ * ] consists of the proper invariant objects of VF * and the category µRV db [ * ] the doubly bounded objects of RV[ * ], all equipped with Γ-volume forms; see § 2.3 for the precise definitions. The nonarchimedean Milnor fiber X f of f is an object of µVF ⋄ [ * ] (with the trivial volume form). Note that µVF ⋄ [ * ] is also graded since, as in classical measure theory, gradation by ambient dimensions is a necessity in the presence of volume forms (a curve has no volume if considered as a subset of a surface). Also, the ideal (P Γ ) is homogenous but is no longer principal. We may again express K µRV db [ * ] as a tensor product of two other Grothendieck rings K µRES[ * ] and K µΓ db [ * ]. Since the objects of µΓ db [ * ] are doubly bounded, the two Euler characteristics coincide and consequently there is only one retraction onto !K RES, which is labeled E ⋄ in (1.2).
The henselian field C((t 1/m )), m ∈ N, is considered as an L RV -substructure ofC and, as such, its value group Γ(C((t 1/m ))) is identified with m −1 Z. Corresponding to each C((t 1/m )) there is a homomorphism h m from a subring K ♮ µRV db directly, but to establish its significance, we need to compare it with the zeta function construction. It is this reason that forces us to work with an integral whose target only involves doubly bounded sets in RV, namely ⋄ , instead of , so as to facilitate the computation of the coefficients of Z f (T ). Without the top row, the diagram (1.2) commutes with the dotted arrows too. The element (Θ • E g • )([X f ]) may be attached to f directly as well, but then its significance is unclear, except in the bottom row. We will say more about this below.
Let RVar be the category of real varieties in the sense of [2]. Taking real points and forgetting theδ-actions, we can specialize H m (X f ) to K RVar and thereby obtain the real motivic Milnor fiber of f in K RVar[[A] −1 ]. However, we are more interested in a subtler construction that is indigenous to the real algebraic environment.
Since f is assumed to be defined over R, it may be realized as a real function (R d , 0) −→ (R, 0). The open and the closed Milnor fibers are constructed as before, but denoted by F + f ,F + f since, in the absence of monodromy, replacing (0, η] with [−η, 0) will, in general, result in different topological types F − f ,F − f . So the qualifiers "positive" and "negative" should be tagged on in the terminology if we are to look at the whole picture. The difference between F + f andF + f is more significant in real geometry.
The sets of real truncated arcs are denoted by L m (R). Replacing L m with L m (R) in X f,m , we get a real variety X 1 f,m . The complexification X 1 f,m ⊗ C of X 1 f,m is a variety over C, which is isomorphic to X f,m , and carries a natural δ m -action, where δ m = µ m ⋊ Gal(C/R). Consequently, f,m inherits a natural µ 2 -action from X 1 f,m ⊗ C. This is indeed how the homomorphism Ξ in (1.2) is constructed.
As a subfield,R inherits fromC a valuation map, a valuation ring, a leading term map, etc. The pair (R, O(R)) forms a henselian valued field. There is a general procedure to specialize the integral to sets in any henselian subfield ofC, in particular, for those inR over R((t)). It only works for constructible sets, that is, quantifier-free definable sets (all definable sets inC are constructible becauseC eliminates quantifiers), since, after all,R is not an elementary submodel ofC. The corresponding homomorphisms between the Grothendieck rings appear in the middle row of arrows in (1.2).
Applying Ξ termwise to Z f (T ) brings about a (positive) motivic zeta function Z 1 f (T ), which belongs to ]; there is of course a negative one too. The rationality of Z 1 f (T ) and hence the existence of the real motivic Milnor fiber S 1 and hence S 1 f may indeed be computed purely in the real algebraic environment as (ΘR f ]). The next step is to justify calling S 1 f a Milnor fiber by recovering invariants ofF + f from S 1 f . Actually the only known additive invariant ofF + f is the topological (or semialgebraic) Euler characteristic χ(F + f ). It is shown in [5,Theorem 4.4] that χ(F + f ) does agree with χ BM (S 1 f ), where χ BM is the Borel-Moore Euler characteristic, also labeled as such in (1.2); note that the real motivic Milnor fiber there is the forgetful image of S 1 f in K RVar[[A] −1 ]. Their method relies on a real analogue of the A'Campo-Denef-Loeser formula, which needs resolution of singularities. Unfortunately, with the absence of monodromy in the real case, we cannot follow the method of [23] outlined above to get a resolution-free proof, at least, perhaps, not without further elucidating the effect of the monodromy action on the complexification of the real Milnor fibers as suggested by [27].
Going through a different route, we use the theory of motivic integration for T -convex valued fields as developed in [34]. This theory is rich in expressive power and hence can handle all the definable objects in the algebraic environment. On the other hand, its expressive power is also its limitation in yielding algebro-geometric information since, in the corresponding categories of definable sets, there are much more morphisms that can cause loss of algebro-geometric data when passing to the Grothencieck rings. Nevertheless, it should retain much of the numerical information.
We work inR, which is now viewed as a real closed field equipped with both a total ordering and a valuation (or more generally a polynomially bounded T -convex valued field). This structure is expressed in a first-order language L T RV , which still has two sorts VF and RV. ], which is understood as a topological zeta function attached to f . The definable set X 1 f may be approximated by a sequence of semialgebraic setsF r , r ∈ R + , whose semialgebraic homology eventually stabilizes. The Euler characteristic of this stabilized semialgebraic homology is equal to, on the one hand, χ(F + f ) and, on the other hand, ( f ]) and hence χ BM (S 1 f ).
The same argument shows that . As a corollary, we get χ([F a,r ]) = (−1) d+1 χ([F a,r ]). This simply comes from an equality at the motivic level (the second and the third rows in (1.2)). However, at that level, it is unclear if there is a geometric interpretation of the image (Θ This approach also works in the complex setting, consideringC asR 2 and hence X f as an object of TVF * . It shows in particular that χ(F f ) is equal to the Euler characteristic of S f , as in [23,Remark 8.5.5], but without using even quasi-unipotence of local monodromy. Note that, over C, the Euler characteristics of the open and the closed Milnor fibers coincide, so if S f encodes information on both of them, one cannot see it at this level.
We can extend Θ • E g • further by composing the Hodge-Deligne polynomial map. It is shown in [30,Proposition 3.23] that this actually gives the Hodge-Deligne polynomial of the limit mixed Hodge structure associated with a variety. Extending ΘR • E b,R • R by composing the virtual Poincaré polynomial map, we get a similar homomorphism into Z [u]. It would be interesting to investigate if it too encodes information on limit structures. But of course we are ahead of ourselves here because such limit structures are not yet available in the real setting.
Finally, in showcasing the potential of the framework underlying (1.2), we describe another main result of this paper, namely a new (local) Thom-Sebastiani formula in mixed variables, extending that in [20] (the results in [19,21] are for separate variables and hence overlap to a much lesser extent with the case we establish here). A precursor of our method has already been used in [25] to recover the Thom-Sebastiani formula of [8,26] To that end, we still work inC, but change the parameter space to C ∪ Q. Unlike the previous situations, the (model-theoretic) automorphism group Aut(C /C ∪ Q) is much larger than the automorphism group Aut(RV / k × ∪ Q). It is this latter group, henceforth abbreviated asτ , that we need. It is isomorphic to the group lim n (C × ) n , where each (C × ) n is just a copy of C × and the transition morphisms are the same as in the limitμ = lim n µ n . More concretely, the elements inτ may be identified as sequencesâ = (a n ) n of nth roots of a, a ∈ C × , satisfying a n kn = a k . Such an element acts onC by the equationâ · t 1/n = a n t 1/n . The Thom-Sebastiani formalism is typically concerned with expressing the Milnor fiber of a compound function h(f 1 , . . . , f l ) in terms of the Milnor fibers of the component functions f 1 , . . . , f l . The classical results and most of the later generalizations can only handle the case of separate variables, that is, h(f 1 , . . . , f l ) is regarded as a function on the product i X i , where X i is the source variety of f i , and often h is just a linear form. Our formula, on the other hand, is much more sophisticated.
Let g : (C d , 0) −→ (C, 0) be another nonconstant polynomial function, singular at 0, and h(x, y) a polynomial of the form (1) are both interpreted as the zero function and ϑ (1) as 1 ∈ Q. Let X ♯ f denote the restriction of f to the set {x ∈ M d | val(f (x)) = 1}, similarly for other functions into the affine line. The fibers of X ♯ f over the set t+t M form precisely the nonarchimedean Milnor fiber X f , so X ♯ f may be called the nonarchimedean The category Var ϑ (ı) C consists of (ϑ (ı) , n)-diagonal varieties over G ı m , for some n ∈ Z + , with good G m -actions; see § 4.2 for the unexplained terms. Each object of Var ϑ (ı) C may be thought of as equipped with aτ -action that factors through, for some n ∈ Z + , the canonical epimorphism τ n :τ −→ (C × ) n . For ı = 1 we write the Grothendieck ring K ϑ (ı) Var C as K 1 Var C ; actually Var 1 C is just the category Var Gm Gm in [20] and hence is equivalent to the category of varieties over C witĥ µ-actions. There is a Kτ Var C -module homomorphism which is referred to as a convolution operator.
Suppose that ; we call it the motivic Milnor fiber of g N ⊕ f (ı) over G ı m . Then our Thom-Sebastiani formula states that, in here the first and the third terms are the motivic Milnor fibers over G m but restricted to the indicated zero sets (in [20] a variant of this is called iterated motivic vanishing cycles). As before, the whole construction can be specialized to the real setting if f , g are defined over R, which enables us to recover the Thom-Sebastiani formula obtained in [3], in a more general form.
A novel perspective behind (1.3) is that S ♯ h(f,g) may be decomposed into terms corresponding to combinatorial data that can be read off of the tropical curve of h(x, y). This actually suggests that our method can handle polynomials more complicated than h(x, y), for instance, those with more variables and even mixed terms. However, the complexity of the combinatorics involved will become quite heavy, perhaps disproportionately so, as it is unclear how the ground gained can shed new light on the geometry and topology of the singularities in question. Thus we have chosen to just present a simple case that is already beyond what is known in the literature.

Hrushovski-Kazhdan style integration
The first-order language L RV has two sorts VF, RV and a cross-sort map rv : VF −→ RV. Let ACVF denote the L RV -theory of algebraically closed valued field of equal characteristic 0. We will not repeat the formal definition of the language L RV or the theory ACVF here, and refer the reader to [32, § 2] for details. Every valued field (K, val) may be naturally interpreted as an L RV -structure and, as such, its structure may be summarized as follows. Let O, M, and k be the corresponding valuation ring, its maximal ideal, and the residue field, respectively. Let VF = K, RV = K × /(1 + M), and rv : K × −→ RV be the quotient map. For each a ∈ K, the valuation map val is constant on the set a + a M and hence there is an induced map vrv from RV onto the value group Γ = val(K). The diagram then commutes, where the bottom sequence is exact.
A cross-section of Γ is a group homomorphism csn : Γ −→ VF × such that val • csn = id. The corresponding reduced cross-section of Γ is the function csn = rv • csn : Γ −→ RV. These are usually augmented by csn(∞) = 0 and csn(∞) = ∞. If such a reduced cross-section exists then it induces an isomorphism RV ∼ = Γ ⊕ k × . In general this is not guaranteed, in other words, the short exact sequence above may not split.
Example 2.1. Let K be the fieldC of complex Puiseux series and val :C × −→ Q the standard valuation. Let RV =C × /(1 + M) and rv :R × −→ RV be the quotient map. This turnsC into an L RV -structure, which is indeed a ACVF-model.
The leading term of a series inC × is its first term with nonzero coefficient. It is clear that two series x, y have the same leading term if and only if rv(x) = rv(y) and hence RV is isomorphic to the subgroup ofC × consisting of all the leading terms. There indeed exists a natural isomorphism given by a q t q −→ (q, a q ) from this latter group of leading terms to the group Q ⊕ C × , through which we may identify RV with Q ⊕ C × (not definably, though).
We may think ofμ as the Galois group Gal(C /C((t))), since they are canonically isomorphic. For each element ξ = (ξ n ) n ∈μ, the assignment n −→ ξ n t 1/n indeed induces a reduced cross-section csn ξ : Q −→ RV, and the map given by ξ −→ csn ξ is indeed a bijection betweenμ and the set Ω of reduced cross-sections csn : Q −→ RV with csn(1) = rv(t); in other words,μ acts freely and transitively on Ω via multiplication in the obvious way.
In this section, following the tradition in the model-theoretic literature, we work in a sufficiently saturated model U of ACVF, together with a fixed parameter space S, which is a substructure of U. This is of course a matter of convenience, otherwise one needs to change the model one is working in whenever compactness is applied. We assume that the map rv is surjective in S (but the value group Γ(S) of S could be trivial) and the definable closure dcl S of S equals S (there is no need to know what a definable closure is beyond this point). Among other things, this latter condition implies that if Γ(S) is nontrivial then the underlying valued field of S is henselian (in fact this is equivalent to the condition dcl S = S). So by a definable set we mean an S-definable set, unless indicated otherwise.
A pillar of the structure of definable sets in U is C-minimality, meaning that every definable subset of VF is a boolean combination of (definable) valuative discs.
Notation 2.2. There is a special element ∞ = rv(0) in the RV-sort. For simplicity, we shall write RV to mean the RV-sort without the element ∞, and RV ∞ otherwise -that is, RV = rv(VF × ) and RV ∞ = rv(VF) -although the difference rarely matters (when it does we will of course provide further clarification). Also write RV •• ∞ = rv(M) and RV •• = RV •• ∞ {∞}. Terminology 2.3 (Sets and subsets). By a definable set in VF we mean a definable subset in VF, by which we just mean a subset of VF n for some n; similarly for other (definable) sorts or even structures in place of VF that have been clearly understood in the context, such as RV ∞ , k, M, or any substructure M of U. In particular, a definable set without further qualification means a definable set in U, that is, a definable subset of VF n × RV m ∞ for some n, m ∈ N.   [i,j] , A (i,j) , A S , etc.; in particular, we shall frequently write A VF and A RV for the projections of A into the VF-sort and the RV-sort coordinates.
Unless otherwise specified, by writing a ∈ A we shall mean that a is a finite tuple of elements (or "points") of A, whose length is not always indicated.
We shall write {t} × A, {t} ∪ A, A {t}, etc., simply as t × A, t ∪ A, A t, etc., when it is clearly understood that t is an element and hence must be interpreted as a singleton in these expressions.
For a ∈ AẼ, the fiber {b | (b, a) ∈ A} ⊆ A E over a is often denoted by A a . Note that the distinction between the two sets A a and A a × a is usually immaterial and hence they may and shall be tacitly identified. In particular, given a function f : A −→ B and b ∈ B, the pullback f −1 (b) is sometimes written as A b as well. This is a special case since functions are identified with their graphs. This notational scheme is especially useful when the function f has been clearly understood in the context and hence there is no need to spell it out all the time.
Another pillar of the structure of definable sets in U is the so-called orthogonality between the k-sort and the Γ-sort, meaning that every definable subset A of U n with pr ≤k (A) in k and pr >k (A) in Γ is a finite union of products A ′ × A ′′ ⊆ k k ×Γ n−k ; in particular, if A is the graph of a function on pr ≤k (A) or pr >k (A) then its image is finite.
Notation 2.5. Semantically, we shall treat the value group Γ as a definable sort (the Γ-sort) consisting of "imaginary" elements (that is, classes of definable equivalence relations). However, syntactically, any reference to Γ may be eliminated in the usual way and we can still work with (much more cumbersome) L RV -formulas for the same purpose.
We shall write γ ♯ , γ ∈ Γ, when we want to emphasize that it is the set vrv −1 (γ) ⊆ RV that is being considered. More generally, if I is a set in Γ then we write

Categories of definable sets.
Definition 2.6. The VF-dimension of a definable set A, denoted by dim VF (A), is the largest natural number k such that, possibly after re-indexing of the VF-coordinates, the projection pr ≤k (A t ) has nonempty interior in the valuation topology for some t ∈ A RV .
It is a fact that if A ⊆ VF n is definable then dim VF (A) equals the Zariski dimension of the Zariski closure of A. Definition 2.7 (VF-categories). The objects of the category VF[k] are the definable sets A of VF-dimension no more than k such that pr VF ↾ A is finite-to-one. Any definable bijection between two such objects is a morphism of VF[k]. Set VF * = k VF[k].
As soon as one considers adding volume forms to definable sets in VF, the question of ambient dimension arises and, consequently, one has to take "essential bijections" as morphisms.
We will not recall the definition of the Jacobian Jcb VF F of a morphism F of VF[k] here since there will be no use of it except in the following definition; see [32, Definition 9.6] for reference. Definition 2.8 (VF-categories with Γ-volume forms). An object of the category µVF[k] is a definable pair (A, ω), where A ∈ VF[k], A VF ⊆ VF k , and ω : A −→ Γ is a function, which is understood as a definable Γ-volume form on A. A morphism between two such objects (A, ω), (B, σ) is a definable essential bijection F : A −→ B, that is, a bijection that is defined outside definable subsets of A, B of VF-dimension < k, such that, for almost every x ∈ A, ω(x) = σ(F (x)) + val(Jcb VF F (x)).
We also say that such an F is Γ-measure-preserving.
For example, there are an essential bijection between the sets M and M 0 and hence a morphism between the objects (M, 0) and (M 0, 0) in µVF [1]. Notation 2.9. In [22], the category µVF[k] is denoted by µ Γ VF[k] to indicate that the volume forms take values in Γ as opposed to RV. Here the subscript "Γ" is dropped since we will not consider RV-volume forms.
In the definition above and other similar ones below, for the cases k = 0, the reader should interpret things such as VF 0 and how they interact with other things in a natural way. For instance, VF 0 may be treated as the empty tuple, the only definable set of VF-dimension < 0 is the empty set, and Jcb VF is always 1 on sets that have no VF-coordinates. So (A, ω) ∈ µVF[0] if and only if A is a finite definable subset of RV n ∞ for some n.  It is certainly more convenient to work with representatives than equivalence classes. In the discussion below, this quotient category µVF[k]/∼ will almost never be needed except when it comes to forming the Grothendieck semigroup or, by abuse of terminology, when we speak of two objects of µVF[k] being isomorphic.  More generally, if f : U −→ RV k ∞ is a definable finite-to-one function then (U, f ) denote the obvious object of RV [≤k]. For example, the inclusion {∞} / / RV ∞ gives rise to an object of RV[0], the inclusion {(1, ∞)} / / RV 2 ∞ gives rise to an object of RV [1], and so on. Often f will be a coordinate projection (every object in RV[ * ] is isomorphic to an object of this form). In that case, (U, pr ≤k ) is simply denoted by U ≤k and its class in K + RV[k] by [U] ≤k , etc. Definition 2.14. Let U ⊆ RV n ×Γ m , V ⊆ RV n ′ ×Γ m ′ , and C ⊆ U × V . The Γ-Jacobian of C at ((u, α), (v, β)) ∈ C, written as Jcb Γ C((u, α), (v, β)), is the element where Σ(γ 1 , . . . , γ n ) = γ 1 + . . . + γ n . If C is the graph of a function then we just write C(u, α) instead of C((u, α), (v, β)).
Definition 2.15 (RV-and RES-categories with Γ-volume forms). An object of the category µRV[k] is a definable triple (U, f, ω), where (U, f ) is an object of RV[k] and ω : U −→ Γ is a function, which is understood as a definable Γ-volume form on (U, f ). A morphism between two such objects (U, f, ω), (V, g, σ) is an RV[k]-morphism F : (U, f ) −→ (V, g) such that, for every u ∈ U, Remark 2.18. If γ ∈ Γ is definable then it is in the divisible hull Q ⊗ Γ(S) of Γ(S), and vice versa. This does not mean, though, that the definable set γ ♯ ⊆ RV contains a definable point unless γ ∈ Γ(S).
The category µΓ fin [k] is the obvious full subcategory of µΓ[k]. where all the arrows are monomorphisms. The map determined by the assignment is well-defined and is clearly K + Γ fin [ * ]-bilinear. Hence it induces a K + Γ fin [ * ]-linear map which is a homomorphism of graded semirings. By the universal mapping property, groupifying a tensor product in the category of K + Γ fin [ * ]-semimodules is, up to isomorphism, the same as taking the corresponding tensor product in the category of K Γ fin [ * ]-modules; the groupification of Ψ is still denoted by Ψ. Similarly, there is a K + µΓ fin [ * ]-linear map We shall abbreviate ⊗ K + Γ fin [ * ] , ⊗ K + µΓ fin [ * ] as "⊗" below when no confusion can arise.  [23] that depend on it. These issues are now resolved in [18], and the modified constructions there are also crucial for this paper, which we shall recall below in due course.  The corresponding principal ideal of K RV[ * ] is thus generated by the element P − 1.
Similarly, let µI sp be the semiring congruence relation on K + µRV[ * ] generated by the pair ( [1], [RV •• ]), which is homogenous, and the corresponding principal ideal of K µRV[ * ] is generated by the element P .  . Since these isomorphisms are obviously compatible with the inductive systems, passing to the colimit of the groupifications, we obtain a canonical isomorphism of rings Similarly, for each k ≥ 0 there is a canonical isomorphism of semigroups Taking the direct sum of the groupifications yields a canonical isomorphism of graded rings This is a combination of two main theorems, Theorems 8.8 and 8.29, of [22]. But it is not enough for our purpose. Another such isomorphism is needed.

2.3.
Integrating doubly bounded sets. We say that a set, possibly with Γ-coordinates, is bounded if, after applying the maps val, vrv, id in the VF-, RV-, Γ-coordinates, respectively, it is contained in a box of the form [γ, ∞] n , and doubly bounded if the box is of the form [−γ, γ] n . We say that an object (A, ω) ∈ µVF[k] is bounded or doubly bounded if the graph of ω is so; similarly in the other categories. In particular, an object (U, f, ω) ∈ µRV[k] is bounded if the graphs of f and ω are both bounded; actually, by [18,Lemma 3.26], if U is doubly bounded then the images of these functions are necessarily doubly bounded.
We shall only be concerned with doubly bounded sets. The corresponding restriction of µΨ is indeed an isomorphism: Terminology 2.29. A VF-fiber of a set A is a set of the form A t , where t ∈ A RV (recall Notation 2.4); in particular, a VF-fiber of a function f : A −→ B is a set of the form f t for some t ∈ f RV (here f also stands for its own graph), which is indeed (the graph of) a function. We say that A is open if every one of its VF-fibers is, f is continuous if every one of its VF-fibers is, and so on.
More generally, for two sets A, B with RV-coordinates, we say that a function f : A −→ B is covariant if every one of its VF-fibers f t is (α t , β t )-covariant for some (α t , β t ) ∈ Γ that depends on t. This is in line with Terminology 2.29. Accordingly, a set is invariant if (every VF-fiber of) its characteristic function is (−, 0)-covariant.
For example, finite sets A ⊆ VF are not invariant (or rather they are ∞-invariant, but ∞ is not allowed in the above definition). The maximal ideal M is α-invariant for every α ∈ Γ + , whereas M 0 is not α-invariant for any α ∈ Γ + , because the radii of its maximal open subdiscs tend to ∞ as they approach 0.
Since identity functions are relatively unary in any coordinate, if a morphism is piecewise a composition of relatively unary morphisms then it is indeed a composition of relatively unary morphisms.
Definition 2.34. The subcategory µVF ⋄ [k] of µVF[k] consists of the proper invariant objects and the morphisms that are compositions of relatively unary proper covariant morphisms whose inverses are also proper covariant.
Remark 2.35. Obviously the composition law holds in µVF ⋄ [k] and hence it is indeed a category. Moreover, every morphism in it is a bijection, as opposed to merely an essential bijection, and is in effect required to admit an inverse. So µVF ⋄ [k] is already a groupoid and there is no need to pass to a quotient category as in Remark 2.10. On the other hand, it does have nontrivial morphisms (see [18, Proposition 6.12, Remark 6.7]).
where t γ ∈ γ ♯ is any definable point. It also stands for the corresponding element in K µRV db [1] (with the constant volume form 0 on each component).
Clearly P γ does not depend on the choice of t γ ∈ γ ♯ . The ideal of K µRV db [ * ] generated by the elements P γ is denoted by (P Γ ). The images of (P Γ ) are contained in (P − 1), (P ) under, respectively, the obvious homomorphisms By [18,Corollary 6.24], the map µL induces a surjective homomorphism, which is still denoted by µL, between the graded Grothendieck rings By [18,Proposition 7.25], the kernel of µL in K µRV db [ * ] is indeed P Γ .
Theorem 2.37 ( [18,Theorem 7.27]). There is a canonical isomorphism of graded Grothendieck rings: 2.4. Uniform retraction to RES. The objects of the category RES are obtained from those of RES[ * ] by forgetting the function f in the pair (U, f ). Any definable bijection between two such objects is a morphism of RES. This is a full subcategory of RV * (see Definition 2.11) Notation 2.38. Denote the half interval (0, ∞) ⊆ Γ simply by H. We can associate two Euler characteristics χ g , χ b with the Γ-sort, which are distinguished by χ g (H) = −1 and χ b (H) = 0; they agree on doubly bounded definable sets, though. The two induced ring homomorphisms K Γ[ * ] −→ Z will also be denoted by χ g , χ b , or both by χ when no distinction is needed. The quotient maps from "K" to "!K" will all be denoted by ι. For simplicity, we will use the same notation for elements when passing from the former to the latter. Proposition 2.40. There are two ring homomorphisms such that • their ranges are precisely the zeroth graded pieces of their respective codomains, For the constructions of E g and E b , see [22,Theorem 10.5] or [18, § 5.1] or [23, § 2.5]. By the second clause above, they induce two homomorphisms on K RV[ * ]/(P − 1), which we shall just denote by E g , E b . . It is then routine to check that this epimorphism is also injective, and hence in the obvious way then it may be compared with E g . They are different since We can equalize them by forcing The composition of µE db and the forgetful homomorphism !K RES[ * ] −→ !K RES is denoted by E ⋄ . By [18,Remark 5.14], the diagram commutes: which may serve as an alternative and more direct construction of E ⋄ .
Remark 2.43. The homomorphism E b will be used in the construction of motivic Milnor fiber in § 3, but not E g , because it does not commute with E ⋄ (by (2.6), For the Thom-Sebastiani formula in § 4 to hold, we must also use E b (otherwise certain terms in the computation would not vanish, see Remark 4.23).
Remark 2.44. We shall only be interested in proper invariant objects of VF * . For those objects, the homomorphisms E g • , E b • only differ by a factor in !K RES[[A] −1 ]. To see this, we first note that the proof of Theorem 2.37 in [18] shows that every proper invariant object A ∈ VF * with dim VF (A) = n is isomorphic to an object of the form LU with (U , 0) ∈ µRV db [n]. So, in light of the isomorphism µΨ in (2.4) and the defining conditions of E g , E b in (2.5), we have In § 5, the T -convex versions of E b • , E g • yield the Euler characteristics of the closed and the open topological Milnor fibers. Then the equality just described may be specialized to one between these two numerical quantities; see Corollary 5.18.
2.4.1. With a reduced cross-section. We can add a reduced cross-section csn : Γ −→ RV to the language L RV , denote by L † RV the extension, and consider the corresponding integration theory; this has been worked out in [33]. We shall, however, only need a few facts about definable sets in RV in this setting. For the next few paragraphs we assume that U carries a reduced cross-section and work in an L † RV -expansion U † of U. Definability, if unqualified, is interpreted accordingly.
So the RV † -dimension dim RV † (A) of a definable set A in RV may be defined as the RV † -dimension of any Γ-partition of A. It may also be shown that there is a definable finite-to-one function The categories RES † [k], RES † are formulated similarly to RES[k], RES.

]). Every definable set in
i is a twistoid and the corresponding twistback is L RV -definable.
, such that its vrv-contraction is a bijection and its graph is a twistoid as well, then we say that (U i ) i is bipolar. Naturally U is called a bipolar twistoid if it admits a trivial bipolar twistoid decomposition.
Obviously a Γ-cohesive twistoid decomposition of a bipolar twistoid is also bipolar.
Lemma 2.52. Every L RV -definable set U ⊆ RV n admits a twistoid decomposition that is both bipolar and Γ-cohesive.
Proof. This follows from [22,Lemmas 3.21,3.25]. In more detail, the proof of [22,Lemmas 3.21] shows that the definable finite partition (D i ) i given by Lemma 2.50 can be refined so as to make the following condition hold: Next, there exists a matrix M i ∈ GL n (Z) such that N i M i is in lower echelon form (in general M i is not a product of "standard" column operations since Z is not a field, but it exists over any principal ideal domain). Observe that if N i M i does not have zero columns then D i must be a singleton. At any rate, the set M −1 i U i must be of the form pr ≤m (U i ) × I ♯ i ⊆ RV n , where n − m is the number of zero columns in N i M i and vrv(pr ≤m (U i )) is a singelton.
Proposition 2.53 ([33, Propositions 3.21, 3.30]). The assignment (2.9) does not depend on the choice of the twistoid decomposition and is invariant on isomorphism classes. The resulting map We consider RV[ * ] as a subcategory of RV † [ * ] and denote the induced homomorphism between the Grothendieck rings by In the current environment, the transition from "K" to "!K" is superfluous since we already where γ is any definable element in vrv(U) and [U γ ] is the indicated class in !K RES which actually does not depend on γ, ). It follows from this and Lemma 2.52 (actually Proposition 2.23 suffices) that we have a commutative diagram (2.10) is that it makes computation easier, essentially because there is no need to (explicitly) decompose K RV[ * ] into a tensor product as before.
Remark 2.54. We may replace χ b with χ g in (2.9) and thereby obtain a ring homomorphism E † g that also vanishes on the ideal (P − 1). The diagram (2.10)

Motivic Milnor fiber
We begin with a brief discussion on "descent" to henselian substructures M, which is based on [22, § 12]. The main cases of interest are M =C (or M =R, the field of real Puiseux series) and its henselian subfields M = C((t 1/m )), m ∈ Z + . The value groups Γ(C), Γ(R) are identified with Q. The value group Γ(C((t 1/m ))) is identified with m −1 Z. But to avoid confusion, we shall not write the residue field k as C (or R) -the latter being regarded as a subfield ofC -even though they are canonically isomorphic.
The parameter space S is going to be R((t)). It may seem at first glance that restricting parameters to R((t)) is unnecessary since every element inR is, after all, definable over R((t)). However, generally speaking, elements inR are definable over R((t)) only inR, not inC, in other words, they are not quantifier-free definable over R((t)) inR (to define them one needs to use the ordering, which is not quantifier-free definable).
3.1. Specialization to henselian subfields. Let M be a substructure of U in which the map rv is surjective. Recall that the substructure S is a part of the language and hence all other substructures contain it. If X ⊆ VF n × RV m is a definable (and hence quantifier-free definable) set then the trace of X in M, denoted by X(M), is the set of M-rational points of X, that is, Such a trace is also called a constructible set in M since it is indeed quantifier-free definable in M. Note that, however, if f : X −→ Γ is a definable function then the image f (X(M)) is not necessarily a set in Γ(M), but rather a set in the divisible hull On the other hand, if X is a set in Γ and f is a piecewise GL k (Z)-transformation on X then f (X(M)) is of course a set in Γ(M); this is the situation in the Γ-categories. , then M is functionally closed, that is, for any definable set X and any definable function f on X, the image f (X(M)) is a set that is definable in M (which then, ex post facto, is constructible in M). This is all we need to deduce the results in this section.
The rest of the material in this subsection will be needed in later sections.
is a constructible bijection in M then obviously there may or may not be a definable bijection between X and Y , let alone one whose trace equals g. But any quantifier-free formula that defines g in M also yields a definable bijection f : Our goal here is to derive a motivic integral that is associated with M. We call K + VF M , etc., M-constructible Grothendieck semirings associated with M.
Since M is functionally closed, it is routine to verify that the following binary relation is welldefined and is indeed a semiring congruence relation: 3.2. Grothendieck rings in real and complex geometry. The complexification of a variety X over R is denoted by X ⊗ R C, which is a variety over C endowed with an antiholomorphic involution coming from the complex conjugation c over C; the Grothendieck ring of the corresponding category is denoted by K c Var C . Conversely, to every quasi-projective variety Y over C endowed with an antiholomorphic involution there corresponds a unique variety X over R such that Y ∼ = X ⊗ R C. So extension of scalars induces an isomorphism K Var R / / K c Var C . Taking the fixed points of the set X(C) of the complex points of a variety X over R under the complex conjugation gives a real variety in the sense of [2]; this is denoted by X(R). Such sets of real points of varieties over R, considered with their sheaves of regular functions over R, form the category RVar of real varieties, and taking real points induces a homomorphism K Var R / / K RVar. We consider also an equivariant version of the Grothendieck ring of complexified varieties over R, taking into account group actions by roots of unity that are compatible with the complex conjugation.
Notation 3.5. Recall that the procyclic groupμ = lim n µ n is canonically isomorphic to the Galois group Gal(C /C((t))). Denote the dihedral group Gal(C/R) ⋉ µ n by δ n , where the Gal(C/R)-action on µ n corresponds to taking the inverse. Setδ = lim n δ n , which is canonically isomorphic to Gal(C /R) ⋉μ, where the action of Gal(C /R) onμ corresponds again to taking the inverse. It may also be identified with the Galois group Gal(C /R((t))).
The conjugation automorphism ofC is also denoted by c. A straightforward computation shows that cσcσ = 1 for any topological generator σ ofμ and hence for every element ofμ. Indeed, for any integer m ≥ 0, cσ m+1 cσ m+1 = cσ m ccσcσσ m = cσ m cσ m , and hence, by a routine induction, cσ m cσ m = 1.
Definition 3.6. Aδ-actionĥ on a complexified variety X over R is good if it factors through some δ n -action and the induced Gal(C/R)-action is the canonical antiholomorphic involution.
The category of complexified varieties over R with goodδ-actions consists of objects of the form X = (X,ĥ), where X is a complexified quasi-projective variety over R andĥ is a goodδ-action on X, andδ-equivariant morphisms between such objects.
The Grothendieck ring of this category is denoted by K ♭,δ Var R . The ring Kδ Var R is the quotient of K ♭,δ Var R by the ideal generated by the elements of the form Remark 3.7. Let X = (X,ĥ) be a complexified variety over R with a goodδ-action. If σ is a topological generator of µ then cσ is another antiholomorphic involution on X and hence gives rise to another complexified variety X ′ = (X ′ ,ĥ) over R with a goodδ-action such that [X] = [X ′ ]. So σ induces a ring involution on K ♭,δ Var R and also Kδ Var R .
] −→ X on a variety X over C may have branches, which are represented by Puiseux series inC. Galois actions over C [ [ t ] ] on these branches encode certain information on the singularity in question and hence are an integral part of the construction in [23]. These Galois actions are gone when we restrict to real branches of real arcs, corresponding to the pairR and R [ [ t ] ], albeit a faint trace remains.
Remark 3.8. Recall the discussion on reduced cross-sections in § 2.4.1. We have seen in Example 2.1 above that there is a natural bijection betweenμ and the set of reduced cross-sections csn : Q −→ RV with csn(1) = rv(t) inC. Similarly, there is such a bijection betweenδ and such a set but with csn(1) = rv(±it). In contrast, there is only one such reduced cross-section inR, which is but another way of saying the fact that Gal(R /R((t))) is trivial. Nevertheless, if n is even then Gal(R((t 1/n ))/R((t))) ∼ = µ 2 , and there are two such reduced cross-sections in R((t 1/n )), determined by the two choices ±t 1/n , and if n is odd then there is only one. Definition 3.9. The category of real varieties with µ 2 -actions consists of objects of the form X = (X, h), where X is a real variety and h is a µ 2 -action on X, and µ 2 -equivariant morphisms between such objects.
The Grothendieck ring of this category is denoted by K ♭,µ 2 RVar. The ring K µ 2 RVar is the quotient of K ♭,µ 2 RVar by the ideal generated by the elements of the form Let [X] = [(X,ĥ)] ∈ Kδ Var R . Theĥ-orbit O x of any (closed) point x ∈ X is finite becausê h factors through some δ n -action. Since µ n is cyclic, the inducedμ-action on O x factors through a faithful µ dx -action with d x |n. Soĥ factors through a δ dx -action h dx . Let x ∈ X(R). For all (3.3) are essentially the same condition and the assignment [X] −→ [X(R)] does respect addition, we have indeed constructed a group homomorphism However, Ξ fails to respect product and hence is not a ring homomorphism: if x, y are two points belonging to X, Y then d (x,y) = gcd(d x , d y ) and hence the µ 2 -action on (x, y) as a point belonging to X × Y is not necessarily the product of the µ 2 -actions on x, y. On the other hand, in light of (3.2) and (3.3), it can be upgraded to an A C -module homomorphism via the obvious ring parameter space is in effect the definable closure S † of R((t)) inC † , and we have VF(S † ) = R((t)) but RV(S † ) ∼ = rv(R). If csn is inR then indeed RV(S † ) = rv(R). Thus, inC † , RV has no symmetries left other than the involution given by the complex conjugation. This also follows from Remark 3.8.
The presence of csn induces an intrinsic isomorphism RV ∼ = k × ⊕ Q. Consequently, K RES † ∼ = K Var R . As in [23, § 4.3], using the twistback function, we construct a homomorphism Remark 3.12. Several variants of Θ will appear below. To show that they are injective, we may simply follow the argument in the proof of [23, Proposition 4.3.1]. For surjectivity, however, some modification is needed, and how much of it is needed varies. For Θ it is quite simple. Let [(X,ĥ)] ∈ Kδ Var R withĥ factoring through a δ n -action. We may assume that X is quasi-projective and irreducible. Considering the induced µ n -action on X, we see that the quotient variety X/µ n , which is also quasi-projective, carries an antiholomorphic involution and hence is defined over R. Then the Kummer-theoretic construction in the proof of [23,Proposition 4 The situation inR is somewhat trickier. Let U ∈ RESR. For each u ∈ U, let d u be the least positive integer such that u is a tuple in RV(R((t 1/du ))), or equivalently, vrv(u) ∈ d −1 u Z. As implied by Remark 3.8, there is a nontrivial µ 2 -action on a two-element set {u, u ′ } ⊆ U if d u is even. Thus, similar to Definition 3.10, we can construct a µ 2 -action on U by u −→ u ′ if d u is even and u −→ u otherwise. If csn is inR then the twistback function yields an isomorphism it is surjective because, for µ 2 -actions, we can apply Kummer theory directly over R. There is also the commutative diagram where ΞR is obtained by taking traces inR, similar to the construction of Ξ in (3.4) (so it is just an A C -module homomorphism). Of course, if we forget the µ 2 -actions then ΞR is indeed the ring homomorphism −/RR with M =R in (3.1), which we shall denote byΞR.
Notation 3.13. We have pointed out above that if U is a bipolar twistoid then E b ([U]) may be written as . The image ω f (U γ ) is finite for every γ ∈ vrv(U), because the k-sort and the Γ-sort are orthogonal to each other. It follows that there is a bipolar twistoid decomposition (U i ) i of U such that every restriction ω f ↾ U i vrv-contracts to a function σ i : We assign to U the expression which is a finite sum and hence belongs to Lemma 3.15. The assignment (3.10) does not depend on the choice of the bipolar twistoid decomposition and is invariant on isomorphism classes.
be bipolar twistoid decompositions of U, V satisfying the condition above. We need to show that h m (U ), which depends on D, and h m (V ), which depends on E, are equal. This is clear if U = V , E is trivial (so U is already a bipolar twistoid and ω f already vrv-contracts to a function on vrv(U)), and D is Γ-cohesive or vrv(U i ) = vrv(U j ) for all i, j. The case that U = V and D is a refinement of E follows easily from this since there is a refinement (U ij ) ij of D such that (U ij ) j is a Γ-cohesive twistoid decomposition of U i and vrv(U ij ) = vrv(U i ′ j ) for all i, i ′ . If ω f , π g both vrv-contract to a function and f vrv-contracts to a bijection then, by Lemma 2.52, we may assume that U, V are already bipolar twistoids. In that case the desired equality follows because C((t 1/m )) is functionally closed (it is henselian, see Remark 3.1).
For the general case, we first remark that the image vrv(f (U γ )) is finite for every γ ∈ vrv(U). It then follows from Lemmas 2.50 and 2.52 that there is an L RV -definable twistoid decomposition (f i ) i of f such that every f i is a µRV db [k]-morphism as in the last special case considered above and, moreover, is compatible with D, E in the obvious sense, in other words, the domains and ranges of these f i induce bipolar refinements of D, E. The result follows.
This means that h m may be viewed as a map on K + µRV db [k]. It is routine to check that we have in effect constructed a ring homomorphism Note that the ideal (P Γ ) of K µRV db [ * ] in Notation 2.36 is now generated by the elements P γ with γ ∈ Z (because the point t γ there needs to be definable, which is possible only if γ ∈ Z in the current setting).
Denote by RES m the full subcategory of RES such that U ∈ RES m if and only if every γ ∈ vrv(U) is a tuple in m −1 Z, or equivalently, U ∈ RES m if and only if the action on U of the kernel of the canonical projectionμ −→ µ m is trivial.
Let β = (β 1 , . . . , β n ) ∈ (m −1 Z) n and A ⊆ O n × RV l be a proper β-invariant definable set such that pr VF ↾ A is finite-to-one. Then there is a set such that, for every t ∈ RV l , the VF-fiber A(C((t 1/m ))) t is the pullback of the VF-fiber A[m; β] t . Note that pr VF ↾ A[m; β] is still finite-to-one, for otherwise, by the β-invariance of A, pr VF ↾ A would fail to be finite-to-one. We may view A[m; β] as a finite disjoint union of objects of RES m (see [23, § 4.2] for detail).  To show this lemma, the statement of Theorem 2.37 itself is not quite enough. We need the fact that there exists a special µVF ⋄ [ * ]-morphism F : (A, 0) −→ (LU , 0), called a proper special covariant bijection, with U ∈ RV db [ * ]. Now both the definition of a proper special covariant bijection and the proof that such a morphism exists are quite involved. It is better that we do not repeat them here and instead refer the reader to [18, Definition 6.6, Proposition 6.12, Lemma 6.14] for a complete discussion. We only note that F may be chosen with high enough "aperture" (a technical notion defined in [18]) so that LU is also β- Proof. We may assume that A is already of the form LU for some U ∈ RV db [ * ]. Let D be a Γ-cohesive twistoid decomposition of U . Since both sides respect finite disjoint union of proper β-invariant definable sets, we may further assume that D is actually trivial or even vrv(U) is a singleton. Then the equality follows from a simple computation. We can now proceed to replicate the construction in [23, § 8] (see also [18] 3.4. Zeta function and motivic Milnor fiber. Let X be a nonsingular connected variety of dimension d over R and f a nonconstant morphism, also over R, from X to the affine line. Let z ∈ f −1 (0) be an R-rational point. Since X, f , and z are fixed, we shall not always carry them in notation and terminology. Note that X may be constructed in an affine neighborhood of z and hence is indeed a (quantifierfree) definable set. Moreover, it is β-invariant for every β ≥ 1. Therefore, X is an object of µVF ⋄ [ * ] equipped with the constant volume form 0. The positive motivic zeta function of f is the power series Now, recall from § 1, for each m ∈ Z + , the set of positive truncated arcs at z: This shows that the coefficients of Z 1 (T ) may also be written as [X 1 m ][A] −md . Remark 3.21. The µ 2 -action on X 1 m considered here (also see Remark 3.12) is in general different from the one in [15], where it is simply induced by t −→ −t for m even. For instance, suppose that X is the affine line, f is the square function, and z = 0, then the µ 2 -action given by Ξ on any two elements of the form ±t 2 + ct 4 and hence induces the obvious nontrivial µ 2 -action on the first factor of {x 2 = 1} × 0 × R, whereas the action induced by t −→ −t is entirely trivial. Actually, the µ 2 -action induced by t −→ −t corresponds to the dotted route from K ♭,δ Var R to K µ 2 RVar in (3.5).
The motivic zeta function Z G (T ) studied in [15] is shown to be a rational series (the rational formula given therein needs to be revised, though) and hence one can take the limit as T goes to infinity, as we shall do to Z 1 (T ) too below. Unfortunately, this process of "taking the limit" kills off the µ 2 -actions on the coefficients of Z G (T ) and, consequently, the limit of Z G (T ) does not actually carry any µ 2 -action. In contrast, the limit of Z 1 (T ) often retains a µ 2 -action and lives in There is the negative counterpart Z −1 (T ) of Z 1 (T ). Since the situation is the same, we shall concentrate on the positive one and drop the qualifier "positive" from the terminology. We do remark that, although complexification has seen success to some extent, for instance, the result on the Euler characteristics in [27] or the fact that the involution defined in Remark 3.7 exchanges Z 1 (T ) and Z −1 (T ) (as observed in [16,Lemma 3.2] for truncated arcs), it is unclear how the duality of "the positive" and "the negative" here works. Applying the homomorphism Φ in (3.5) termwise to the coefficients of Z 1 (T ), we obtain a zeta functionZ 1 (T ). It is known from [14], using resolution of singularities, thatZ 1 (T ) belongs to K RVar[[A] −1 ][T ] † and, letting "T go to infinity" as described in [23, § 8.4], we get a limit with l ∈ Z and consider the zeta function ].
If Without loss of generality, we may assume that U is already a bipolar twistoid and the function l f : U −→ Γ (recall Notation 2.22) vrv-contracts to a function σ :  is not injective, we cannot really take the motivic Milnor fiber S 1 of f in K µ 2 RVar, at least not if S 1 is viewed as something obtained through Z 1 (T ). It is this point of view that forces us to work with an integral whose target only involves doubly bounded sets in RV, namely ⋄ , instead of , so as to facilitate the computation of the coefficients of Z 1 (T ), and consequently with the nonarchimedean Milnor fiber X , which is proper invariant (Definition 2.34), instead of, perhaps, the more obvious set which is not proper invariant. This set is closely related to the analytic Milnor fiber introduced in [31] and does play a role in [23] (but not in this paper).
On the other hand, in light of Theorem 3.24, we can forego the zeta function point of view and recover S 1 directly as (Ξ ). In that case there is truly no need to invert [A]. In fact, we can also recover S 1 directly as Vol µ 2 ([X ]), where Vol µ 2 is short for and the result is the same because the diagram (3.17) Remark 3.27. The real nonarchimedean Milnor fiber X 1 of f is the set X (R) ofR-rational points of X . Appending (3.1) and then (3.8) to (3.17) with M =R (this fulfills the requirement that Γ(M) be divisible) and writing Vol µ 2 R = ΘR • E b,R • R , we can calculate S 1 as Vol µ 2 R ([X 1 ]). This is sometimes much simpler than working with the complex nonarchimedean Milnor fiber X ; see Example 3.29 below. The reason is thatR is real closed (and indeed o-minimal). This additional structure does give rise to a variant of the Hrushovski-Kazhdan construction, which we shall discuss in § 5.
Remark 3.28. We have seen in Remarks 2.43 and 2.44 that (3.17) almost commutes if E b is replaced by E g . At any rate, one can still define a homomorphism Vol µ 2 g using E g instead of E b . It would be interesting to give a geometric interpretation of the class Vol µ 2 g ([X ]) and relate it to the motivic Milnor fiber Vol µ 2 ([X ]). As has been mentioned earlier, we can indeed establish such a relation for the Euler characteristics, see Remark 5.17.
Example 3.29. Consider the polynomial function f (x, y) = x 6 + x 2 y 2 + y 6 on the affine plane and take z to be the origin. We decompose the real nonarchimedean Milnor fiber X 1 into the following sets in RV(R): . Also observe that if we work with the complex nonarchimedean Milnor fiber X then the first term in C should be {1/2 < val(x 6 ) = val(x 2 y 2 ) ≤ 1}, but the only possibility for C is the indicated condition because the leading terms of x 6 and x 2 y 2 cannot cancel inR. This simplifies the computation tremendously. In comparison, we shall perform a similar decomposition inC for a simpler polynomial (no mixed terms) in § 4. In The assignment (x, y) −→ (xy, y) gives a definable bijection between D and where the µ 2 -action is given by (x, y) −→ (−x, y) for the first term and x −→ −x for the second term. Then, applying the realization map β µ 2 in Remark 3.11, we get If we take further the virtual Poincaré polynomial then it becomes 0, since {x 6 + x 2 y 2 = 1} has the same virtual Poincaré polynomial as the unit circle minus two points.

3.5.
Concerning the virtual Poincaré polynomial. Let R be a real closed field. An R-variety is defined in the same way as a real variety, but with R replaced by R. The corresponding category of R-varieties is denoted by RVar and its Grothendieck ring by K RVar; we have seen the special case R = R in § 3.2.
The virtual Poincaré polynomial is an invariant of RVar, which is defined in [28]. The proof for its existence there relies on the weak factorization theorem of [1] and Poincaré duality; the former is valid over any field of characteristic 0 and the latter is available for singular homology of compact nonsingular real algebraic varieties with F 2 -coefficients. Replacing singular homology with semialgebraic homology H sa with F 2 -coefficients (see [7] or [2, § 11.7]), Poincaré duality still holds (in the semialgebraic setting "compact" means "closed and bounded"). Thus the proof goes through almost verbatim for RVar: Theorem 3.30. There exists a unique homomorphism β R : K RVar −→ Z[u] that assigns to each compact nonsingular R-variety X its Poincaré polynomial i∈N dim H sa i (X, F 2 )u i . If R = R then we denote β R simply by β as in Remark 3.11.
Remark 3.31. Let R / / R ′ be a real closed field extension. Let X be an R-variety. Then the virtual Poincaré polynomial of the extension X(R ′ ) of X to R ′ is equal to the virtual Poincaré polynomial of X. Actually, for X compact and nonsingular, this follows immediately from the invariance of semialgebraic homology under real closed field extension. The general case follows from additivity, expressing the class of X in terms of classes of compact nonsingular R-varieties via resolution of singularities.
Write VolR for the composition of Vol µ 2 R with the forgetful homomorphism K µ 2 RVar −→ K RVar. SinceRVar is a subcategory of VFR, there is a natural homomorphism KRVar −→ K VFR. Composing this with VolR and then the virtual Poincaré polynomial map β, we obtain a homomorphism β lim : KRVar −→ Z[u]. Thus we have found two homomorphisms β lim , βR from KRVar to Z[u].
Remark 3.32. Over the algebraic closure of a henselian discretely valued field, it is shown in [30,Proposition 3.23] that the analogue of β lim , defined with the Hodge-Deligne polynomial instead of the virtual Poincaré polynomial, gives the Hodge-Deligne polynomial of the limit mixed Hodge structure associated with a variety. It would seem interesting to also compare β lim with a similar map on limit structures, but such structures have yet to be constructed in the real framework.
Also, the duality of E b and E g described in Remarks 2.43 and 3.28 of course yields another homomorphism β lim  Combining this lemma with Remark 3.31, we get the following equality: However, the two homomorphisms do not coincide in general. Here is a counterexample: Example 3.35. Consider the polynomial f (x, y) = x 6 + x 2 y 2 + y 6 again. Let X ⊆R 2 be theRvariety given by the equation f (x, y) = t. Observe that we actually have X ⊆ M(R) 2 and hence X is closed and bounded.
For any t ′ ∈ VF with rv(t ′ ) = rv(t), there is an immediate automorphism σ ofC over R with σ(t ′ ) = t, where "immediate" means that σ fixes RV pointwise. Therefore, changing t to t ′ in the definition of X does not change the value [X(C)]. It follows from compactness that (3.20) [

Thom-Sebastiani formula
Let X be a smooth connected variety and f , g nonconstant functions from X to the affine line, all defined over C. In this section we aim to establish a local motivic Thom-Sebastiani formula for composite morphisms on X of the form h(f, g), where h(x, y) is a polynomial of the form here we may take N = m 1 , but it plays a special role and hence is denoted differently. The actual condition we shall assume is somewhat weaker than this, see Hypothesis 4.21.

4.1.
Combinatorial data and Galois actions of the torus. The said formula expresses the motivic Milnor fiber of h(f, g) as a sum of (iterated) motivic Milnor fibers of morphisms derived from f , g and their convolution products. Before diving into technicalities, we first describe how the various terms in the sum are singled out based on certain combinatorial data that is read off from the tropical curve of h(x, y).
Consider the planes in (Q + ) 3 defined by the following equations: z = 1, z = Ny, and z = m ı x for 2 ≤ ı ≤ ℓ. The lowest points on these planes form the surface of a convex polyhedron whose edges are the pairwise intersections of the three planes z = 1, z = Ny, and z = m 2 x. The tropical curve H of h(x, y) is the orthogonal projection of these edges in the (x, y)-plane. Thus H consists of two rays H 1 , H 2 and a line segment H 3 , all emanating from the point (1/m 2 , 1/N), see the illustration on the left in Figure 1. Both H 1 and H 2 contribute a term of (iterated) motivic Milnor fiber to the formula.
In this section, we choose to work with varieties over C with G m -actions instead ofμ-actions. To that end, we shall mainly work in the ACVF-modelC with S = C ∪ Q. Even though Γ ∼ = Q is only a definable sort ofC, the Hrushovski-Kazhdan integration theory still goes through. This is not explicitly stated in [22] but is included in the more general assumption of "effectiveness" there; in [32] and its sequels, val(VF(S)) is assumed to be nontrivial, but this is merely for convenience and is by no means an essential requirement.
Let z ∈ f −1 (0) be a C-rational point. As before, since the discussion below will be of a local nature, we may assume that X is actually affine (hence a definable subset of VF n for some n) and, without loss of generality, z = 0. Write X ∩ M n as X(M). We shall consider definable sets of the form X ♯ γ = {x ∈ X(M) | val(f (x)) = γ}, γ ∈ Γ + ; for simplicity, X ♯ 1 shall just be written as X ♯ , which is of primary interest, and the restriction f ↾ X ♯ γ just as X ♯ γ (this will become a general notational scheme below). For each u ∈ RV and each a ∈ u ♯ ⊆ VF, let X a = {x ∈ X(M) | f (x) = a} and X u = {x ∈ X(M) | rv(f (x)) = u}, which are a-definable sets; so X rv(t) is just the set called the nonarchimedean Milnor fiber of f above. The following equality relating [X u ] and [X a ] generalizes (3.20), and shall be used frequently (and often implicitly); the argument for it is the same as the one given thereabout.
For each a ∈ C × , there is an automorphism C((t)) −→ C((t)) sending t to at. Thus there is a subgroup of Gal(C((t))/C) that may be identified with C × ; the preimage of C × along the canonical surjective homomorphism Gal(C /C) −→ Gal(C((t))/C) is denoted byτ . A moment reflection shows thatτ ∼ = lim n (C × ) n , where each (C × ) n is just a copy of C × and the transition morphisms are the same as in the limitμ = lim n µ n ; so for each n there is a canonical epimorphism τ n :τ −→ (C × ) n , which is a part of the limit construction (in the category of groups, say). More concretely, the elements inτ may be identified as sequencesâ = (a n ) n of nth roots of a, a ∈ C × , satisfying a n kn = a k . Such an element acts onC byâ · t 1/n = a n t 1/n . We have a short exact sequence 1 −→μ −→τ −→ C × −→ 1. This sequence does not split, though.
Here is a different perspective onτ . By the structural theory of valued fields, an element σ ∈ Gal(C /C((t))) is in the ramification subgroup if and only if it fixes RV pointwise (see [13,Lemma 5.3.2]). But it can be easily checked that every σ ∈ Gal(C /C((t))) moves some element of RV unless σ = id. So Gal(C /C((t))) ∼ =μ may be identified with Aut(RV / RV(C((t)))), where RV(C((t))) is equal to the subgroup generated by rv(t) over k × .
For each u ∈ k × , there is an automorphism Aut(RV / k × ) sending rv(t) to u rv(t); observe that an automorphism in Aut(RV / k × ) fixes Q ∼ = RV / k × pointwise if and only if it is of this form. Sô τ may also be identified with a subgroup of Aut(RV / k × ), namely Aut(RV / k × ∪ Q). Remark 4.3. From yet a different perspective, recall from Remark 3.8 that there is a natural bijection betweenμ and the set of reduced cross-sections csn : Q −→ RV with csn(1) = rv(t). Of course this is still the case if we change rv(t) to any other element of the form u rv(t), u ∈ k × . Consequently, we may identifyτ with the set of all such reduced cross-sections.
Since every reduced cross-section csn determines a reduced angular component ac : RV −→ k × via the assignment u −→ tbk(u) and, conversely, every reduced angular component ac determines a reduced cross-section csn with csn(Q) = ac −1 (1), we see thatτ may also be identified with the set of all such reduced angular components.
Intuitively, as we have seen above, all this is just saying that if any reduced angular component or reduced cross-section is chosen and added to the structure of RV then we have an intrinsic isomorphism RV ∼ = k × ⊕ Q, and hence if both k × and Q are fixed pointwise then RV has no symmetries left other than the trivial one.
Therefore, similar to the case S = C((t)), elements in !K RES now carry goodτ -actions, that is, thoseτ -actions that factor through some τ n and hence may be considered as G m -actions. To emphasize this and to distinguish it from the similar ring with goodμ-actions, we shall denote !K RES by !Kτ RES over S = C ∪ Q and by !Kμ RES over S = C((t)). Let Y be a variety over C with a good G m -action h. We say that h is n-weighted for some n ∈ Z + if there is a morphism π : Y −→ G m such that π(c · y) = c n π(y) for all c ∈ G m and all y ∈ Y . We also say that h is 0-weighted if it is trivial, and the only witness to this is the morphism Y −→ 1. Observe that if there is a G m -equivariant isomorphism between (Y, h) and (Y ′ , h ′ ) then h is n-weighted if and only if h ′ is n-weighted. Furthermore, if h is n-weighted with a withness π and h ′ is n ′ -weighted with a withness π ′ then π(c · y)π ′ (c · y ′ ) = c n+n ′ π(y)π ′ (y ′ ) for all c ∈ G m , all y ∈ Y , and all y ′ ∈ Y ′ and hence the good diagonal G m -action h×h ′ on Y ×Y ′ is (n+n ′ )-weighted.
The category Var τ,n C consists of the varieties over C with n-weighted G m -actions and the G mequivariant morphisms between them. Let Varτ C denote the colimit of the inductive system of these categories Var τ,n C , n ∈ Z + , in which transition functors correspond to multiplication of integers (so there are no functors between Var τ,0 C and other Var τ,n C ). We may and do think of an object of Varτ C as equipped with aτ -action that factors through some τ n , hence the notation. The Grothendieck groups K τ,n Var C , Kτ Var C are constructed subject to the usual condition on trivializing G m -actions on affine line bundles analogous to (3.2). Clearly Kτ Var C is the colimit of K τ,n Var C , n ∈ Z + , and is indeed a commutative ring, with the product operation induced by that in Varτ C .
Remark 4.5. Choose a reduced cross-section csn : Q −→ RV; the point rv(t) ∈ RV is not special in the present setting (the object of interest shall be X ♯ , not X rv(t) ) and hence we no longer demand csn(1) = rv(t). Let U ⊆ γ ♯ ⊆ RV n be an object of RES, where γ 1 ≤ . . . ≤ γ n , and d the least positive integer d such that U is a set in RV(C((t 1/d ))); note that any other such integer is a multiple of d. Consider the function π : U −→ RV given by u −→ u d n . Then π(c · u) = c ed π(u) for all c ∈ G m and all u ∈ U, where e ∈ Z + with γ n = e/d. So if tbk(U) is a variety over C then it is ed-weighted, which is witnessed by tbk(π).
Recall that the isomorphism in [ indeed commutes, where the first vertical arrow is induced by the subcategory relation and the second vertical arrow is induced by the obvious forgetful functor (μ is a subgroup ofτ ).
Remark 4.6. The modified argument for the surjectivity of Θτ is not as straightforward as that in Remark 3.12. Some model theory is needed. Let (Y, h) ∈ Var τ,n C and π : Y −→ G m witness that h is n-weighted. We may assume that Y is irreducible and quasi-projective. By elimination of imaginaries in the first-order theory of algebraically closed fields, there is a definable surjection ω : Y −→ Z, where Z is a set in G m , such that each fiber ω −1 (z) contains precisely one h-orbit. Then ω × π is a definable finite-to-one surjection from Y onto Z × G m each of whose fibers inherits a µ n -action. By Kummer theory and compactness, there are a definable function η : Z −→ G m and a definable bijection ζ 1 from the fiber (ω × π) −1 (Z × 1) to the set If ζ 1 (y) = (z, 1, v) and c ∈ G m then set ζ(c · y) = (z, c n , cv). It can be readily checked that if c · y = c ′ · y ′ then ζ(c · y) = ζ(c ′ · y ′ ) and hence ζ is a G m -equivariant bijection from Y onto a set V ⊆ Z × G m × G m , where the G m -action on V has just been given.
Following the notational scheme introduced in Remark 3.26, let us denote the composites Θτ • E b • , Θμ • E b • by Volτ , Volμ. Relative to the chosen reduced cross-section csn, the fiber X csn (1) gives the motivic Milnor fiber S f as constructed in [23, § 8.5], that is, Volμ([X csn (1) ]) = S f , but for any v ∈ k × other than 1, S v f := Volμ([X v csn(1) ]) is not equal to S f in general. Theμ-action on S v f corresponds to a coset ofμ inτ , which in turn corresponds to the various reduced cross-sections csn ′ : Q −→ RV with csn ′ (1) = v csn(1). All the relevant constructions above still go through if we replace S = C((t)) with S = C((t q )) for any q ∈ Q + .

Categories with angular components.
To define convolution operators, we need to consider objects equipped with angular component maps and equivariant morphisms between them, as follows.
Ultimately we are only interested in the points (α ı , β ı ) ∈ (Q + ) 2 described above and the corresponding elements m i /m ı , 2 ≤ i ≤ ı, in the interval (0, 1] ⊆ Q + . But it is conceptually clearer to work in a more general setting. Thus let ϑ = (ϑ 1 , . . . , ϑ ℓ ) be a sequence of elements in (0, 1] such that if ℓ > 1 then ϑ 1 = ϑ 2 < . . . < ϑ ℓ . Let λ be the least positive integer such that λϑ i /ϑ ℓ is an integer for every 2 ≤ i ≤ ℓ; so any other integer that has this property must be a multiple of λ. Let Y be a variety over C with a good G m -action. Let π : Y −→ G ℓ m be a morphism and π i : Y −→ G m , 1 ≤ i ≤ ℓ, its coordinate projections. Suppose that there exists a morphism π * : Y −→ G m such that π i = (π * ) λϑ i /ϑ ℓ for every 2 ≤ i ≤ ℓ.
Definition 4.7. An object of the category Var ϑ,n C is a pair (Y, π) such that Y is a variety over C with a good G m -action and π : Y −→ G ℓ m is a (ϑ, n)-diagonal morphism. A morphism between two such objects is a morphism between the (ϑ, n)-diagonal varieties over G ℓ m that is equivariant with respect to the G m -actions.
The category Var ϑ C is the colimit of the inductive system of the categories Var ϑ,n C , n ∈ Z + , in which transition functors correspond to multiplication of integers.
The Grothendieck groups K ϑ,n Var C are constructed as before. Fiber product (reduced) over G ℓ m with diagonal action induces a product operation on K ϑ,n Var C , which turns the latter into a commutative ring. Set Observe that if ℓ = 1, 2 then the entries in ϑ do not really have any bearing on the definitions of (ϑ, n)-diagonality and Var ϑ,n C , in which case we shall just say "n-diagonal" and write Var 1,n C , etc., when ℓ = 1 and Var 2,n C , etc., when ℓ = 2. Also note that Var 1,n C is just the category Var Gm,n Gm as defined in [20, § 2.3] and hence, by [20,Lemma 2.5], it is equivalent to the category of varieties over C with µ n -actions, in particular, K 1,n Var C ∼ = K µn Var C . Assume ℓ > 1. Let (Y, π) ∈ Var ϑ,n C . For each 1 ≤ i ≤ ℓ, write π i = pr i •π, π >i = pr >i •π, etc. The compositionπ It is clear from (4.1) that every fiber ofπ inherits a good G m -action from Y and hence F n (Y, π) :=π × π * is an object of Var Gm,n/λ G 2 m , where π * is the morphism that comes with the (ϑ, n)-diagonality of π. Conversely, let ξ : Z −→ G 2 m be an object of Var Gm,n/λ G 2 m and consider the morphismξ on Z given by z −→ (ξ 1 (z), 1, . . . , 1)ξ 2 (z) λϑ/ϑ ℓ ∈ G ℓ m , where ξ(z) λϑ/ϑ ℓ = (ξ(z) λϑ i /ϑ ℓ ) 1≤i≤ℓ . Then G n (ξ) := (Z,ξ) is an object of Var ϑ,n C . It is straightforward to extend the two assignments F n , G n to functors with G n (F n (Y, π)) ∼ = (Y, π) and F n (G n (ξ)) ∼ = ξ, that is, they are quasi-inverse to each other. So where the new Grothendieck groups are constructed as before. Consequently, all the operations on K ϑ,n Var C that will appear below may be considered as defined on K µ n/λ Var Gm via this isomorphism. As is the case with [20, Proposition 2.6], the point here is that K µ n/λ Var Gm is much closer to objects that have been studied extensively in the literature and hence for which we have a deeper understanding.
If Z ∈ Varτ C and (Y, π) ∈ Var ϑ,n C then (Y × Z, π • pr Y ) ∈ Var ϑ,n C , where the good G m -action on Y × Z is given by c · (y, z) = (c · y, c n · z) for all c ∈ G m . This is compatible with the inductive systems in question and hence, after passing to the colimits, we see that K ϑ Var C is indeed a Kτ Var C -module. Definition 4.9. Assume ℓ > 1. We construct a Kτ Var C -module homomorphism For (Y, π) ∈ Var ϑ,n C , let (π 1 + π 2 ) −1 (0) and Y (π 1 + π 2 ) −1 (0) denote, in Var C , the pullbacks of π ≤2 : Y −→ G 2 m along the antidiagonal of G 2 m and its complement, respectively. It is clear from (4.1) that both varieties inherit a good G m -action from Y .
For the base case ℓ = 2, we consider the good G m -action on (π 1 + π 2 ) −1 (0) × G m whose second factor is given by c · z = c n z. Then the expressions designate two elements in K 1,n Var C ; they only depend on the class of (Y, π) and hence may be denoted byΨ 2,n ([(Y, π)]),Ψ 2,n ([(Y, π)]), respectively. These assignments respect the defining relations of K 2,n Var C and hence may be extended uniquely to two group homomorphismsΨ 2,n , Ψ 2,n . These group homomorphisms in turn are compatible with the inductive systems in question and hence, after passing to the colimits, we obtain two group homomorphismsΨ 2 ,Ψ 2 , which also respect the Kτ Var C -module structure. Set Ψ 2 = −(Ψ 2 −Ψ 2 ). For the inductive step ℓ > 2, let ϑ ′ = (ϑ 3 , ϑ 3 , . . . , ϑ l ) and λ ′ be the least positive integer such that λ ′ ϑ i /ϑ ℓ is an integer for every 3 ≤ i ≤ ℓ. Then λ ′ divides λ. We consider the good G m -action on G m × (π 1 + π 2 ) −1 (0) whose first factor is given by c · z = c nϑ 3 /ϑ ℓ z. It follows that the expression designates an object of Var ϑ ′ ,n C whose class only depends on that of (Y, π) and hence may be denoted by Ψ ϑ ′ ϑ,n ([(Y, π)]). The assignments Ψ ϑ ′ ϑ,n , n ∈ Z + , may be extended to group homomorphisms and their colimit Ψ ϑ ′ ϑ : The negative sign at the front is inherited from the literature.
Remark 4.10. The case ℓ = 2 is of course special. For (X, π X ) ∈ Var 1,m C and (Y, π Y ) ∈ Var 1,n C , let π X ⊕ π Y be the obvious morphism X × Y −→ G 2 m . Then (X × Y, π X ⊕ π Y ) is an object of Var 2,mn C whose class only depends on those of (X, π X ) and (Y, π Y ). We may then define a binary map on Although the category Var 2 C is not the same one used in [20, § 5.1], the proof of [20, Proposition 5.2] still goes through verbatim, which justifies referring to (4.4) as a convolution product. • ac is a function U −→ ϑ ♯ , which is referred to as an angular component map on U , such that, for each r = (r 1 , . . . , r ℓ ) ∈ ran(ac), • the pair (ac −1 (r), f ↾ ac −1 (r)) is an object of RV[k] (of course the category RV[k] here is formulated relative to the additional parameters r), . The category RES ac ϑ is formulated in the same way, but with RV[k] replaced by RES. The ring structure of K RV ac ϑ [ * ] is induced by fiberwise disjoint union and fiberwise cartesian product in RV ac ϑ [ * ]; similarly for other such categories. We may also think of K RV ac ϑ [ * ] as a K RV[ * ]-module and !K RES ac ϑ as a !Kτ RES-module (the extra defining condition for "!K" in !K RES ac ϑ is in effect imposed fiberwise). If ℓ = 1 and ϑ = 1 then the subscript ϑ shall be dropped from the notation. For the base case ℓ = 2, that is, ϑ = (1, 1), let f 1 : U −→ RV k+1 be the function given by u −→ (f (u), ac 1 (u)), similarly for f 2 . The pairs (U, f 1 ), (U U ′ , f 1 ), (U ′ , f 1 ) are more suggestively denoted, respectively, by here the second term is such that each fiber of pr 1 ♯ is a copy of (ac 1 + ac 2 ) −1 (0), and hence is indeed an element in K RV ac [k+1]. These two assignments do not depend on the representative (U , ac) or the choice between f 1 and f 2 , and hence may be extended uniquely to two K RV[ * ]module homomorphismsΠ (1,1) ,Π (1,1) (the gradation has been shifted by 1). Then set Π (1,1) = −(Π (1,1) −Π (1,1) ). For the inductive step ℓ > 2, let ϑ ′ = (ϑ 3 , ϑ 3 , . . . , ϑ ℓ ). Then the triple ) designates an object of RV ac ϑ ′ [k] whose class only depends on that of (U , ac) and the assignment . Thus we may seṫ There is a similar construction resulting in a !Kτ RES-module homomorphism which is denoted by Π ϑ as well. Its construction is actually simpler since the categories RES ac ϑ , RES ac are not graded and the function f is irrelevant. Also, in light of the ring homomorphism E b : K RV[ * ] −→ !Kτ RES, this Π ϑ may be viewed as a K RV[ * ]-module homomorphism.
For (U , ac U ) ∈ RV ac [k] and (V , ac V ) ∈ RV ac [l], let ac U ⊕ ac V be the obvious function from U ×V into (1, 1) ♯ . Then the class (1,1) [k+l] only depends on the classes [(U , ac U )], [(V , ac V )], not their representatives. Set which may be thought of as a convolution product of the two classes.
The following lemma only serves to confirm the structural resemblance of the binary map * here to the convolution product (4.4). It will not be of any use beyond this point. Proof. The formal computations involved are essentially the same as those in the proof of [20,Proposition 5.2]. We shall just write down some details for the second claim since the expected convolution identity 1 is actually off by a factor, namely [1], in this setting.
The obvious function on U × 1 ♯ induced by ac is still denoted by ac, similarly for f and id ↾ 1 ♯ . Write (U × 1 ♯ , f ⊕ ac) as U × 1 ♯ . Then [(U , ac)] * 1 ∈ K RV ac [k+1] is given by Of course this is just [((ac − id) −1 (0), pr 1 ♯ )]. Since we now have, for every r ∈ 1 ♯ ,  . By Proposition 2.23 and compactness, there is a definable finite partition (B i ) i of ϑ ♯ such that every ac −1 (r) is r-definably bijective, uniformly over each B i , to a disjoint union of products U rij × D ♯ rij , where U rij ∈ RES[ * ] and D rij ∈ Γ[ * ]; actually, we may write D rij as D ij since it must be the case that D rij = D r ′ ij for any other r ′ ∈ B i . Let U ij = r∈B i U rij × r. The obvious coordinate projection U ij −→ ϑ ♯ is denoted by ac ij . Remark 4.15. Keeping the notation of Remark 4.14, we see that, over each B i , there are elements . This is an equality in !Kμ RES with S = C((t ϑ ℓ /λ )), not with S = C((t)) unless ϑ is a sequence of integers or the isomorphism Θμ is applied on both sides (in which case the equality happens in does not depend on the choice of U ij and D ij . Setting (4.8) [ yields a ring homomorphism , ac)]) may be understood as a function into ϑ ♯ whose fibers are of the form E b ([ac −1 (r)]), which has nothing to do with the partition (B i ) i . The point of the partition (B i ) i here is just to show the existence of such a finite sum as in (4.8).
The map Π ϑ on K RV ac ϑ [ * ] is indeed related to the map Π ϑ on !K RES ac ϑ via E ac b,ϑ : , similarly forΠ ϑ and hence for Π ϑ .
Proof. Although the case of Π ϑ follows immediately from those ofΠ ϑ andΠ ϑ , we shall show this for Π ϑ directly using the same argument. It is enough to consider elements in K RV ac ϑ [ * ] of the form [(U , ac)], since the general case would follow from K RV[ * ]-linearity. We proceed by induction on ℓ and, for simplicity, assume ϑ ℓ = 1.
For the base case ℓ = 2, by Remark 4.14, using the notation there, we may write By the construction of Π (1,1) , we have Let n ij = χ b (D ij ). Then, since the gradation is forgotten by E ac b , we have where (U ij , ac ij ) stands for the obvious object of RES ac (1,1) in relation to (U ij , ac ij ). The right-hand side of this equality also equals (Π (1,1) • E ac b,(1,1) )([(U , ac)]). For the inductive step ℓ > 2, let ϑ ′ be as in Definition 4.12. Remark 4.15 and the construction of Π ϑ ′ ϑ together imply that (4.9) So the desired equality follows from the definition of Π ϑ and the inductive hypothesis.
Let RES ac ϑ,n be the full subcategory of RES ac ϑ of those objects for which this condition holds. Thus we have obtained an inductive system of categories RES ac ϑ,n such that RES ac ϑ = colim n RES ac ϑ,n and !K RES ac ϑ = colim n !K RES ac ϑ,n ; Lemma 4.18. For each n there is a ring isomorphism Θ ac ϑ,n : !K RES ac ϑ,n −→ K ϑ,n Var C , determined by the assignment [(V, ac)] −→ [tbk(V, ac)] for vrv(V ) a singleton, and hence a ring isomorphism Θ ac ϑ : !K RES ac ϑ −→ K ϑ Var C . Moreover, under the ring homomorphism Θτ • E b , we have Θ ac •Π ϑ =Ψ ϑ • Θ ac ϑ as K RV[ * ]-module homomorphisms, similarly forΠ ϑ ,Ψ ϑ and hence for Π ϑ , Ψ ϑ .
Note that the set tbk(V, ac) is definable without using the implicit reduced cross-section csn; in other words, varying csn will not change tbk(V, ac), but does change the bijection in question, and that is why tbk(V, ac) inherits theτ -action on (V, ac).
Proof. The situation here is very similar to that in [23, § 4.3] or in Remarks 4.5 and 4.6, so we shall be brief. If vrv(V ) is a singleton then the graph of tbk(ac) is just a constructible set, in fact uniformly so fiberwise. So the assignment induces a homomorphism Θ ac ϑ,n at the semiring level and hence at the ring level. If tbk(V, ac) and tbk(V ′ , ac ′ ) are isomorphic in Var ϑ,n C then the isomorphism may be twisted to one between (V, ac) and (V ′ , ac ′ ) in RES ac ϑ,n . Thus Θ ac ϑ,n is injective. On the other hand, since objects in Var ϑ,n C and RES ac ϑ,n are all endowed fiberwise with µ n/λ -actions via restriction, the argument for surjectivity in Remark 4.6 can be easily modified to work for Θ ac ϑ,n . The second claim follows from an inductive argument, completely similar to the one in the proof of Lemma 4.16.

4.4.
Decomposing the composite Milnor fiber. We shall make use of the homomorphism E † b • Λ in (2.10). The composite homomorphism E † b • Λ • shall be abbreviated as Vol † b below when we (tacitly) work inC † .
Remark 4.19. Assume ℓ = 1. Let A be a definable set in M and λ : A −→ ϑ ♯♯ a definable function. Note that λ must be surjective because no nonempty proper subset of γ ♯♯ is definable for any γ ∈ Q + . For each r ∈ ϑ ♯ , by Lemma 4.1, we have There is usually no need to carry ϑ in the notation, since it is implicit in the integrand λ.
Notation 4.20. From here on, write the set X ♯ γ as X ♯ f,γ , etc., to emphasize the dependency on the morphism f as given in § 4.1. Also keep in mind the convention that if γ = 1 then it is dropped from the notation.
For any definable set A ⊆ X(M), the restriction f ↾ (X ♯ f,γ ∩ A) is just denoted by X ♯ f,γ ∩ A. We here f (1) , f (1) are both interpreted as the zero function. Naturally f (ℓ) − f (ı) denotes the function ı<i≤ℓ f m i . Let g be another complex regular function from X to the affine line with g(0) = 0 and N another positive integer. We may think of N as m 1 , but its role will be somewhat different and hence is denoted differently. For each 2 ≤ ı ≤ ℓ, set Note that val •(g N + f (ı−1) ) = val •f mı < 1 is implied in the first line and val •f mı = 1 is implied in the third line. Also set If ı = 1 then Z g N +f (ı) is also written as Z g N . The set Z g N will not play a role. So, for 2 ≤ ı ≤ ℓ, The restrictions of g N + f (ℓ) to the sets denoted by the union terms on the right-hand side are all definable functions onto 1 ♯♯ , and hence Remark 4.19 may be applied to them; to curb excess of notation, these functions and other similar ones below shall just be denoted by their respective domains, as we have done for sets of the form X ♯ f,γ . Hypothesis 4.21. From here on we assume that, in the sequence (m 2 , N, m 3 , . . . , m ℓ ), each number is sufficiently large relative to the data in question that involve only the numbers before it. This condition will become clear and precise in the discussion below when it is needed, so we will not labor further here to explain it. We do note, however, that it is not necessarily the case that each number is greater than the numbers before it.
denoting both the set and the corresponding function), etc., be defined as above. If M is a sufficiently large positive integer then Proof. For the first equality, let X ♯ φ M ,ψ denote the restriction of φ M to the set X ♯ φ M ∩ Z ψ , which is also a definable function onto 1 ♯♯ . Clearly for every r ∈ 1 ♯ , (4.12) (X ♯ φ M ∩ Z ψ ) −1 (r ♯ ) = (X ♯ φ M ,ψ ) −1 (r ♯ ) and hence, by the construction of ac , the two integrals in question are equal. So it is enough to . We consider φ instead of φ M . For γ, β ∈ Q + , denote by X ♯ φ,γ,ψ,β the restriction of φ to the set X ♯ φ,γ ∩ X ♯ ψ,β , which is a definable function onto γ ♯♯ , and write By Lemma 2.52, there is a (γ, β)-definable finite partition (D γ,β,i ) i of vrv(W γ,β ) such that, for each i, the set W γ,β ∩ D ♯ γ,β,i is a bipolar twistoid. By compactness, there is a γ-definable finite partition (E γ,j ) j of Q + such that, over each piece E γ,j , the partitions (D γ,β,i ) i may be achieved uniformly and, for each i, the corresponding twistbacks are the same. So each class ac [ β∈E γ,j X ♯ φ,γ,ψ,β ] is indeed represented by a finite disjoint union of bipolar twistoids W γ,i,j ∈ RV ac [ * ] and each vrv(W γ,i,j ) is of the form β∈E γ,j D γ,β,i × β, where χ b (D γ,β,i ) ∈ Z is constant over E γ,j .
Working over S = C, by compactness, these partitions (E γ,j ) j may be achieved uniformly over γ ∈ Q + ; in other words, they form a definable finite partition of (Q + ) 2 whose pieces are cones based at the origin. This implies that there are a j and a p ∈ Q + such that (pγ, ∞) ⊆ E γ,j for all γ ∈ Q + . Since M is sufficiently large, we have ( ∞). A moment of reflection shows that the class ac [ β∈(1,∞) X ♯ φ M ,ψ,β ] must admit a representative of this form as well, which then is annihilated by E ac b because χ b ((1, ∞)) = 0. This leaves only Z ψ in the computation. The first equality follows. For the second equality, since the roles of φ M , ψ are not exactly symmetric, a slightly different argument is needed. Let the restrictions X ♯ ψ,φ M , X ♯ ψ,γ,φ,β of ψ and the partitions (D γ,β,i ) i , (E γ,j ) j be defined as expected. Actually we will only need the case γ = 1 and hence will write X ♯ ψ,φ,β , D β,i , E j instead. Therefore, it is enough to show Vol ac ([X ♯ ψ,φ M ]) = S ♯ ψ . Since M is sufficiently large, (0, 1/M] ⊆ E j for some j. The class ac [ β∈(0,1/M ] X ♯ ψ,φ,β ] is represented by a finite disjoint union of bipolar twistoids satisfying the "regularity" condition in question, hence so is the class . This is again annihilated by E ac b because χ b ((0, 1]) = 0. Since X ♯ ψ is the union of X ♯ ψ,φ M and β∈(0,1] X ♯ ψ,φ M ,β , the lemma follows. Remark 4.23. It is not essential to use the bounded Euler characteristic χ b for the second equality as the interval (0, 1] vanishes under both, but χ b is needed for the first equality.
Corollary 4.24. Substituting suitable functions for φ, ψ in Lemma 4.22, we obtain the following equalities: . Proof. The second and the third equalities need additional explanation. For the second equality, recall (4.10). Similar to (4.12), for any ı (including ı = 1) and any r ∈ 1 ♯ , ) for any definable set A ⊆ X(M). For the same reason, , where as before the intersection on the right-hand side denotes the restriction of g N + f (ı+1) to the eponymous domain. Thus if ı > 1 then the first equality of Lemma 4.22 may be applied with φ = f , M = m ı+1 , and ψ = g N + f (ı) , and if ı = 1 then the second equality of Lemma 4.22 may be applied with ψ = f m 2 , φ = g, and M = N.
Next we turn to the remaining terms in (4.11). To compute their values under Vol ac , we need to make use of the convolution operators introduced in § 4.2.
Remark 4.25. Here we extend the discussion in Remark 4.19. So assume ℓ > 1. Let A be a definable set in M and λ : A −→ ϑ ♯♯ a definable function of the form φ ⊕ 2≤i≤ℓ ψ m i .
Observe that (pr ı •λ)(A) = ϑ ♯♯ ı for every 1 ≤ ı ≤ ℓ, because no nonempty proper subset of γ ♯♯ is definable for any γ ∈ Q + (this fact has been used in Remark 4.19 and will be used implicitly several times below).
For each a ∈ M 2 , write [(φ ⊕ ψ) −1 (a)] = [U a ]/(P − 1). This equality makes sense over the larger substructure S a . However, unlike in Remark 4.19, U a does depend on the parameter a now. Anyway, by compactness, there is a definable finite partition (B i ) i of M 2 such that the objects U a are defined uniformly over each B i . Since each val(B i ) ⊆ (Q + ) 2 is a cone based at the origin (because S = C), there are a B i and a p ∈ Q + such that α ♯♯ × (pα, ∞) ♯♯ ⊆ B i for all α ∈ Q + . Let ξ(x, y, . . .) be a quantifier-free formula that defines the object U a over a ∈ B i . Then this p ∈ Q + may be chosen so that for any a = (a 1 , a 2 ) ∈ B i and every term in ξ(x, y, . . .) of the form rv(F (x, y)), where F (x, y) ∈ C[x, y], if p val(a 1 ) < val(a 2 ) then rv(F (a)) = rv(F (a ′ )) for any a ′ ∈ rv(a) ♯ , and hence U a = U a ′ .
In particular, the foregoing discussion may be applied to the functions in (4.11), to which we shall return presently.
Theorem 4.31. In conclusion, we have derived a local Thom-Sebastiani formula in K 1 Var C : The special case ℓ = 2 and m 2 = 1 is related to the local Thom-Sebastiani formula in [20,Corollary 5.16] as follows. In terms of motivic Milnor fibers instead of motivic vanishing cycles, this latter formula may be written as Here z ∈ f −1 (0) is a C-rational point, which is implicit in Theorem 4.31 (recall the simplification made at the beginning of this section). The (local) motivic Milnor fibers S f,z and S g N +f,z are constructed via motivic zeta functions with coefficients in M Gm Gm ; see [20, § 3.6] for details. The meaning of the term S g N ,z ([f −1 (0)]) is established in [20,Theorem 3.9], and it belongs to M Gm Gm . According to the nearby cycles formalism of [20, § 4.6] , are denoted by M Gm Gm , Mμ therein, respectively. It is routine to check that a similar construction via "taking the fiber at csn(1)" also yields an isomorphism !K RES ac −→ !Kμ RES, which shall also be denoted by Υ, and indeed Θμ • Υ = Υ • Θ ac . Consequently, by [20,Remark 3.13] and the complex version of Theorem 3.24 (see [18,Theorem 8.7]), we have . This implies that, for any sufficiently large N ∈ Z + , . The methodology of [20] offers a geometric interpretation of "sufficiently large N ∈ Z + " in terms of log-resolutions. Our interpretation lies in the proof of Lemma 4.22 and Remark 4.29, and is not as informative since it depends on compactness. It is not clear how to relate the two thresholds. Also note that the left-hand side of (4.21) is obviously commutative in the sense that S ♯ g N ⊕f = S ♯ f ⊕g N , and perhaps this can be translated into an expression on the right-hand side through a resolutionbased analysis of the motivic zeta functions involved.
The setup for the motivic Thom-Sebastiani formula in [8] involves a morphism f ′ on another smooth variety X ′ and the obvious morphism f + f ′ on the product Y = X × X ′ . This formula is a special case of [20,Corollary 5.16], as demonstrated in [20,Theorem 5.18], and hence can be recovered from Theorem 4.31 as well, although we do need to check that it holds for N = 1 in that situation. Anyway, we can give a more direct proof. To begin with, write (4.11) as Observe that the conclusion of Remark 4.29 already holds for the function f ⊕ f ′ on Y (M) = X(M) × X ′ (M) and indeed To compute the other two terms, now symmetric, the key is the following equality.
Proof. We actually show a more general claim: It is a basic model-theoretic fact that the Γ-sort is orthogonal to the k-sort, which implies that val(g( i LU i )) is finite and hence only 0 and ∞ can occur in its coordinates; in the case we are interested in, that is, A ⊆ M n for some n, only ∞ can occur, but then A must contain the point 0. So g( i LU i ) is either empty or is the singleton 0, which means that i [U i ] is either 0 or 1, respectively. Since The real case. If we work in the ACVF-modelC with S = R ∪ Q and let the variety X, the morphism f , etc., be defined over R then the preceding discussion is still valid. In more detail, there is a subgroup of Gal(C((t))/R) that may be identified with Gal(C/R) ⋉ C × ; its preimage along the canonical surjective homomorphism Gal(C /R) −→ Gal(C((t))/R) is denoted by cτ , which may in turn be identified with lim n (Gal(C/R) ⋉ C × ) n . There is again an isomorphism !K cτ RES ∼ = K cτ Var R (for surjectivity, combine the arguments in Remarks 3.12 and 4.6).
The categories in Definition 4.7 and the corresponding Grothendieck groups are now written as Var ϑ,n R and K ϑ,n Var R . Note that, as in Definition 3.6, for an object (Y, π) of Var ϑ,n R , the good Gal(C/R) ⋉ C × -action on Y and the morphism π : Y −→ G ℓ m are required to be compatible with the antiholomorphic involution in question; in particular, for the generator c ∈ Gal(C/R), the condition (4.1) should read (4.23) π 1 (c · y) = c · π 1 (y) and π * (c · y) = c · π * (y), so if y is a real point then π 1 (y), π * (y) must be real points as well. We can construct a K cτ Var Rmodule homomorphism Ψ ϑ : K ϑ Var R −→ K 1 Var R as in Definition 4.9. Then Theorem 4.31 holds in K 1 Var R as well. However, as in § 3.4, we are more interested in a statement that is indigenous to the real algebraic environment. In addition, we shall point out how to deduce the real Thom-Sebastiani formula in [3] from ours.
Let Kρ RVar be the real analogue of Kτ Var C , that is, the Grothendieck ring of the category of real varieties with weighted good R × -actions. A morphism π : Y (R) −→ (R × ) ℓ on a real variety Y (R) with a good R × -action is (ϑ, n)-diagonal if the obvious analogue of (4.1) holds. The categories RVar ϑ,n , RVar ϑ , etc., are defined accordingly. The Kρ RVar-module homomorphism Ψ ϑ in the bottom row of (4.24) is constructed as in Definition 4.9 again.
Given any n-weighted good cτ -actionĥ on Y ⊗ R C, by considering the induced δ n -action in each fiber and the orbit size of each real point as in Definition 3.10, one sees thatĥ gives rise to an n-weighted good R × -action on Y (R). Consequently, as in (3.5), taking real points yields A C -module homomorphisms Ξ ϑ , Ξ 1 in (4.24) (also one K cτ Var R −→ Kρ RVar). (4.24) By an inductive argument similar to the one in the proof of Lemma 4.16, noting also that, by (4.23), fibers of π over genuinely complex points make no contributions to fibers over real points in (4.2) and (4.3), we deduce that the first square of (4.24) indeed commutes. It follows that Theorem 4.31 holds in K 1 RVar too, as a direct specialization of the same equality in K 1 Var R via Ξ ϑ and Ξ 1 .
This state of affairs may seem somewhat unsatisfactory as the supposedly real formula is in actuality computed from the complex objects in (4.11) and the volume operators Θ ac ϑ • E ac b,ϑ • ac overC. To remedy this, we can start the specialization procedure earlier, using the technique in § 3.1, as has been done in Remark 3.27, and obtain the same formula using theR-trace of (4.11) and the corresponding volume operators overR. No new perspective lies herein and hence we shall not labor further on it.
Remark 4.33. The second square of (4.24) also commutes, where the two horizontal arrows are constructed via taking the fiber at 1 as in (4.19). However, as another manifestation of the duality of the sign, taking the fiber at −1 yields a genuinely different ring homomorphism Neither Υ 1 nor Υ −1 is injective, not even taken as a pair (for instance any even power function on the torus gives the same class). Now, the said formula in [3] is formulated in a specialization M AS of K 1 RVar[[A] −1 ], which is constructed using arc-symmetric (semialgebraic) sets and maps. In more detail, adapting the method of [20], the (generalized) real motivic Milnor fiber S × f of f is the limit of a motivic zeta function Z × (T ) whose coefficients are given by sets of truncated arcs of the form {ϕ ∈ X(R[t]/t m+1 ) | f (ϕ) = at m mod t m+1 with a ∈ R × and ϕ(0) = z} together with the built-in angular component map sending ϕ to a. Then an equality similar to the special case (4.22) may be established in M AS ; see [3,Corollary 6.20]. Here we point out that the process of "taking the limit" forces the R × -actions on the coefficients of Z × (T ) to factor through a R + -action and, consequently, the negative part of R × does not really figure in S × f ; this is but another manifestation of what has been said in Remark 3.21 about the construction in [15].
Let us rather consider the same construction at the level of K 1 RVar (hence finer, since full R × -actions are retained). In order to show that [3, Corollary 6.20] can be obtained from the specialization of (4.22) to K 1 RVar, one needs to check that S × f can indeed be recovered as Vol ac (X ♯ ) overR, similar to (4.20). We may attempt to reproduce the argument given there. To begin with, taking the fiber at 1 coefficientwise, we recover from Z × (T ) the motivic zeta function Z 1 (T ) in (3.13) (taking the fiber at −1 gives its negative counterpart Z −1 (T )), and it is straightforward to check that this operation commutes with the operator "− lim T →∞ " in (3.15); actually this is just an analogue of [20,Remark 3.13], which we have also gone through in § 3.4. However, this is as far as we can go since, unlike Υ in (4.19), Υ 1 is not an isomorphism. In other words, although we know that the images of S × f , Vol ac (X ♯ ) under Υ 1 coincide in K µ 2 RVar, we cannot conclude that they themselves coincide in K 1 RVar.
Thus the apparent shortcut is blocked in the real environment, and we shall have to revert back to the zeta function point of view, that is, we need to show a version of Theorem 3.24 with respect to Z × (T ) and X ♯ f (R). Although some extra care is needed concerning the use of the integral ⋄ , there is no new insight arising in this endeavor and, as above, we choose not to labor further on technicalities.

In T -convex valued fields
It is also shown in [23, § 8] that one can recover, in a localization of Kμ Var C [A −1 ], the motivic zeta function and then the motivic Milnor fiber S of f from its nonarchimedean Milnor fiber X . In [23,Remark 8.5.5], these results yield a proof, without using resolution of singularities but still using other sophisticated algebro-geometric machineries, that the Euler characteristic of S equals that of the topological Milnor fiber of f (whether finer invariants such as the Hodge-Deligne polynomial can be recovered this way is still unknown). In this section, we aim to prove this equality and its real analogue using a geometric argument at the level of T -convex sets instead. Moreover, as is already mentioned in Remark 2.44, in the real environment, the difference between the bounded and the geometric Euler characteristics in the Γ-sort is manifested as an equality relating the Euler characteristics of the closed and the open topological Milnor fibers. 5.1. The universal additive invariant. We first summarize the main result of [34]. To begin with, let T be a complete polynomially bounded o-minimal L T -theory extending the theory RCF of real closed fields. It is not necessary in [34], but here we assume that R is a T -model. Let R := (R, <, . . .) be a nonarchimedean T -model containing R and O ⊆ R be the convex hull of R. Then O is a proper T -convex subring of R in the sense of [12], that is, O is a convex subring of R such that, for every definable (no parameters allowed) continuous function f : R −→ R, we have f (O) ⊆ O. According to [12], the theory T convex of the pair (R, O), suitably axiomatized in the language L convex that extends L T with a new unary relation symbol, is complete. We further assume that T admits quantifier elimination and is universally axiomatizable, which can always be arranged through definitional extension. Then T convex admits quantifier elimination too. It also follows that R is an elementary L T -substructure of R.
We may also view R as an L RV -structure. To construct Hrushovski-Kazhdan style integrals in this environment, however, we need to work with a different language, which extends L RV . Since 1 + M is a convex subset of R × , the total ordering on R × induces a total ordering on RV. This turns RV into an ordered group and k into an ordered field. By the general theory of T -convexity, there is a canonical way of turning k further into a T -model, which is isomorphic to the T -model R, with the isomorphism given by the residue map res. Let k + be the set of positive elements of k (similarly for other totally ordered sets with a distinguished element), which forms a convex subgroup of RV.
Notation 5.1. Denote the quotient map RV −→ Γ := RV / k + by vrv. The composition val := vrv • rv : R × −→ Γ is referred to as a signed valuation map. The corresponding value group is a "double cover" of the traditional value group. Consequently, the Euler characteristics, still denoted by χ g and χ b , are slightly different from the ones in Notation 2.38.
All of this structure can be expressed in a two-sorted first-order language L T RV , in which R is referred to as the VF-sort and RV is taken as a new sort. The resulting theory TCVF (see [34,Definition 2.7]) is complete and weakly o-minimal, and admits quantifier elimination. Informally and for all practical purposes, the language L T RV may be viewed as an extension of the language L convex .
Henceforth we work in the unique (up to isomorphism, of course) TCVF-model R rv that expands the T convex -model (R, O), with all parameters allowed.
Example 5.2. If T = RCF then we can turnR into a model of TCVF, with signed valuation, as follows. First note that rv is just the leading term map described in Example 2.1, and we may identify RV with Q ⊕ R × . Then the ordering on RV is the same as the lexicographic ordering on Q ⊕ R + or Q ⊕ R − (but not both of them together due to the issue of sign). The quotient group Γ = (Q ⊕ R × )/R + is naturally isomorphic to the subgroup ±e Q := e Q ∪ −e Q of R × , where e = exp(1), so that Q is identified with e Q via the map q −→ e q . Adding a new symbol ∞ to RV, now it is routine to interpretR as an L T RV -structure, with the signed valuation given by x −→ rv(x) = (q, a q ) −→ sgn(a q )e −q , where sgn(a q ) is the sign of a q . It is also a model of TCVF: all the axioms in [34,Definition 2.7] are more or less immediately derivable from the valued field structure, except (Ax. 7), which holds since RCF is polynomially bounded, and (Ax. 8), which follows from [ which is an isomorphism of graded rings.
Remark 5.5 (Explicit description of K TRES). The semiring K + TRES is actually generated by isomorphism classes [U] with U a set in k + . We have the following explicit description of K + TRES. The dimensional part is lost in the groupification K TRES of K + TRES, that is, K TRES ∼ = Z, which is of course much simpler than K + TRES.
The elements [1], P , and [A] in K TRV[ * ], the lifting map L, and the semiring congruence relation I sp are all defined as before.
Proposition 2.40 still holds in the current environment: Here we can also write the last two equalities in a form that is not simplified so to make the similarity to Proposition 2.40 apparent (the classes are replaced by their Euler characteristics in the residue field): E T g (x ⊗ y) = χ g (y)x(−1) l (−1) −(k+l) and E T b (x ⊗ y) = χ b (y)x(−1) l 1 −(k+l) . Note that −1 in the expression (−1) l is the Euler characteristic of the half torus (think R + ), not the torus (think R × ); this is related to the use of signed valuation map, see Notation 5.1. Both E T b and E T g will be relevant to our construction below.  where the vertical arrows are all induced by the subcategory functors. Of course E b,R , E T b may be replaced by E g,R , E T g and the diagram still commutes; however, as we have pointed out in Remark 3.28, doing so would not extend (3.1) properly (off by a factor).

5.2.
Link with the topological Milnor fiber. Denote by Def T the category of L T -definable sets and L T -definable bijections. So Def T is a subcategory of TVF * and we have an induced homomorphism i : K Def T −→ KTVF * . Let χ : K Def T −→ Z be the o-minimal Euler characteristic, which is of course an isomorphism; see [11]. On the other hand, K Def T is also canonically isomorphic to K TRES (Remark 5.5). Since χ, χ T g • i, and χ T b • i all agree on the class of the singleton {1}, they must be equal.

5.2.1.
The real case. In the case T = RCF, that is, in semialgebraic geometry, the Borel-Moore homology is defined for locally compact semialgebraic sets and satisfies a long exact sequence, which gives rise to an additive (and multiplicative) Euler characteristic χ BM . It is equal to the Euler characteristic of the singular cohomology with compact supports, also defined only for locally compact semialgebraic sets. One can compute χ BM on a cell decomposition, and the formula obtained can be used to extend the definition of χ BM to any semialgebraic set; see [6, § 1.8].
Consequently, χ BM coincides with χ. This holds in general for any o-minimal theory, but we do not know a reference that contains a complete account of it.
Notation 5.11. Let X, f , and z be as in § 3.4. Recall that the (positive) closed topological Milnor fiber is instantiated by L T -definable sets (in R) of the form F a,r = {x ∈ X(R) | x − z ≤ r and f (x) = a}, 0 < a ≪ r ≪ 1, where · : VF d −→ VF denotes the Euclidean norm restricted to R. The (positive) open topological Milnor fiber is similarly instantiated by L T -definable sets F a,r , but with x − z ≤ r replaced by x − z < r. Fix a t ∈ M + . For each r ∈ VF + , the setF r is defined asF a,r , but with X(R) replaced by X(VF) and a by t (since t does not vary anymore, we drop it from the notation); similarly for F r . SoF r is the topological closure of F r . Let ∂F r be the boundary ofF r , that is, ∂F r =F r F r = {x ∈ X(VF) | x − z = r and f (x) = t}.
Set F = r∈U +F r = r∈U + F r , where U = O M, or equivalently, Since O is the convex hull of R, we can also write F = r∈R +Fr = r∈R + F r . Note that F is definable but is in general not L T -definable. Note that all the occurrences of χ here stand for the o-minimal Euler characteristic, but on one side of the equality it is taken in R, and in R on the other side.
Proof. ConsideringF a,r as a definable set in R, it has the same Euler characteristic (since any cell decomposition in R is also a cell decomposition in R) and, by o-minimal trivialization, there is a t ′ ∈ M + such that χ([F a,r ]) = χ([F ′ r ]), whereF ′ r is defined asF r but with t replaced by t ′ . Since t, t ′ make the same cut in R, there is an automorphism of R over R mappingF r toF ′ r . The first equality follows. The second equality is similar. ]. This series is, up to sign, the positive topological zeta function considered in [24]. In more detail, for each m ≥ 1, let X + m be the following set of truncated arcs at z: {ϕ ∈ X(R[t]/t m+1 ) | f (ϕ) = at m mod t m+1 with a ∈ R + and ϕ(0) = z}.

5.2.2.
The complex case. We may consider the complex geometry ofC over C((t)) in the TCVF-modelR, sinceR may also be viewed as an L RV -structure. Thus VF * , RV[ * ], etc., refer to the categories in § 2.1 with S = C((t)) and VFR, RVR[ * ], etc., refer to the categories in § 3.1 with M =R. Also X, f are defined over C and the point z is C-rational.
The new perspective is that, as an L RV -structure, there is an obvious interpretation, in the model-theoretic sense, ofC inR; this is just a fancy way to say that, after fixing a square root √ −1 of −1,C may be identified withR 2 , C((t)) with R((t)) 2 , RV(C) with RV(R) 2 , and so on. For convenience, we shall callC a complex field, (R, 0) ⊆C the real line inC, and (0,R) ⊆C the imaginary line inC. SinceC is now interpreted inR, there is an induced faithful functor VF * −→ VFR, which in turn yields a homomorphism D : K VF * −→ K VFR. , although a similar functor is available, we need to be more careful since these categories are graded.
To illustrate the concern, consider the object RV ∞ (C) = RV ∞ (R) 2 . Since the real and the imaginary lines have only one nonzero coordinate, this object has nonempty components in all of the three categories RVR[0], RVR [1], and RVR [2]. This interpretation leads to an issue since, for instance, the complex points (1, 0) and (1, 1) should certainly be isomorphic objects, but they cannot be since they do not even belong to the same graded piece.
To resolve this issue, we can work with a dimension-free version of the Grothendieck ring and hence it makes no difference which one of the generalized Euler characteristics χ T g , χ T b is used to relate X and F . We recover thus the result in [23,Remark 8.5.5]: Theorem 5.24. The Euler characteristic of the topological Milnor fiber of f equals χ C ([X ]).