COMBINATORIAL STUDY OF MORSIFICATIONS OF REAL UNIVARIATE SINGULARITIES

. We study a broad class of morsiﬁcations of germs of univariate real analytic functions. We characterize the combinatorial types of the resulting Morse functions via planar contact trees constructed from Newton-Puiseux roots of the polar curves of the morsiﬁcations.


Morsifications
In this paper, by a singularity we mean a germ of real or complex analytic function with an isolated critical point.By a Morse function on a compact manifold with boundary we mean a smooth function having only non-degenerate critical points, all of them interior to the manifold, and pairwise distinct critical values.A powerful method for analyzing a singularity is to deform it in a suitable way and relate it to the various resulting simpler singularities.This method has been extensively used for complex singularities.For instance, a generic holomorphic deformation of a complex singularity with Milnor number µ produces a Morse function with exactly µ critical points (see [5, page 150]).However, similar morsifications of real singularities have been much less studied, even in the case of one variable.
In this paper we examine the combinatorial types of morsifications of univariate real singularities.This problem is inspired by Arnold's papers [1], [2], which studied the combinatorial types of real Morse univariate polynomials, and by Ghys' book [9], which examined the combinatorial types of real plane curve singularities (see also Ghys' paper [8] and Ghys and Simon's paper [10]).Let us mention also two very recent related articles.In [18], Teissier describes some open problems about real morsifications and in [19], Vassiliev describes the possible real Morsification types in the case of a real simple singularity in any number of variables.

Bi-ordered critical sets as measures of the combinatorial types of morsifications
We encode the combinatorial type of a Morse function defined on a compact interval by a bi-ordered set: its critical set endowed with the total order induced by its inclusion in the source interval and with the total order of the corresponding critical values.Let F 0 (y) ∈ R{y} be a convergent power series defining a univariate real singularity.Fix a compact interval I around y = 0 on which F 0 (y) is defined and has a single critical point at the origin.Let F x (y) ∈ R{x, y} be a morsification of F 0 (y).This means that for every small enough x 0 > 0, the functions F x0 : I → R are Morse and have the same combinatorial type.Moreover, this combinatorial type is independent of the choice of interval I, it is therefore canonically attached to the morsification F x (y) ∈ R{x, y}.
Our central problem is to compute this combinatorial type starting from the series F x (y).
We solve this problem under a suitable hypothesis, the injectivity condition.Our answer is governed by the contact tree T R (f ) of the real Newton-Puiseux roots ξ i of f (x, y) := ∂ y F x (y).It is a rooted planar tree whose leaves correspond bijectively to the series ξ i .As an abstract tree, it is determined by the valuations of the pairwise differences of those series, that is, by the initial exponents of those differences.In turn, its planar structure is given by the total order < R on its set of leaves such that ξ i < R ξ j if and only if ξ i (x 0 ) < ξ j (x 0 ) for x 0 > 0 small enough.
Under the injectivity condition, we construct canonically from F x (y) a second planar structure on the abstract rooted tree T R (f ).This second planar structure determines a new total order on the set of leaves of T R (f ).In Theorem A, we prove that: Theorem A. Assume that f satisfies the injectivity condition.Then, for x 0 > 0 small enough, the bi-ordered critical sets of the Morse functions F x0 : I → R are isomorphic to the set of leaves of the contact tree T R (f ), endowed with the total orders determined by the two planar structures above.
The previous result led us to ask whether T R (f ), endowed with its second planar structure, may also be interpreted as a contact tree.In Theorem B we prove that this is indeed the case: Theorem B. Assume that f satisfies the injectivity condition.Then, with its second planar structure, T R (f ) is isomorphic to the contact tree of the real Newton-Puiseux roots of the discriminant curve of the morphism (x, y) → (x, F x (y) ).
This discriminant curve is the critical image of this morphism, also called apparent contour in the target.The apparent contour in the source is the curve f (x, y) = 0, that is, the polar curve of F x (y) relative to x.
Let us assume that the injectivity condition is satisfied.Then, as a consequence of Theorem A, the structure of the real contact tree T R (f ) strongly constrains the combinatorial type of the morsifications F x (y) (see Remark 6.7) and, as a consequence of Theorem B, the real contact trees of the apparent contours in the source and in the target of the morphism (x, y) → (x, F x (y)) are isomorphic as abstract rooted trees (see Remark 7.1).

The meaning of the injectivity condition
Assume that the real Newton-Puiseux roots of f are numbered such that ξ 1 < R • • • < R ξ n .The bi-ordered critical set of a Morse function F x0 : I → R, for small enough x 0 > 0, is determined by the signs of all the differences F x0 (ξ j (x 0 )) − F x0 (ξ i (x 0 )) of its critical values.If i < j, we may write: where S r := F x (ξ r+1 ) − F x (ξ r ).For small enough x 0 > 0, the sign of F x0 (ξ j (x 0 )) − F x0 (ξ i (x 0 )) is thus equal to the sign of the initial coefficient of the sum S i + • • • + S j−1 of real Newton-Puiseux series.We meet the precise situation in which Newton introduced the method of turning ruler, which led to the notion of Newton polygon (see [9, pages 51-53]): denoting by ν(S l ) the valuation of S l , we know that the initial coefficient of the sum S i + • • • + S j−1 is the sum of the initial coefficients s r of the series (S r ) i r<j achieving the minimum min{ν(S r ) | i r < j}, provided that this last sum of initial coefficients s r is non-zero.The injectivity condition of Definition 6.1 is equivalent to the fact that this non-vanishing condition is satisfied for every pair (i, j) with i < j.

Structure of the paper
As the function which controls the combinatorial types of the Morse functions y → F x0 (y) is f (x, y) := ∂ y F x (y) rather than F x (y), we prefer to start from a real analytic series f (x, y) and integrate it relative to y in order to get the series F x (y).In Section 2 we recall the factorization of f (x, y) via its Newton-Puiseux roots, we distinguish between real and non-real roots and we define the notions of right-reduced series and of primitive of f (x, y).In Section 3 we explain the needed notions about univariate Morse functions and their bi-ordered critical sets, as well as about morsifications of univariate singularities.In Section 4 we explain basic facts about rooted and planar trees and we introduce the types of rooted trees used in the paper: the real contact tree T R (f ) mentioned above, and the complex contact tree T C (f ), which is an abstract rooted tree containing T R (f ).Section 5 contains our main technical results.In it, we introduce the area series S l mentioned in Subsection 1.3, we compute their valuations in terms of the embedding T R (f ) ֒→ T C (f ) (see Proposition 5.1) and we deduce the valuations of the sums S i + • • • + S j−1 under the non-vanishing hypothesis mentioned in Subsection 1.3 (see Lemma 5.4).In Section 6 we define the injectivity condition (Inj), we give examples in which it is not satisfied (see Examples 6.3 and 6.4), we prove our first main result, Theorem A, and we explain that Lemma 5.4 allows to get a weaker statement even if the injectivity condition is not satisfied (see Remark 6.8).In Section 7 we define real polar and discriminant curves and we prove our second main result, Theorem B. We conclude the paper by an example with parameters, explained in Section 8.

Related works
In this paper, we generalize results of the PhD thesis [17] of the last author, published in [14], [15], [16].In those works, the polar curve f (x, y) = 0 and the series F x (y) had to respect some hypotheses: -all the branches of the polar curve were real, distinct, smooth and transverse to the vertical axis x = 0; -the real contact tree T R (f ) was a rooted binary tree (then the injectivity condition is automatically satisfied); -F x (y) had a strict local minimum at (0, 0).
The aim was to describe the asymptotic shape of the level curves F x (y) = ε when ε > 0 converged to 0. This description was done in terms of a Poincaré-Reeb tree measuring the non-convexity of the interior of the topological disk bounded by the level curve F x (y) = ε, relative to the direction x (see also [3] for a general study of level curves of real bivariate polynomials).Here we replace all the former hypotheses by two much less restrictive conditions, namely that: -the real Newton-Puiseux roots ξ i of f (x, y) are pairwise distinct; -the injectivity condition is satisfied.
1.6.An explanatory picture Figure 1 introduces the main geometric objects studied in this paper.It corresponds to Whitney's classical cusp singularity from [20] of a map between real planes.This example will also illustrate our main Theorems A and B (see Examples 6.6 and 7.2).We start from the real plane curve germ f (x, y) = 0 at (0, 0) represented in the real plane R 2 x,y at the bottom, where f (x, ).It has two Newton-Puiseux roots ξ 1 = −x 1 2 and ξ 2 = x 1 2 .Both are real.We define F x (y) as a primitive of f w.r.t. the variable y.Here we choose F x (y) := y 3 − 3xy.The graph of the function (x, y) → F x (y), for positive x, is the surface depicted in the central part of the figure.By intersecting this surface with a vertical plane defined by x = x 0 , we get the graph of y → F x0 (y).In our example, it is a Morse function with one local maximum and one local minimum, for every x 0 > 0. It is possible to follow these local extrema when x 0 tends to 0: they trace the two orange curves on the surface.These two curves project to the real plane R 2 x,y exactly onto the graphs of the roots ξ 1 and ξ 2 .
On the other hand, these orange curves form the apparent contour in the source of the projection π of the surface above onto the vertical plane R 2 x,z .The apparent contour in the target plane R 2 x,z is the discriminant curve of π, which in this example consists of the graphs of two real Newton-Puiseux series δ 1 and δ 2 (at the top of Figure 1).The graphs of ξ 1 and ξ 2 in the real plane R 2 x,y form the polar curve of F x (y) with respect to x; it is defined by the equation f (x, y) = 0.By the projection π, the series ξ 1 corresponds to δ 1 , and ξ 2 to δ 2 .In the plane R 2 x,y , ξ 1 appears before ξ 2 (we will define the real total order on the ring of real Newton-Puiseux series in Section 2), while in the plane R 2 x,z , δ 1 appears after δ 2 .This permutation ( 1 2 2 1 ) encodes the combinatorial type of the Morse function y → F x0 (y).Theorem A explains that whenever the injectivity condition is satisfied, the corresponding permutation may be read from the embedding T R (f ) ֒→ T C (f ).
The results of our paper allow therefore to make pictures analogous to that of Figure 1, representing correctly the combinatorial types of the Morse functions y → F x (y) whenever f satisfies the injectivity condition.

Real Newton-Puiseux series, right semi-branches and primitives
In this section we explain our notations about Newton-Puiseux series, we define right semi-branches as the germs of graphs of real Newton-Puiseux series and we introduce the notion of primitive of a bivariate series.

Newton-Puiseux series
For K = C or R, let K{x} and K{x, y} denote the ring of convergent power series in one and two variables respectively, with coefficients in the field K. Consider also the ring of Newton-Puiseux series in the variable x, with coefficients in K. Then R{x We may write uniquely: +∞) and the remainder hot (which stands for higher order terms) gathers the terms of γ whose exponents are greater than σ.The number s ∈ C * is the initial coefficient of γ, denoted by χ(γ), and σ ∈ Q ∩ [0, +∞) is the initial exponent of γ, denoted by ν(γ).By convention, ν(0) = ∞.The function ν : C{x ring valuation, which will play a crucial role in the sequel.
The ring R{x 1 N } of real Newton-Puiseux series is naturally totally ordered: Definition 2.1.The real total order < R on the ring R{x 1 N } is defined as follows: for any two distinct Newton-Puiseux series Note that ξ 1 < R ξ 2 if and only if ξ 1 (x 0 ) < ξ 2 (x 0 ) for x 0 > 0 small enough.

Right semi-branches
Consider ξ ∈ R{x 1 N } with ξ(0) = 0.In the sequel it will be often needed to turn the formal series ξ into a real-valued function.This will be performed by choosing a real number ε ∈ (0, ∞) such that the series with real terms ξ(x 0 ) converges for every x 0 ∈ [0, ε].For simplicity, we still denote by ξ : [0, ε] → R the resulting function.We will say that it is the sum of the series ξ.The sum of the series ξ depends on the chosen interval of convergence [0, ε], but its germ at the origin is well-defined.Therefore, the germ at (0, 0) ∈ R 2 of the graph of the function ξ is also well-defined.We call it the right semi-branch of the series ξ ∈ R{x 1 N }.If f ∈ R{x, y} is such that f (0, 0) = 0 but f (0, y) ≡ 0, then by the Weierstrass preparation theorem (see [6, page 107]), together with the Newton-Puiseux theorem (see [7,Theorem 1.2.20], [4, Section 8.3]), we can write in a unique way: ( 1) f (x, y) = u(x, y) such that u ∈ C{x, y} is a unit (i.e.u(0, 0) = 0) and γ i ∈ C{x 1 N } for all i ∈ {1, . . ., k}.Since f ∈ R{x, y}, we have that u ∈ R{x, y}.The Newton-Puiseux series γ i are called the Newton-Puiseux roots of f .We will make below (see Formula (3)) a distinction between roots having only real coefficients (denoted by ξ i ) and the others (denoted by η l ).We denote by R K (f ) the multi-set of roots γ ∈ K{x 1 N } of f (that is, each root is counted with its multiplicity).The set of right semi-branches of f is by definition the set of right semi-branches of the elements of R R (f ).

Primitives
Consider f ∈ R{x, y} with f (0, 0) = 0 and f (0, y) ≡ 0. A primitive of f is a series F x (y) ∈ R{x, y} such that: (2) Primitives of f always exist.They are of the form g(x) + h(x, y), where g ∈ R{x} is arbitrary and h ∈ R{x, y} is obtained by termwise integration of the series f , that is, by replacing each non-zero term c p,q • x p • y q of it by c p,q • x p • y q+1 q + 1 .

Morsifications
In this section we give basic vocabulary about univariate Morse functions and we introduce their bi-ordered critical graphs.Then we define morsifications of univariate singularities and their combinatorial types.

Morse functions
Let us first introduce standard definitions from Morse theory, particularized to our context of univariate functions: Definition 3.1.Let I ⊂ R be a compact interval and let ϕ : I → R be a smooth function.We say that c ∈ I is a critical point of ϕ if ϕ ′ (c) = 0; it is called non-degenerate if ϕ ′′ (c) = 0. We say that ϕ is a Morse function if: -all its critical points are non-degenerate; -they lie in the interior of I; -its critical values are pairwise distinct.The critical graph of ϕ is the graph of the restriction of ϕ to its set of critical points: Non-degenerate critical points being isolated, a Morse function on a compact interval has only a finite number of critical points.In the literature, what we call Morse functions are sometimes called excellent Morse functions, the attribute referring to the third condition above, which is equivalent to the condition that no two critical points lie on the same level set.As we do not consider non-excellent Morse functions, we prefer to use the simplified terminology of Definition 3.1.

The canonical bi-order on the critical graph of a Morse function
In this paper, by an order we mean either a strict or non-strict partial or total order on a given set, depending on the context.We will denote by ≺ the strict order associated to an order .
A finite set S is bi-ordered if it is endowed with a pair of total orders.The critical graph Crit(ϕ) (see Definition 3.1) of any Morse function ϕ : I → R defined on a compact interval I is canonically bi-ordered: Definition 3.2.Let I ⊂ R be a compact interval and ϕ : I → R be a Morse function.The source order < s and target order < t are the total orders on the critical graph Crit(ϕ) defined as follows for any two distinct points p = (y 1 , z 1 ), q = (y 2 , z 2 ) ∈ Crit(ϕ): p < s q if and only if y 1 < y 2 , p < t q if and only if z 1 < z 2 .
(1) The bi-ordered set (Crit(ϕ), < s , < t ) may be thought as a measure of the combinatorial type of the Morse function ϕ.Indeed, let ϕ 1 : I 1 → R and ϕ 2 : I 2 → R be two Morse functions on compact intervals.Then the associated bi-ordered critical sets are isomorphic if and only if the restrictions of ϕ 1 and ϕ 2 to the minimal intervals containing all their critical points are right-left equivalent by orientation-preserving diffeomorphisms.Without restricting ϕ 1 and ϕ 2 in this way, one should also take into account their boundary values in order to construct a complete invariant of right-left equivalence.( 2) As explained in [9, pages 17-18] (see also [17,Section 3.2.6]), the comparison between the two total order relations on a bi-ordered set naturally gives rise to a permutation.The permutations coming from Morse functions were called snakes by Arnold (see [1], [2], [14, Definition 1.4]).We will use again this terminology in Section 8 (see Figure 16).Its critical graph Crit(ϕ) has 4 elements p 1 , . . ., p 4 .Since y 1 < y 2 < y 3 < y 4 and z 2 < z 4 < z 1 < z 3 , the two orders on it are : The associated snake is It encodes the relation between the two orders for the points

Right-reduced functions, morsifications and their Morse rectangles
We define now the notion of morsification of a univariate singularity, paying attention to the intervals of definition of the associated Morse functions: Definition 3.5.Let f ∈ R{x, y} be such that f (0, 0) = 0 and f (0, y) ≡ 0. Let F x (y) ∈ R{x, y} be a primitive of f in the sense of Formula (2).A Morse rectangle of F x (y) is a product [0, ε] × I, where ε > 0 and I is a compact interval neighborhood of the origin in the y-axis such that: We say that F x (y) is a morsification (of F 0 (y)) if it admits a Morse rectangle.Let us introduce now a notion of reducedness of real series adapted to their study in the right half-plane x 0. Geometrically, this means that we assume that the right semi-branches of f are reduced in the divisor of f .
Example 3.7.The series (y 2 + x) 3 (y 2 − x) is right-reduced, but it is not reduced as an element of the ring R{x, y}.Proposition 3.8.Let f ∈ R{x, y} be a right-reduced series and F x (y) be a primitive of f .Assume that the series F x (ξ i ) ∈ R{x Proof.We first choose a rectangle [0, ε] × I included in the convergence disk of f (x, y) and of F x (y).We may reduce I in order that F 0 has a single critical point at y = 0. We may then diminish ε such that the roots ξ i : [0, ε] → I of f converge on [0, ε] and that for every x 0 ∈ (0, ε], one has ξ i (x 0 ) = ξ j (x 0 ) whenever i = j.

Hence one may assume that
] and ξ i (x 0 ) = ξ j (x 0 ) whenever i = j, we see that the roots of f (x 0 , y) on I are exactly the real numbers ξ i (x 0 ).By the same condition, these numbers are pairwise distinct, which proves our claim.
(1) Notice that if G x (y) is another primitive of f , then G x (y) = g(x) + F x (y).Therefore, for a fixed x 0 > 0, the graph of G x0 : I → R is a vertical translation of the graph of F x0 : I → R, hence they have equivalent bi-ordered critical graphs.The bi-ordered critical graph is also independent of the choice of a Morse rectangle.(2) Because the critical points of F x0 : I → R are (ξ i (x 0 )) 1 i n and the critical values are (F (ξ i (x 0 ))) 1 i n , the "right-reduced" hypothesis implies that the critical points of F x0 : I → R are non-degenerate; the hypothesis on distinct F x (ξ i ) implies that the critical values of F x0 (y) are pairwise distinct when x 0 is small enough.
Definition 3.10.Let F x (y) ∈ R{x, y} be a morsification.Its combinatorial type is the isomorphism class of the bi-ordered critical graphs of the functions F x0 : I → R chosen as in Proposition 3.8.
Our goal is to describe the combinatorial types of morsifications starting from the series F x (y) ∈ R{x, y} defining them.This goal will be achieved in Theorem A, under the hypothesis that f = ∂ y F x satisfies the so-called injectivity condition, explained in Subsection 6.1.

Considerations about trees
Since trees play a key role in our results, in this section we explain basic facts concerning them, partly following [17, Section 1.4.1] and the references therein.

Abstract trees
A tree is a topological space homeomorphic to a finite connected graph without cycles.Except when it is reduced to a point, a tree has an infinite number of points.The valency of a point of a tree is the number of connected components of T \ {P }.Its vertices are its points of valency different from 2 and its edges are the closures of the connected components of the complement of its set of vertices.Given two points P, Q of a tree, we denote by [P, Q] the unique segment joining them.
For us, a rooted tree has a marked point O of valency 1, called the root.We choose this hypothesis about valency because all the rooted trees considered in this paper, namely the contact trees of Subsection 4.4, satisfy it.Every rooted tree T is endowed with a natural partial order T : given two distinct points P and Q of T , A leaf of T is a maximal element for the partial order T .
Denote by V(T ) the set of vertices, by L(T ) the set of leaves and by V • (T ) = V(T ) \ (L(T ) ∪ {O}) the set of internal vertices of T .If P is an internal vertex of T , then an outgoing edge of T at P is by definition an edge [P, Q] that is not contained in a segment of the form [O, P ].We denote by E + T (P ) the set of outgoing edges of T at P .These sets will be used in Definition 4.1 for the formulation of the notion of planar tree.
To any two points P and Q of T we associate their greatest lower bound P ∧ Q relative to the partial order T .That is (see Figure 4):

Planar trees
In this subsection we explain the notion of planar tree, which is essential in the sequel, as one may associate canonically such a tree to any finite set of real Newton-Puiseux series (see Subsection 4.4): Definition 4.1.A planar structure on a rooted tree T is a choice of a total order < P on each set E + T (P ) of outgoing edges, when P varies among the internal vertices of T .A planar tree is a rooted tree endowed with a planar structure.
The terminology planar structure is motivated by the fact that such a structure is equivalent to the choice of an isotopy class of embeddings of the rooted tree in any given oriented plane.This equivalence would not be true any more if the root were of valency at least 2. Indeed, in that case an isotopy class of embeddings in an oriented plane would only be fixed if one chooses moreover a cyclic order of the edges adjacent to the root.
When embedding canonically a planar tree T in an oriented plane, one sees that its set of leaves L(T ) is canonically totally ordered (see Figure 5).This associated total order may also be defined intrinsically (without mentioning an embedding into a plane) as follows: if ℓ 1 , ℓ 2 are two distinct leaves of T and P := ℓ 1 ∧ℓ 2 , then ℓ 1 < ℓ 2 if and only if e 1 < P e 2 , where e 1 , e 2 ∈ E + T (P ) are the outgoing edges at P going to ℓ 1 and ℓ 2 respectively.
. .Not every total order on its set L(T ) of leaves comes from a planar structure on a rooted tree T , as shown by the following proposition: Proposition 4.2.Let < be a total order on the set of leaves of a rooted tree T .The necessary and sufficient condition for < to come from a planar structure on T is that for any two incomparable vertices P, Q of T (that is, vertices such that P T Q and Q T P ), the leaves T -greater than P are either all <-smaller or all <-bigger than the leaves T -greater than Q.In this case, the total order < determines the planar structure uniquely.
Proof.Let us assume first that T is endowed with a planar structure.Denote by < the associated total order on L(T ).For each vertex P of T , let D(P ) be the set of leaves T -greater than P (which may be thought as the set of descendants of P , if T is imagined as a genealogical tree).Consider two incomparable vertices P, Q of T .Denote R := P ∧ Q / ∈ {P, Q}.Let e P be the outgoing edge at R directed towards P and define similarly e Q .We may assume, possibly after permuting P and Q, that e P < R e Q .Choose ℓ P ∈ D(P ) and ℓ Q ∈ D(Q).Then e P is the outgoing edge at R directed towards ℓ P , and similarly e Q goes towards ℓ Q .The definition of the total order < on L(T ) and the fact that e P < R e Q imply that ℓ P < ℓ Q .Therefore, all leaves in D(P ) are <-smaller than all the leaves in D(Q).
Let us assume now that L(T ) is endowed with a total order < verifying the given condition.Consider a vertex P of T .We want to show that < determines a canonical total order < P on E + T (P ).If P is a leaf, there is nothing to prove.Assume therefore that P is not a leaf.Let e 1 and e 2 be two distinct outgoing edges at P .Let us write e i = [P, P i ].As the vertices P 1 and P 2 are incomparable, we know that the elements of D(P 1 ) are either all <-smaller or all <-bigger than the elements of D(P 2 ).In the first case we set e 1 < P e 2 and in the second one e 2 < P e 1 .We get an antisymmetric binary relation < P on the set E + T (P ).As < is a total order, this is also the case for < P .Proposition 4.2 motivates the following definition, which will be used in the formulation of Proposition 4.7: Definition 4.3.Let T be a rooted tree.A total order on the set L(T ) of leaves of T is called planar relative to T if it is determined by a planar structure on T .
Example 4.4.Consider the abstract rooted tree T of Figure 6.Take the following total order on L(T ) = {ℓ 1 , ℓ 2 , ℓ 3 }: The vertices P := ℓ 1 and Q := ℓ 2 ∧ ℓ 3 are incomparable, but ℓ 1 , which is the only leaf T -greater than P is neither <-smaller nor <-bigger than both ℓ 2 and ℓ 3 , which are the leaves T -greater than Q.Therefore this total order is not planar relative to T .This example shows also that it is important to allow the vertices P and Q appearing in Proposition 4.2 to be leaves.

The wedge map of a planar tree
Let (X, <) be a finite totally ordered set.Denote its elements by of two successive elements of X. Denote by BI(X) the set of basic intervals of (X, ).This set is empty if and only if n 1.
Let T be a planar tree.The sets L(T ) and E + T (P ), where P is an internal vertex of T , are therefore naturally totally ordered, as explained in Subsection 4.2.Let {ℓ, ℓ ′ } be a basic interval of L(T ).Denote P := ℓ ∧ ℓ ′ .Let e and e ′ be the outgoing edges going from P to the leaves ℓ and ℓ ′ (see Figure 7).Then {e, e ′ } is a basic interval of E + T (P ), by the definition of the total order on the set of leaves of a planar tree.This construction defines the wedge map of the planar tree T : Note that when P ∈ V • (T ), one has BI E + T (P ) = ∅.The following proposition will be crucial in Subsection 5.3, as well as in Subsection 6.1, in order to define the injectivity condition: Proposition 4.5.The wedge map W of a planar tree is bijective.
Proof.The source and target of the wedge map W have the same number of elements, as may be proved by induction on the number of leaves of T .Therefore, in order to prove that W is bijective, it is enough to prove that it is surjective.Consider a vertex P ∈ V • (T ) and a basic interval {e j , e j+1 } of E + T (P ), with e j < P e j+1 .Let ℓ ι(j) be the <-biggest leaf among the descendants of e j and ℓ ι(j+1) be the <-lowest leaf among the descendants of e j+1 .By the construction of the total order < on L(T ) explained in Subsection 4.2, we have ℓ ι(j) < ℓ ι(j+1) .By Proposition 4.2, we see that {ℓ ι(j) , ℓ ι(j+1) } is a basic interval of (L(T ), <).As results from the definition of the wedge map, its image by W is the basic interval {e j , e j+1 } of E + T (P ).This shows that W is surjective, therefore bijective.Proposition 4.5 generalizes [14, Corollary 2.20], which concerned only the case where the rooted tree was binary, that is, where all its vertices had valency 1 or 3.

Contact trees
Let us consider a finite set of Newton-Puiseux series N = {γ 1 , . . ., γ n } ⊂ K{x 1 N }, such that γ i (0) = 0 for all i = 1, . . ., n.The contact tree of the set N , denoted by T K (N ) or by T K (γ 1 , . . ., γ k ), is a rooted tree encoding the valuations of pairwise differences of the elements of N .It is canonically determined by the ultrametric distance d : N × N → (0, +∞) defined by: whenever γ i = γ j .For details, we refer the reader to [13, Section 9.4] and references therein.The contact tree is a version of the so-called Kuo-Lu tree, introduced in [12] (see [7, Section 1.6.6]).
Let us explain informally how to construct T K (N ) by gluing compact segments identified to [0, ∞], one segment per series.Associate a copy I i of the interval [0, ∞] to each series γ i ∈ N .The point of I i whose coordinate is a ∈ [0, ∞] represents the formal monomial x a .If γ i , γ j ∈ N are such that γ i = γ j , then glue the segments [0, ν(γ j − γ i )] of the intervals I i and I j by identifying the points having the same coordinate in [0, ∞].This gluing process leads to a tree which is by definition the contact tree T K (N ).All the points of coordinate 0 of the intervals I i get identified to a point O ∈ T K (N ), which is chosen as the root.As ν(γ j − γ i ) > 0 whenever i = j, the root is of valency 1.The set L(T K (N )) of leaves of T K (N ) is in canonical bijection with the set N .We will identify them using this bijection: Example 4.6.Consider the set N consisting of the following real Newton-Puiseux series: γ 1 = −x, γ 2 = x, γ 3 = x + x 3 and γ 4 = x + 2x 3 .The corresponding intervals I i are drawn on the left of Figure 8, the marked points being those whose coordinates are exponents of monomials appearing in γ i .The contact tree T K (N ) is drawn on the right.The monomial x r corresponding to a vertex is written between brackets as a decoration.The corresponding term c i x r in each series γ i is written as a decoration of the edge going towards γ i , seen as a leaf of T K (N ).Note that in this example all Newton-Puiseux series have integral exponents.One could restrict to this situation throughout the paper by making a change of variable of the form x = x N 1 , for a value N ∈ N * divisible by the denominators of all the exponents appearing in the complex Newton-Puiseux roots of f (x, y).Suppose now that the finite set N consists only of real Newton-Puiseux series ξ i ∈ R{x 1 N }.It acquires then a canonical total order, by restriction of the real total order < R of Definition 2.1.Therefore, we also call it the real total order on N .It is planar relative to the rooted tree T R (N ): Proposition 4.7.Let N be a finite subset of R{x 1 N }.The real total order on N = L(T R (N )) is planar relative to the tree T R (N ), in the sense of Definition 4.3.
Proof.Let ξ i and ξ j be two distinct elements of N .By definition of the real total order, ξ i < R ξ j if and only if χ(ξ j − ξ i ) > 0 (see Subsection 2.1).Denote k := ν(ξ j − ξ i ) > 0. Then ξ i = ξ + a k x k + hot and ξ j = ξ + b k x k + hot, where a k = b k and ξ is a Newton-Puiseux polynomial of degree < k.Therefore χ(ξ j − ξ i ) = b k − a k , which shows that ξ i < R ξ j if and only if a k < b k .This implies easily the condition for planarity described in Proposition 4.2 (for more details, see [17,Section 1.7

.2]).
A different version of real contact tree of a finite set of real Newton-Puiseux series was introduced in [11, Section 6.3] by Koike and Parusinski.

The real and complex contact trees of right-reduced series
Let f ∈ R{x, y} be a right-reduced series.We will distinguish the real Newton-Puiseux roots of f from the non-real ones.Therefore, relation (1) becomes: where: - . ., n, are the real roots of f , which are pairwise distinct by the hypothesis that f is right-reduced; η l , η l ∈ R C (f ) \ R R (f ), l = 1, . . ., m, are the non-real roots of f ; they are not necessarily pairwise distinct.
Recall that u is a unit (i.e.u(0, 0) = 0) and since f ∈ R{x, y} we have that u ∈ R{x, y}.Denote by T C (f ) the contact tree of the set of all the Newton-Puiseux roots (real or complex) of f .Similarly, denote by T R (f ) the contact tree of the set of real Newton-Puiseux roots of f , namely T R (ξ 1 , . . ., ξ n ).Note that T R (f ) is a rooted sub-tree of T C (f ) and that T R (f ) is canonically planar, by choosing the real total order on its set of leaves (see Proposition 4. 7).We say that this planar structure is the real planar structure of T R (f ).We will define in Section 6 below a second planar structure on it, the integrated planar structure.

Area series and their valuations
Throughout this section, we assume that f ∈ R{x, y} is a right-reduced series and that its real Newton-Puiseux roots ξ i satisfy: y} be a primitive of f .Its area series are the successive differences F x (ξ i+1 ) − F x (ξ i ).We compute their valuations and we prove that they may be described using a strictly increasing function on the real contact tree T R (f ), the integrated exponent function.Then we deduce the valuations of the differences F x (ξ j ) − F x (ξ i ) for j − i 2, whenever a non-vanishing hypothesis is satisfied.

Area series
Consider a Morse rectangle [0, ε] × I of F x (y) (see Definition 3.5).The numerical series F x0 (ξ i (x 0 )) converges for every x 0 ∈ [0, ε] and its sum is a critical value of the function F x0 : I → R. For this reason, we say that F x (ξ i ) is a critical value series.
In order to compare two critical values F x0 (ξ i (x 0 )) and F x0 (ξ j (x 0 )) of F x0 when x 0 ∈ (0, ε], we will first evaluate the initial terms of the differences ( 4) of consecutive critical value series.For all i = 1, . . ., n − 1, we have: by the definition of F x (y).Therefore, S i (x 0 ) is the signed area of the region contained between the interval [ξ i (x 0 ), ξ i+1 (x 0 )] of the y-axis and the graph of y → f (x 0 , y) (see Figure 9).For this reason, we say that S i is the i-th area series of f .Notice that the signs of the areas S i (x 0 ) alternate when x 0 ∈ (0, ε], since the hypothesis of right-reducedness of f implies that y → f (x 0 , y) has only simple roots ξ i (x 0 ).Hence the critical points of y → F x0 (y) alternate between local minima and local maxima.In order to compare two critical values F x0 (ξ i (x 0 )) and F x0 (ξ j (x 0 )) when j − i 2, we need to determine the sign of the initial coefficient of the difference: Denote by s r x σr the initial term of the area series S r : (6) S r = s r x σr + hot.

The valuations of the area series
The following proposition generalizes [14, Proposition 3.1], which concerned the case when f had only real Newton-Puiseux roots with integer exponents, and that the associated real contact tree was binary.It shows that the valuations of the area series S l may be computed combinatorially on the real contact tree T R (f ), using the embedding T R (f ) ⊂ T C (f ): E(P ∧ γ).

Signs of differences of critical values
Let P be an internal vertex of T R (f ) and let (e 1 , e 2 , . . ., e p ) be the strictly increasing sequence of outgoing edges of P (see Figure 10  Take {e r , e r+1 } ∈ BI(E + T R (f ) (P )).By Proposition 4.5, there exists a unique basic interval {ξ ι(r) , ξ ι(r In this way we can associate to the basic interval {e r , e r+1 } the area function S ι(r) and also its initial coefficient s ι(r) = χ(S ι(r) ) (see Equation ( 6)).
Denote by the sign function.The following lemma enables to determine the signs of the differences appearing in Formula ( 5) via the knowledge of certain sums of initial coefficients s r of the area series S r : . .Lemma 5.4.Assume that f is right-reduced.Let ξ i < R ξ j be two real roots of f .Consider the vertex P := ξ i ∧ξ j of the real contact tree T R (f ).Let s ι(a) , s ι(a+1) , . . ., s ι(b) be the initial coefficients associated to the outgoing edges from P , in between the edges going to the leaves ξ i and ξ j (see Figure 11).If s ι(a) +s ι(a+1) + • • • +s ι(b) = 0, then there exists ε > 0 such that: for every x 0 ∈ (0, ε].
Proof.The statement is equivalent to the fact that the sign of the initial coefficient of . In order to prove this property, we use the fact that for every r ∈ {1, . . ., n−1}, the valuation σ r of S r only depends on the vertex P := ξ r ∧ ξ r+1 (see Proposition 5.1) and that it is a strictly increasing function on the real contact tree (see Lemma 5.3).Therefore: One concludes using the non-vanishing hypothesis s ι(a) + s ι(a+1

The injectivity condition and the combinatorial types of morsifications
We start this section by defining the injectivity condition on real contact trees of right-reduced series, which is a crucial hypothesis for our main Theorems A and B. Then we define a second planar structure on those contact trees under the hypothesis that the injectivity condition is satisfied: the integrated planar structure.Finally, we state and prove Theorem A, which describes the combinatorial types of morsifications whenever the injectivity condition is satisfied.

The injectivity condition
We explained in Subsection 1.3 how our valuation-theoretical approach leads naturally to the injectivity condition.In the present subsection we formulate it in a way which explains its name.The equivalence of this formulation and that of Subsection 1.3 results from Lemma 5.4.
We keep the notations ι(k) introduced in Subsection 5.3.The injectivity condition on f will involve all the sums of initial coefficients s ι(k) taken on consecutive basic intervals of E + T R (f ) (P ): we impose that all these sums are non-zero.This may be also expressed as the condition that the following discrete integration map at P is injective: (8) Definition 6.1.The injectivity condition on the right-reduced series f ∈ R{x, y} is: For each internal vertex P , the discrete integration map ∫ P is injective.
The injectivity condition is equivalent to: For each internal vertex P, each partial sum s ι(a) + s ι(a+1) + • • • + s ι(b) of consecutive terms is non-zero.
Example 6.2.The injectivity condition is automatically satisfied when the rooted tree T R (f ) is binary, that is, when all its internal vertices have valency 3 (as in [14]).
Example 6.3.The injectivity condition is never satisfied for series f which are odd in the variable y (that is, such that f (x, −y) = −f (x, y)) and verify ord y (f (0, y)) 3. Indeed, in this case F (x, −y) = F (x, y), which implies that F x0 : [−h, h] → R is even and has at least three critical points for every Morse rectangle [0, ε] × [−h, h] of F x (y) and every x 0 ∈ (0, ε].Therefore, F x0 is not Morse.For instance, the injectivity condition is not satisfied if f (x, y) = y(y 2 − x 3 ).

Integrated contact trees
Recall from Proposition 4.7 that the real total order < R on the set R R (f ) of leaves of T R (f ) is planar relative to the abstract tree T R (f ).We will define now a second total order < on the set R R (f ), whenever f satisfies the injectivity condition.By contrast with the real total order, the total order < will not be defined directly on the set of leaves, but it will be associated to a planar structure in the sense of Definition 4.1.
Let ξ i , ξ j (with i < j) be two real roots of f .Let P := ξ i ∧ ξ j and s ι(a) , s ι(a+1) , . . ., s ι(b) denote the initial coefficients associated with the outgoing edges at P , in between the edges going to the leaves ξ i and ξ j .Then one may define a binary relation < ,P on the set of outgoing edges at P by: In terms of the discrete integration map of formula (8), this equivalence may be reformulated as follows: if e a is the outgoing edge going from P to ξ i and e b+1 is the outgoing edge going from P to ξ j (see Figure 11).
The fact that this binary relation is a strict total order results from the injectivity condition (Inj).The set of these total orders, when P varies among the internal vertices of T R (f ), defines a planar structure.Therefore, there is an induced total order < on the set R R (f ) of leaves of T R (f ).Definition 6.5.Assume that f satisfies the injectivity condition (Inj).The collection of all total orders < ,P , when P varies among the internal vertices of T R (f ), is the integrated planar structure on T R (f ).We say that the abstract rooted tree T R (f ) endowed with this planar structure is the integrated contact tree T (f ) of f .The associated total order < on the set R R (f ) of leaves of T R (f ) is the integrated order.
The attribute "integrated" in the previous definition is motivated by the fact that the integrated orders are defined using the discrete integration maps ∫ P of formula (8).The chosen primitive is F x (y) = y 3 − 3xy.Thus we obtain Fix x 0 > 0 and denote by p 1 = (ξ 1 (x 0 ), F x0 (ξ 1 (x 0 ))) and p 2 = (ξ 2 (x 0 ), F x0 (ξ 2 (x 0 ))) the elements of the critical graph of F x0 (see the right of Figure 12).We have p 1 < s p 2 .As S 1 (x 0 ) < 0, then F x0 (ξ 2 (x 0 )) < F x0 (ξ 1 (x 0 )), therefore p 2 < t p 1 .On the other hand ξ 1 < R ξ 2 and since s 1 < 0, we have ξ 2 < ξ 1 .Conclusion: the bi-orders on the critical graph and on the set of real roots of f are isomorphic.Remark 6.7.As a consequence of Theorem A, the combinatorial type of the primitives F x (y) of a right-reduced series f (x, y) which satisfies the injectivity condition is constrained by the structure of the real contact tree T R (f ) of f .For instance, if T R (f ) is isomorphic to the planar tree with three leaves ℓ 1 , ℓ 2 , ℓ 3 from Example 4.4, then the combinatorial type of F x (y) cannot be (ℓ Remark 6.8.Even if the right-reduced series does not satisfy the injectivity condition, Lemma 5.4 allows to get constraints on the combinatorial types of its primitives F x (y), when these primitives are morsifications.More precisely, for each pair of real roots ξ i < R ξ j , the lemma gives the order relation of the critical values F x0 (ξ j (x 0 )) and F x0 (ξ i (x 0 )) for x 0 small enough, whenever the sum s ι(a) + s ι(a+1) + • • • + s ι(b) is non-zero.For instance, this sum is non-zero if in the planar tree T R (f ) there is no other outgoing edge at ξ i ∧ ξ j in between the edges going to the leaves ξ i and ξ j .Indeed, then the sum above contains only one term, which is by definition non-zero.

The contact tree of the apparent contour in the target
Assume again that the series f R{x, y} is right-reduced, satisfies the injectivity condition (Inj), and that F x (y) ∈ R{x, y} denotes a primitive of f .In this section, we identify the real contact tree of the apparent contour in the target of the morphism (x, y) → (x, F x (y)) with the integrated contact tree of f from Definition 6.5 (see Theorem B).

Real polar and discriminant curves
By Theorem A, F x (y) is a morsification.Let [0, ε] × I be a Morse rectangle for it.Denote as before by ξ 1 , . . ., ξ n the real Newton-Puiseux roots of f .We will consider in full generality three geometric objects which appeared already in Figure 1 of the introduction: -The graph of the function (x, y) → F x (y), that is, the surface: -The projection π : G → R 2 x,z given by π(x, y, z) = (x, z).The critical image ∆ ⊂ R 2 x,z of π is called the discriminant curve or the apparent contour in the target of π.The real Newton-Puiseux roots of ∆ in the coordinate system (x, z) are denoted by δ 1 , . . ., δ n ∈ R{x x,y of π is the projection of the critical locus of π to the horizontal real plane R 2 x,y .Then Γ is defined by f (x, y) = 0 and its real Newton-Puiseux roots are exactly ξ 1 , . . ., ξ n .By construction we have, for all i = 1, . . ., n: δ i = F x ξ i .
Remark 7.1.Note that both Γ and ∆ are semi-analytic germs.As a consequence of Theorem B below, their real contact trees are isomorphic as abstract rooted trees whenever the injectivity condition is satisfied.
In the real plane R 2 x,y , the right semi-branches Γ ξ1 , . . ., Γ ξn are ordered by the real total order < R of Definition 2.1.Our goal is to determine the total order of their projections Γ δ1 , . . ., Γ δn in the plane R 2 x,z .This order is encoded in the contact tree T R (δ 1 , . . ., δ n ) of the real Newton-Puiseux roots of the apparent contour in the target ∆.Theorem B below describes the isomorphism type of this planar tree.

The second main theorem
Our second main theorem shows that the planar tree T (f ) from Definition 6.5 is isomorphic to a real contact tree: Theorem B. Let f ∈ R{x, y} be a right-reduced series satisfying the injectivity condition (Inj).The integrated contact tree T (f ) is isomorphic to the real contact tree T R (δ 1 , . . ., δ n ) of the real roots δ i = F x (ξ i ) of the apparent contour in the target of the projection π.
Proof.Since f is right-reduced, the Newton-Puiseux series ξ i are pairwise distinct.The injectivity condition implies that the Newton-Puiseux series δ i are also pairwise distinct.
We will prove that there exists a unique homeomorphism from T (f ) to T R (δ 1 , . . ., δ n ) which respects the labels in {1, . . ., n} of their leaves and sends the integrated exponent function of T (f ) to the exponent function of T R (δ 1 , . . ., δ n ).Its uniqueness comes from the fact that the constraint of respecting the labels obliges to identify the segments [O, ξ i ] and [O, δ i ] for every i ∈ {1, . . ., n}, and that there is only one such identification which transforms the integrated exponent function on [O, ξ i ] into the exponent function on [O, δ i ].It is therefore enough to prove the existence of such a homeomorphism.
To prove that this homeomorphism identifies the planar structures of T (f ) and T R (δ 1 , . . ., δ n ), we need to prove that the leaves are ordered in the same way: δ i < T R (δ1,...,δn) δ j ⇐⇒ ξ i < ξ j .

An example with three cusps
Let us consider the following f ∈ R{x, y}: where c > 1 is a parameter.Its Newton-Puiseux roots are (here ρ = e 2iπ/3 ): There are therefore 5 real right semi-branches Γ ξi and 2 non-real ones (which we may define similarly to the real ones, as the germs Γ η and Γ η of the graphs of x → η and x → η, where x 0), as represented in Figure 14.We will show that f satisfies the injectivity condition (Inj) and we will apply Theorem A to compute the combinatorial type of the associated morsifications F x0 (y) (where x 0 > 0 is small enough).We will see that the result depends on the parameter c.The real contact tree T R (f ) is depicted with solid edges in Figure 15.The remaining edges of the complex contact tree T C (f ) are dotted.We have the following real total order on R R (f ): We compute by termwise integration a primitive of f in the sense of Equation ( 2 This formula enables to compute the area series S i defined in Equation ( 4), giving the following expressions:  1).We see that the initial coefficients s i are polynomials in the variable c.Now we may compute the series δ i = F x (ξ i ).We get: x 16/3 + hot.By our numbering, we have for small x 0 > 0: ξ 1 (x 0 ) < ξ 2 (x 0 ) < • • • < ξ 5 (x 0 ).What is the order of the critical values δ i (x 0 )?Since σ 4 is strictly smaller than the other valuations, we have δ 5 (x 0 ) < δ i (x 0 ), for i = 1, . . ., 4. Also, the relation s 1 (c) = s 3 (c) imposes constraints on the order of the critical values.However, depending on c > 1, several outcomes are still possible (see Figure 16).Let us denote, for c > 1: -Case 1: 0 < λ(c) < 1 2 (take for instance c = 2).In this situation s 1 + s 2 < 0, s 2 + s 3 < 0 and s 1 + s 2 + s 3 < 0 so that for the critical values we obtain δ 3 (x 0 ) < δ 4 (x 0 ) < δ 1 (x 0 ) < δ 2 (x 0 ) (δ 5 (x 0 ) being smaller than all).In other words: ξ 5 < ξ 3 < ξ 4 < ξ 1 < ξ 2 .The corresponding snake, i.e. the permutation associated to the bi-ordered critical set, is represented on the left of Figure 16.

Figure 1 .
Figure 1.The graph of a morsification (x, y) → F x (y) of y → y 3 , its source and target projections and sections of the graph by the planes defined by x = 0 and x = x 0 .

Example 3 . 4 .Figure 2 .
Figure 2. A Morse function and the two total orders on its critical set.

Figure 3 .
Figure 3.A Morse rectangle of f .

1 N
} are pairwise distinct when ξ i varies among the real Newton-Puiseux roots of f .Then a Morse rectangle [0, ε] × I of F x (y) exists.Moreover, the bi-ordered critical graphs (Crit(F x0 ), < s , < t ) of the Morse functions F x0 : I → R are isomorphic for all x 0 ∈ (0, a].

Figure 4 .
Figure 4.A set of outgoing edges and the greatest lower bound of two vertices.

Figure 5 .
Figure 5. Canonical total order on the leaves of a planar tree.

Figure 6 .
Figure 6.The abstract rooted tree from Example 4.4.

Figure 7 .
Figure 7.A basic interval of L(T ) and its image in E + T (P ) by the wedge map .

Figure 8 .
Figure 8. Construction of a contact tree.

Figure 10 .
Figure 10.The outgoing edges e l at the internal vertex P and a basic interval of their totally ordered set.
which identifies the restrictions to those segments of the integrated exponent function σ on T R (ξ 1 , . . ., ξ n ) and of the exponent function E on T R (δ 1 , . . ., δ n ).This is a consequence of Lemma 5.3 and of the fact that, by Definition 5.2, the restrictions [O, ξ i ] → [0, ∞] of the function σ are increasing homeomorphisms.-Letus consider distinct series ξ i and ξ j , which are represented by two leaves of the tree T R (ξ 1 , . . ., ξ n ).

Figure 15 .
Figure15.The real contact tree T R (f ) of the series f is depicted with solid edges; the remaining edges of T C (f ) are dotted.