Asymptotics of multivariate sequences in the presence of a lacuna

We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions d at least 4. This result depends on computations in the homology of the algebraic variety where the generating function has a pole. These computations are similar to, and inspired by, a thread of research in applications of complex algebraic geometry to hyperbolic PDEs, going back to Leray, Petrowski, Atiyah, Bott and Garding. As a consequence, we give a topological explanation for certain asymptotic phenomenon appearing in the combinatorics and number theory literature. Furthermore, we show how to combine topological methods with symbolic algebraic computation to determine explicitly the dominant asymptotics for such multivariate generating functions, giving a significant new tool to attack the so-called connection problem for asymptotics of P-recursive sequences. This in turn enables the rigorous determination of integer coefficients in the Morse-Smale complex, which are difficult to determine using direct geometric methods.


Introduction
Let k ≥ 1 be an integer, and for P and Q coprime polynomials over the complex numbers let be a rational Laurent series converging in some open domain D ⊂ C d . The field of Analytic Combinatorics in Several Variables (ACSV) describes the asymptotic determination of the coefficients a r via complex analytic methods. Let V = V Q denote the algebraic set {z : Q(z) = 0} containing the singularities of F (z). The methods of ACSV, summarized below, vary in complexity depending on the nature of V. When V is a smooth manifold, for instance when Q and ∇Q do not vanish simultaneously, explicit formulae may be obtained that are universal outside of cases when the curvature of V vanishes [PW02,PW13]. When V is the union of transversely intersecting smooth surfaces, similar residue formulae hold [PW04,PW13,BMP22]. The next most difficult case is when V has an isolated singularity whose tangent cone is quadratic, locally of the form x 2 1 − d j=2 x 2 j . These points, satisfying the cone point hypotheses [BP11, Hypotheses 3.1], are called cone points; the necessary complex analysis in the vicinity of a cone point singularity, based on the work of [ABG70], is carried out in [BP11].
Let |r| = |r 1 | + · · · + |r d |. In each of these cases, asymptotics may be found of the form a r ∼ C(r)|r| β z * (r) −r (1.2) where C and z * depend continuously on the directionr := r/|r|. A very brief summary of the methodology is as follows. The multivariate Cauchy integral formula gives where T ⊆ D is a torus in the domain of convergence and dz/z is the logarithmic holomorphic volume form z −1 1 · · · z −1 d dz 1 ∧ · · · ∧ dz d . Expand the chain of integration T so that it passes through the variety V, touching it for the first time at a point z * where the logarithmic gradient of Q is normal to V, and continuing to at least a multiple (1 + ε) times this polyradius. Let I be the intersection with V swept out by the homotopy of the expanding torus. The residue theorem, described in Definition 3.4 below, says that the integral (1.3) is equal to the integral over the expanded torus plus the integral of a certain residue form over I. Typically, z −r is maximized over I at z * , and integrating over I yields asymptotics of the form (1.2).
In the case of an isolated singularity with quadratic tangent cone, Theorem 3.7 of [BP11] gives such a formula but excludes the case where d = 2m > 2k + 1 is an even integer and d − 1 is greater than twice the power k in the denominator of (1.1). In that paper the asymptotic estimate obtained is only a r = o(|r| −m z −r * ) for all m, due to the vanishing of a certain Fourier transform. This leaves open the question of what the correct asymptotics are, and whether they are smaller by a factor exponential in |r|.
In [BMPS18] it is shown via diagonal extraction that, for k = 1 and a class of polynomials Q with an isolated quadratic cone singularity, in fact a n,...,n has strictly smaller exponential order than expected. Diagonal extraction applies only to coefficients of monomials precisely on the diagonal -i.e., where every variable has the same power -leaving open the question of behavior in a neighborhood of the diagonal 2 , and leaving open 2 To see why this is important, consider the function (x − y)/(1 + x + y) that generates differences of binomial coefficients the question of whether this behavior holds beyond the particular class, for all polynomials with cone point singularities. The purpose of the present paper is to use ACSV methods to show that indeed the behavior is universal for cone points, to prove that it holds in a neighborhood of the diagonal, and to give a topological explanation.
2 Main results and outline
(2.1) Let C * := C \ {0} and let M := C d * \ V be the domain of holomorphy of z −r F (z) for sufficiently large r. Let amoeba denote the amoeba of Q, defined by amoeba := {L(z) : z ∈ V}. It is known [GKZ94] that the components of the complement of the amoeba are convex and correspond to Laurent series expansions for F , each component being a logarithmic domain of convergence for one series expansion. Let B denote the component of amoeba c such that the given series r a r z r converges whenever z = exp(x + iy) with x ∈ B.
We refer to the torus T (x) := L −1 (x) as the torus over x. For any r ∈ R d we denoter := r/|r| and For a subset A ⊂ C d , whenr and z * are understood, we use the shorthand A(−ε) := A ∩ {z : hr(z) < hr(z * ) − ε} . (2.2) Assume that V intersects the torus {exp(x * + iy) : y ∈ (R/(2π)) d } at the unique point z * = exp(x * ). We will be dealing with the situation where V has a quadratic singularity at z * . More specifically, we will assume that Q has a real hyperbolic singularity at z * .
Definition 2.1 (quadric singularity). We say that Q has a real hyperbolic quadratic singularity at z * if Q(z * ) = 0, the gradient ∇Q(z * ) = 0, and the quadratic part q 2 of Q = q 2 (z) + q 3 (z) + . . . at z * is a real quadratic form of signature (1, d − 1); in other words, there exists a real linear coordinate change so that q 2 (u) = u 2 d − d−1 j=1 u 2 j + O(|u| 3 ) for u a local coordinate centered at z * .
We denote by T x * (B) the open tangent cone in R d to the component B of amoeba(Q) c , namely all vectors v at x * := L(z * ) such that x * + εv ∈ B for sufficiently small ε. The inequality defining T x * (B) is the same as the inequality Q(v) > 0 where Q is the leading (homogeneous quadratic) term of Q(exp(x * + v + iy * )), along with an inequality specifying T x * (B) rather than −T x * (B).
Definition 2.2 (tangent cone; supporting vector). The vector r is said to be supporting at z * if h r attains its maximum on the closure of B at x * and if {dh r = 0} intersects the tangent cone T x * (B) only at the origin. The open convex cone of supporting vectors is denoted N and the set of unit vectors over which it is a cone is denoted N .
Theorem 2.3 (main theorem). Let P be holomorphic in C d , Q a Laurent polynomial, k a nonnegative integer, B a component in the complement of the amoeba of Q, and r∈E a r z r the corresponding Laurent series expansion of F = P/Q k .
Suppose that Q has real hyperbolic quadratic singularity at z * = exp(x * ) such that x * belongs to the boundary of B and z * is the unique intersection of the torus T(x * ) with V.
Let K ⊆ N be a compact set and suppose that d is even and 2k < d.
(i) If ε > 0 is small enough then for anyr ∈ K there exists a compact cycle Γ(r), of volume uniformly bounded inr, such that for allr ∈ K the cycle Γ(r) is supported on M(−ε) and for all but finitely many r ∈ E, where γ(r) is a compact (d − 1) cycle in V(−ε) of volume uniformly bounded inr and Res V is the residue operator defined below in Section 3.
The heuristic meaning of this result is that, for purposes of computing the Cauchy integral, the chain of integration in (1.3) can be slipped below the height h r (z * ) of the singular point z * .

Motivation
Our motivating example for this theorem comes from the Gillis-Reznick-Zeilberger family of generating functions [GRZ83], which has origins in the work of Szegö [Sze33] and Askey and Gasper [AG77] and is further discussed in [BMPS18, Theorems 9 -12].
Example 2.4 (GRZ function at criticality). In four variables, let F λ (z) := 1/(1−z 1 −z 2 −z 3 −z 4 +λz 1 z 2 z 3 z 4 ) and, for convenience, write F (z) = F 27 (z). It is shown in [BMPS18] via ACSV results for smooth functions that the diagonal exponential growth rate |a n,n,n,n | 1/n of the power series coefficients of F λ is a function of λ that approaches 81 as λ → 27. At the critical value 27, however, the denominator Q of F has a real hyperbolic quadric singularity at z * := (1/3, 1/3, 1/3, 1/3). Theorem 2.3 has the immediate consequence that the exponential growth of a r forr in a neighborhood of the diagonal is strictly less than that of z −r * = 81 |r| . Thus there is a drop in the exponential rate at criticality.
With a little further work, understanding of the drop can be sharpened considerably. In Section 8 we state a result for general functions satisfying the conditions of Theorem 2.3. The result, Theorem 8.1, sharpens Theorem 2.3, quantifying the exponential drop by pushing the contour Γ down all the way to the next critical value. It is a direct consequence of Theorem 2.3 together with a deformation result of [BMP22]. In the case of the GKZ function at criticality, the following explicit asymptotics result.
Theorem 2.5. The diagonal coefficients a n,n,n,n in the power series expansion of F (z) from Example 2.4 have an asymptotic expansion in decreasing powers of n, beginning a n,n,n,n = 3 · 4i (2.5) +O 9 n n −5/2 .
More generally, as r → ∞ andr varies over some compact neighborhood of the diagonal, there is a uniform estimate a r = p n r n −3/2 · Cr cos(nαr + βr) + O(p n r n −5/3 ) , where pr, Cr, αr, and βr vary continuously withr and specialize to produce (2.5) whenr is on the diagonal.

Heuristic argument
Our plan is to expand a torus T of integration representing series coefficients via the Cauchy integral theorem using a homotopy H that takes it through the point z * and beyond. Let V * = V ∩ C d * denote the points of V with no coordinate vanishing. A classical construction, due to Leray, Thom and others, shows that T is homologous in H d (M) to a cycle Γ which coincides above height h(z * ) − ε with a tube around a cycle σ; the height h r is maximized on σ at the point z * and the chain σ is the intersection of H with V * . We would like to see that σ is homologous to a class supported on V(−ε).
To do this, we compute the intersection σ directly in coordinates suggested by the hypotheses of the theorem. In particular, we use local coordinates where, after taking logarithms, V is the cone {z 2 1 − d j=2 z 2 j = 0}, and select a homotopy H from x + iR d to x + iR d with x ∈ B so that the line segment xx is perpendicularly bisected by the support hyperplane to B at x * . In these coordinates, the intersection class I is the cone {iy : y ∈ R d and y 2 1 = d j=2 y 2 j }. The residue is singular at the origin (in new coordinates) but converges when d > 2k + 1. Inside the variety V, the cone I may be folded down so as to double cover the cone {x + iy : y ∈ R d , x > 0, y 1 = 0 and x 2 = |y| 2 }, as shown in Figure 1. The two covering maps have opposite orientations when d is even. The critical points of h r restricted to V are obstructions for deforming the contour of integration downwards, and in this case the residue integral vanishes and the contour may be further deformed until it encounters the next highest critical point.

Outline of actual proof
The proof cannot precisely follow the heuristic argument because the intersection cycle construction and the residue integral theorem work only when V * is smooth. We remark that the same trouble arose in the setting of [BP11]. There, the authors adopt the method of [ABG70] to reduce the local integration cycle to its projectivized, compact counterpart: the so-called Petrowski or Leray cycles. That path required significant investment into analytic auxiliary results and, more importantly, would not immediately prove that the integration cycle in the presence of lacuna (i.e., when d is even and the denominator degree not too high) allows one to "slide" the integration cycle below the height of the cone point.
Thus we use a different strategy, first perturbing the denominator so that the perturbed varieties on which Q(z) = c for small c become smooth. This kind of regularization also has the advantage, compared to what was used in [BP11], that we obtain information about the behavior of coefficients of the generating functions P/(Q − c) k . Next, we study the behavior of the coefficients of the resulting generating functions as c → 0. We denote the zero set of Q(z) − c by V c , write (V c ) * for the points of V c with non-zero coordinates, and denote the restriction of V c to its points of height at most h(z * ) − ε by V c (≤ −ε). It is easiest to work in the lower dimensional setting, with σ c on (V c ) * rather than Γ c on M c , and to work in relative homology of (V c ) * with respect to V c (≤ −ε).
Section 4 lays the theoretical groundwork by computing the explicit intersection cycle in a limiting case of the perturbed variety as c ↓ 0; this is a rescaled limit, and is smooth, in contrast to the variety at c = 0. Although the results of Section 4 are subsumed by later arguments, its focus on explicit computation allows for valuable intuition and visualization. Properties of our family of perturbations are given in Section 5. Section 6 uses this approach to complete a relative homology computation in M c for sufficiently small c. It turns out that σ c is not null-homologous in H d−1 ((V c ) * , V c (≤ −ε)) but is instead homologous to an absolute cycle S c which is homeomorphic to a (d − 2)-sphere and lies in a small neighborhood of z * . Under the hypotheses of Theorem 2.3, the integral over S c is easily seen to converge to zero as c ↓ 0. The remaining outline of the proof is as follows.
). This is accomplished in Section 6.
2. Pass to a tubular neighborhood to see that T in (1.3) may be replaced by the sum of tubular neighborhoods of S c and a second chain γ, not depending on c, whose maximum height is at most h(z * ) − ε. This is accomplished in Section 7.
3. Dimension analysis shows that the integral over the tubular neighborhood of S c goes to zero as c ↓ 0. This is accomplished in Section 7, proving Theorem 2.3.
of Q (the product of its distinct irreducible factors) does not vanish. The well known Tubular Neighborhood Theorem (for example, [MS74,Theorem 11.1]) states the following.
Proposition 3.1 (Tubular Neighborhood Theorem). The normal bundle over K is trivial and there is a global product structure of a tubular neighborhood of This implies the existence of operators • and o, respectively the product with a small disk and with its boundary, mapping k-chains in V * respectively to (k + 2)-chains in M and (k + 1)-chains in M, well defined up to a natural homeomorphism as long as the radius of the disk is sufficiently small. We refer to oγ as the tube around γ and •γ as the tubular neighborhood of γ. Elementary rules for boundaries of products imply The class I can be represented by the manifold H ∩ V for any manifold H with boundary T − T in C d * that intersects V transversely, with appropriate orientation (or, alternatively, by the image of the fundamental class of H ∩ V under the natural embedding).
We remark that if V is not smooth but its singularities (where Q and the gradient of its square-free part vanish) have real dimension less than d − 2 then H generically avoids the singularities of V, so I(T, T ) is well defined. Although the singular set does not always satisfy this dimensional condition, it does so in our applications, where the singular set is zero dimensional.
For our purposes, the natural cycles to consider are the tori T(x) for x in the complement of the amoeba of Q. In this case, there is an especially convenient choice of cobordism between T(x) and T(x ), namely the L-preimage of the straight segment connecting x and x (or its small perturbation). We will be referring to this cobordism as the standard one.
What are the good choices of x ? We would like to make the integrand F (z)z −r dz/z exponentially small in |r| when L(z) = x , which happens if we can take −r · x to have arbitrarily small modulus. When Q is a Laurent polynomial the feasibility of this follows from known facts about cones of hyperbolicity, as we now demonstrate.
First, recall that the Newton polytope of Q is the convex hull of the exponents m of the monomials of Q, The following result is well known; see, e.g. [BMP22, Proposition 2.14].
Definition 3.4 (residue form). There is a homomorphism Res : In general, Res (ω) can be derived locally from a form representing ω (we also use the notation Res for the corresponding operator on differential forms). When F = P/Q is rational with Q squarefree, Res commutes with multiplication by any locally holomorphic function and satisfies More generally, if F = P/Q k , then (see, e.g. [Pha11]) the residue can be expressed in coordinates as where σ is the natural area form on V (characterized by dQ ∧ σ = dz), and the partial derivatives with respect to c are taken in the coordinates where c is one of the variables.
Putting this together with the definition and construction of the intersection class and Cauchy's integral formula yields the following representation of the coefficients a r .
Moreover, if x is a descending component B with respect to B, and G is a polynomial, then for all but finitely many r ∈ E, Proof: The first identity is Cauchy's integral formula, the definition of the intersection class, and (3.3). The second identity follows from the fact that sup T(x ) |G/Q k | and the volume of T(x ) grow at most polynomially in |x | on the torus over x . For r ∈ N (Q) m large enough in size, the degree of the decay of |z −r | overtakes that polynomial growth, so that the last term in (3.5) can be made arbitrarily small. As it is independent of x as long as x varies in the same component B , it vanishes identically. 2

The limiting quadric
In this section we focus on the properties of the particular smooth quadratic function Note that unlike the local behaviour of a function Q near a real hyperbolic quadratic singularity, the quadric q has constant term −1. The zero setṼ of q can be viewed as the solution set V c to the equation Q(z) = c near the quadratic singularity of Q, after the variables are scaled by c 1/2 . Our first statement deals with the gradient-like flow onṼ with respect to the function h := x 0 .
Lemma 4.1. The function h has two critical points z ± = (±1, 0, . . . , 0) onṼ, both of index d − 1. The stable manifold for z + is the unit sphere and its unstable manifold is the upper lobe of the 2 sheeted real hyperboloid The stable manifold for z − is the lower lobe H − of this hyperboloid, while the unstable manifold of z − is still the sphere S.
Proof: The critical points can be found by a direct computation. Their indices are necessarily d−1, as h is the real part of a holomorphic function on a complex manifold [GH78]. Similarly, direct computation shows that the tangent spaces to S, H ± are the stable/unstable eigenspaces for the Hessian matrices of h restricted toṼ at the critical points. Lastly, as the gradient vector field is invariant with respect to symmetries y → −y , leaving H ± and S invariant, they are the invariant manifolds for the gradient flow. Proof: Writing z j := x j + iy j , the equations for z such that z is in the range of Φ and q(z) = 0 become The solutions to (4.2) -(4.5) form the union of two sets, one obtained by solving (4.2) -(4.4) when x 1 = 0 and the other by solving (4.2) -(4.4) when y 1 = 0; these intersect along the solutions to (4.2) -(4.4) when x 1 = y 1 = 0. The first of these is the one-sheeted hyperboloid H ⊆ iR d given by The second is the sphere S ⊆ R × i(R d−1 ) given by These intersect at the equator of the sphere S, which is the neck of the hyperboloid H. The intersection set The intersection class is given by the intersection ofṼ with any homotopy intersecting it transversely. While Φ does not intersectṼ transversely, it is the limit of the intersections ofṼ with arbitrarily small perturbations of Φ that do intersectṼ transversely. Let γ n be a sequence of such transverse intersection cycles converging to γ := H ∪ S. BecauseṼ is smooth, the global product structure on a neighborhood ofṼ from the Thom lemma implies that as d-chains, and hence that γ represents the intersection class.
Finally, we determine the orientation via a different perturbation argument. Choose a point p ∈ S , say for specificity p = (0, . . . , 0, i). The tangent space T p (S ) is the span of the vectors ie k for 2 ≤ k ≤ d − 1. The tangent space T p (S) is obtained by adding the basis vector e 1 , while the span of the tangent space T p (H) is obtained by adding instead the basis vector ie 1 . We see that near S , γ has a product structure S × W, where W is diffeomorphic to two crossing lines, with tangent cone xy = 0 in the plane e 1 , ie 1 , as in the black lines in Figure 2.
Now perturb the homotopy as follows. Let u : [−1, 1] → R be a smooth function that is equal to 1 on [−1/4, 1/4] and vanishes outside of (−1/2, 1/2). Define Φ ε (y, t) := te 1 + εu(t)e d + iy where ε is a real number whose magnitude will be chosen sufficiently small and whose sign could be either positive or negative. Because S and H intersect only on the subset of Φ where t = 0, their Hausdorff distance on the set t / ∈ (−1/4, 1/4) is positive; it follows that for sufficiently small |ε|, the intersection of Φ ε withṼ is in the subset of Φ ε where −1/4 ≤ t ≤ 1/4. There u = 1 and the equations for the intersection γ ε are modified from (4.2) -(4.5) as follows: Although Φ ε andṼ still do not intersect transversely, the intersection set γ ε := Φ ε ∩Ṽ is now a manifold. We now fix y 2 , . . . , y d−1 at a value y inside the unit ball, setting x 2 = ε and solving (4.4) and (4.5) for y 1 and y d as a function of x 1 . For x 2 1 < 1 − |y| 2 , as ε ↓ 0, there are two components of the solution, with y d → ± 1 − x 2 1 − |y| 2 respectively. These correspond to different points on the sphere. Fixing one, say with y d > 0, locally γ ε has a product structure S × W ε , where W ε is a hyperbola in quadrants II and IV; see the blue curve in Figure 2. The (oriented) chains W ε converge to W as ε ↓ 0, therefore the possible orientations for W are one of the four shown on the right of Figure 2. The (oriented) chains W ε also converge to W as ε ↑ 0, narrowing the choices to the second and third choices in Figure 2, and proving the desired result. 2 Theorem 4.3. Let n be the chain given by S with orientation reversed in the southern hemisphere; in other words, n is a sphere, oriented the same as the northern hemisphere of S. When d is even, the chain γ is homotopic to n in H d−1 (Ṽ).
Proof: Let X 1 := R × S and ι 1 : S → X 1 be the embedding y → (0, y). Let X 2 = [−π/2, π/2] × S and ι 2 : S → X 2 be the embedding y → (0, y). Let X denote the space obtained by gluing X 1 to X 2 modulo the identification of ι 1 and ι 2 (which conveniently identifies identically named points (0, y) in X 1 and X 2 ). If for j ∈ {1, 2} there are homotopies T j : X j × [0, 1] →Ṽ making the maps in Figure 4 commute, then their union modulo the identification is a homotopy T : Figure 3: commuting homotopies define a homotopy on the identification space X To prove the lemma, it suffices to construct these in such a way that T 2 is a homotopy from S to n and T 1 is a homotopy from H to a null homologous chain. On X 1 , let ρ denote the R coordinate on X 1 and σ denote the S coordinate. Let z = (z 2 , . . . , z d ) and let x and y denote respectively the real and imaginary parts of z. Let t denote the [0, 1]-coordinate of X 1 × [0, 1]. We may then define the homotopy T 1 via the equations and check that T 1 ((ρ, σ), 0) parametrizes H via Next we define the map τ : This is a linear homotopy from the identity to the map ρ → min(2ρ, 0) − π/2, pictured in Figure 4. Define T 2 by the equations x 0 = sin(τ (ρ, t)), y = cos(τ (ρ, t))σ . (4.9) Again, we verify that T 2 ((ρ, σ), 0) parametrizes the chain S via the parametrization x 0 = sin(ρ) and y = cos(ρ)σ. The parametrization is not one to one, mapping the set {−π/2}×S to the south pole and {π/2}×S to the north pole, however it defines a singular chain homotopy equivalent to a standard parametrization of S .
Thirdly, we check that the diagram in Figure 4 commutes, mapping (y, t) in both cases to the point (sin(tπ/2), i cos(tπ/2)y 2 , . . . , i cos(tπ/2)y d ). Fourthly, we check that T 2 is a homotopy from S to n. This is clear because the homotopy T 2 leaves the (generalized) longitude component alone while pushing all the southern latitudes to the south pole and stretching the northern latitudes to cover all the latitudes.
Finally, we check that T 1 is a homotopy from H to a null-homologous chain. The map T 1 (·, 1) maps the imaginary hyperboloid H parametrized by (ρ, σ) into the {x 1 > 0} branch of the real two sheeted hyperboloid H defined by x 2 1 = 1 + d j=2 x 2 j and parametrized by cylindrical coordinates (r, σ ). The parametrization is a double covering, with (ρ, σ) and (−ρ, σ) getting mapped to the same point. We need to check that the − π π π π /2 /2 /2 /2 − Figure 4: the linear homotopy τ orientations at (ρ, σ) and (−ρ, −σ) are opposite. We may parametrize H by its projection x onto the last d − 1 coordinates, then, still preserving orientation, by polar coordinates (r, σ ) where r > 0 is the magnitude and σ (x) = x/r when r > 0 (σ can be anything when r = 0). In these coordinates, the point (ρ, σ) ∈ H gets mapped to the point Recalling that the orientation form on H is given by sgn (ρ)dρ ∧ dσ, the Jacobian is therefore given by (4.10) The central symmetry flips the orientation exactly on even-dimensional spheres, so that (4.10) changes signs with the sign of ρ exactly when d − 2 is even. This implies that for d even, the two branches locally covering the sheet {x 2 0 = |x| 2 + 1, x 0 > 0} receive opposite signs and the chain T 1 (·, 1) is homologous to zero. 2 In the next section we prove perturbed versions of these results leading to identification of certain homology and cohomology classes. To pave the way, we record some further facts about the intersection of the explicit homotopy Φ with the quadric. Invariance of H ± follow from the fact that the gradient is a real map (the gradient at real points is real) and therefore the real subspace, of which H ± are connected components, is preserved by gradient flow. Invariance of S follows from the same argument after reparameterizing via (x 1 , . . . , x d ) = (s 1 , i s 2 , . . . , i s d ). (ii) For any index r, the coefficient of the power series expansion for F c = P/Q k c given by (1.3), is holomorphic in the disk |c| < |Q(0)|. In particular, any given coefficient is continuous at c = 0 as a function of c.
Proof: The first statement is follows from the Bertini-Sard theorem (the values of c that make V c singular is a finite algebraic set). The second follows from the fact that each term in the (converging, under our assumptions) expansion of is holomorphic and thus integrable over any torus in the domain of holomorphy of F , and the modulus of each term is bounded. 2 We will need to understand the local behavior of h r on the smooth varieties V c near z * . The following proposition shows that the perturbed varieties have the same geometry as the limiting quadric described in Section 4.
Proposition 5.2 (local behavior). Assume thatr strictly supports the tangent cone T x * (B). Then (i) There is a δ > 0 such that for sufficiently small |c| = 0, there are precisely two critical points of hr on the variety V c in the ball B δ (z * ). These points tend to z * as c → 0.
(ii) If c is positive and real, these critical points z ± c are real, and can be chosen such that Proof: By part (i) of Proposition 5.1, V c is smooth. The function hr is the real part of the logarithm of the locally holomorphic function zr near z * , hence it has a critical point on the smooth complex manifold V c if and only if z r does, i.e., if dz r is collinear with dQ. This latter condition defines the so-called log-polar variety. A local computation implies that under our conditions this is a smooth curve, intersecting V with multiplicity 2 at z * as long as r is not tangent to the tangent cone T L(z * ) (log V).
Indeed, one can find a real affine-linear coordinate change such that in the new coordinates, centered at z * , where our conditions on r imply k a 2 k < 1. In these coordinates, the log-polar variety is given by the equation z k = −a k z 0 + O(|z| 2 ). Thus, the log-polar curve intersects V c transversely for |c| = 0 small, and consists of 2 geometrically distinct points. A similar computation implies the second statement for real c. 2 The main work in proving Theorem 2.3 will be to prove the following result.  (iv) For fixed r as c → 0, Let h * := h(z * ). What we require for our results is a description of the relative homology group ), together with explicit generators. To compute these we start with the homogeneous situation and then perturb. Denote by q the quadratic part of Q at z * . This is a real quadratic form, invariant with respect to conjugation, with signature (1, d − 1) on the real part of the tangent space at z * . We denote the two convex cones where q ≥ 0 as C ± , and extend Definition 2.2 by considering supporting vectors to C + as well as C − . Lemma 6.2. If r is supporting then v intersects S ∩ H transversely.
Proof: By the hypothesis that r is supporting, one can choose h as one local coordinate, changing the rest of the coordinates so that the quadratic form q preserves its Lorentzian form. In these new coordinates it remains to prove that the functions have independent differentials at their common zeros outside of the origin. This can be checked directly.
Proof: For a given ρ > 0, introduce new coordinates in which z * is the origin and the ρ-ball around z * becomes the unit ball in C d , while rescaling Q by ρ −2 and h by ρ −1 . The resulting functions become small perturbations (decreasing with ρ) of the quadratic and linear functions in Lemma 6.2, and their zero sets become small deformations Q ρ and H ρ of the corresponding varieties.
In particular, the determinants whose nonvanishing witnesses the transversality of the varieties of Q ρ , H ρ and S are small deformations of the determinants witnessing the transversality in Lemma 6.2, and therefore are non-vanishing on some open neighborhood U of the set of solutions to H ρ = Q ρ = 0 intersected with the spherical shell where the distance to the origin is between, say, 1 and 1/2 for small enough ρ.
We will need one more result on the local geometry of V and {h = const}.
Lemma 6.4. For ε = 0, the intersection of the real hyperplane x 1 = −ε with the quadric is homotopy equivalent to a (d − 2)-dimensional sphere for |c| small enough.
For nonzero c it can be verified that the manifolds given by (6.1) are transverse, and therefore remain transverse for small c, meaning the intersections are homeomorphic.
Proof: We can choose coordinates in which the quadratic part of Q and hr are given by z 2 1 − z 2 2 − . . . − z 2 d and x 1 , respectively. Then, repeating the argument in Corollary 6.3, we can view a rescaled Q and h as small perturbations of the quadratic and linear functions in Lemma 6.4, and apply transversality.
We will be referring to the intersection for ρ, satisfying the conditions of Corollary 6.3, as the (ρ, )-slab. We call the intersection of the slab with the boundary ∂B ρ its vertical boundary, and the intersection with h = h * − ε its bottom. Use a partition of unity to glue together the vector fields v (z) , ensuring that the partition gives weight 1 to points in a ρ/3 neighborhood of z * and zero weight outside the ρ/2 neighborhood. This ensures conclusions 1, 2 and 3. If the partition is chosen invariant with respect to conjugation, the last conclusion will be true as well.
Proposition 6.7. Assume again that r is supporting at z * , where z * is a quadratic singularity of Q with signature (1, d − 1). Fix ρ and satisfying conditions of Corollary 6.3 and the corresponding (ρ, ε)-slab.
Letting bottom denote V c ∩ slab ∩ {h − h * = −ε}, the relative homology group is free of rank 2 for small enough |c| = 0. For small real c > 0, its generators are given by • an absolute cycle, the image of the generator of H d−1 (V c ∩ B r ) under the natural homomorphism into H − , and • the relative cycle corresponding to the lobe of the real part of V c located in {h ≤ h * }.
Proof: The trajectories of the flow along the vector field v(·) constructed in Corollary 6.6 starting on V c,slab either converge to the critical points of h on V c,slab , or reach bottom. Indeed, the value of h is strictly decreasing outside of the critical points, and cannot leave the slab through its side due to conclusion 3 of Corollary 6.6. All trajectories therefore remain in the slab or reach the bottom.
The homology of the pair (V c ∩ slab, bottom) is generated by classes represented by the unstable manifolds of the Morse function h at critical points on V c ∩ slab; this is the fundamental theorem of stratified Morse theory, for example [GM88,Theorem B]. In our situation, there are exactly two such critical points, z − and z + , both in the real part of V c and both of index d − 1. This proves the statement about the rank of the group.
The long exact sequence of the inclusion of the bottom into V c ∩ slab gives an exact sequence containing the maps Using Corollary 6.5, the first of these groups vanishes because V c ∩ bottom is homotopy equivalent to S d−2 . It follows that the absolute cycle generating is therefore a generator of H − .
For c > 0, the real part of V c located within the lower half of the slab, {h < h * }, contains the critical point z − (by Proposition 5.2), and the vector field v is tangent to it (thanks to the reality property mentioned above). Hence it coincides with the unstable manifold of z − .
Of course, the same argument applies to the Morse function −h on V c , implying that the group also has rank 2 and, for positive real c, is generated by the same absolute cycle together with the analogous relative cycle (the lobe of the real part of V c located in {h ≥ h * }). For small positive c, the situation we will restrict ourselves to from now on, we will denote the generators in H − as S − and H − , where S − is the absolute class represented by the small sphere in V c and H − is the relative class represented by the corresponding component of the real part of Q c . In the same way we define classes S + and H + generating H + .
A general duality result implies that the relative groups H − and H + are dual to each other, with the coupling given by the intersection index. Briefly, the reason is that the vector field in Corollary 6.3 may be used to deform slab until the boundary of the top flows down to the boundary of the bottom; this makes the space into a manifold with boundary satisfying the hypotheses of [Hat02, Theorem 3.43]. The conclusion of that theorem is an isomorphism between a homology group and a cohomology group, which, combined with Poincaré duality, proves the claim. In fact, we won't use this argument because we need to compute this coupling explicitly, as follows.
Proposition 6.8. The intersection pairing between H − and H + is given by Remark. We pedantically distinguish between S + and S − , although they are the image of the same absolute class, or even chain, in V c . Also, we note that our orientations of the spheres and their tangent spaces can be in disagreement with the standard orientations induced by the complex structure. By changing the orientation of the chain S, one can suppress the annoying sign factor in the second and third equalities, but not in the last one.
Proof: We can work (after rescaling) in the setup of Lemma 4.1. The cycles representing H ± c are disjoint, explaining the first line. Each intersect S in precisely one point. Denoting ∂/∂x k as ξ k and ∂/∂y k as η k , the tangent spaces to S = S ± at z ± are spanned by the vectors ±η 2 , . . . , ±η d , and the tangent spaces to H ± at z ± are spanned by ±ξ 2 , . . . , ±ξ d , respectively.
In the standard orientation of the complex hypersurfaceṼ, the frame (ξ 2 , η 2 , . . . , ξ d , η d ) is positive. Hence, the intersection index of H + and S is the parity of the permutation shuffling (ξ 2 , . . . , ξ d , η 2 , . . . , η d ) into that standard order, giving the second line. The third line is obtained similarly, taking the signs into account.
The last pairing can be observed by noting that the self-intersection index of a class represented by a manifold of middle dimension in a complex manifold is equal to the Euler characteristics of the conormal bundle of the manifold, under the identification of the collar neighborhood of the manifold with its conormal bundle. This gives χ(S) = (1 + (−1) d−1 )(−1) d(d−1)/2 , where again the mismatch between the standard orientation of the conormal bundle and the ambient complex variety contributes the factor (−1) d(d−1)/2 . 2 Importance of the local homology computation lies in the following localization result. Let u * := L(z * ) ∈ R d be a point on the boundary of amoeba(Q) (recall L is the logarithmic map z → log |z|).
Theorem 6.9. Assume that the quadratic critical point z * is the only element of T(u * ) ∩ V, that z * lies on the boundary of a component of amoeba(Q) c and that r is supporting. Then for any ρ > 0 there exist ε, c * > 0 such that for all c ∈ (0, c * ) the intersection class I(T) ⊆ V c can be represented by a chain supported on Proof: Choose ρ small enough so that the conclusions of Corollary 6.3 hold. As the intersection of V with the torus T(u * ) containing z * is a single point, standard compactness arguments imply that for sufficiently small positive δ the intersection of V with the L-pre-image of B(u * , δ) is contained in B ρ (z * ). Pick a torus T(x) where x is a point in the intersection of B with the component of the complement to the amoeba defining our power series expansion. Choose ε > 0 such that {h ≤ h * − ε} intersects B ρ (z * ). Let y be a point in the component B defined at the end of Section 3, such that hr(y) < hr(z * ) − ε. Choose any smooth path {α(t) : 0 ≤ t ≤ 1} from x to y that passes through B ρ (z * ) and along which hr decreases. Then the L-preimage of that path is a cobordism between T and a torus T in {h ≤ h(z * ) − ε}. The transversality conclusion of Corollary 6.3 means that this cobordism, or a small perturbation of it, produces a chain realizing the intersection class I(T) and satisfying the desired conclusions. 2 We now come to the main result of this section, which completes Step 1 of the outline at the end of Section 3.
Theorem 6.10. For even d the intersection class ε)), up to sign.
Proof: Let e denote the class of I(T) in the relative homology group H − . Then, by Lemma 6.7, we have e = aH − + bS − for some integers a and b. We claim that The construction of the chain representing the intersection class I(T) in Theorem 6.9 implies that it meets the chain representing H + at precisely one point z c . The point z c is not necessarily the point z + c , but it is characterized by being the unique point where the homotopy intersects the real variety V c,R ⊆ V c .
The intersection class is represented by a chain that is smooth near z c . We need to check that its intersection with the "upper lobe" H + is transverse within V c . Indeed, one can linearly change coordinates centered at z c so that in the new coordinates z the homotopy segment runs along the x 1 axis, and thus the equations defining the cobordism are x 2 = . . . = x d = 0. Then V c is given by z 1 = R(z 2 , . . . , z d ) with dR| z + * = 0. By direct computation, the intersection is transversal and the tangent space to the chain representing the intersection class at z + * is the tangent space to V c at z + * multiplied by i.
Because H + and e intersect transversely at a single point, the first identity in (6.2) is proved. For the second identity, we again rely on perturbations of the cobordism defining the intersection class. If the path defining the cobordism avoids z * , for c small enough, the chain realizing I(T) constructed in Theorem 6.9 will completely avoid the chain representing S, implying that the intersection number of e with S + is zero.
When d is even the Euler characteristic of the (d − 1)-dimensional sphere vanishes together with a. 2 7 Proof of the main theorem and Theorem 5.3 We are now ready to prove Theorem 5.3, and thus obtain our main Theorem 2.3. At each stage it is easiest to prove the result for fixedr and then argue by compactness that the conclusion holds for allr ∈ K. We start with a localization result. Use the notation I c to denote intersection class with respect to the perturbed variety V c .
Lemma 7.1. Fixr ∈ K. Under the hypotheses of Theorem 6.10, there is an ε > 0 such that the intersection class I c (T, T ) is where the cycle γ c (r) representing the class [γ c ] ∈ H d−1 (V c ) is supported in V c (< −ε) with respect to hr.
Proof: By Theorem 6.10, I c − S c is mapped to zero in the second map of the exact sequence Let Σ denote the singular locus of V, that is, the set {z ∈ V : ∇Q(z) = 0}. The point z * is a quadratic singularity, thus isolated, and we may write Σ = {z * } ∪ Σ where Σ is separated from z * by some positive distance.
Corollary 7.2. If the real dimension of Σ is at most d − 2 then, for some δ > 0, the cycles {γ c (r) : 0 < |c| < δ,r ∈ K} may be chosen so as to be simultaneously supported by some compact Ξ disjoint from Σ.
Remark. In the case where Σ is the singleton {z * } or when any additional points z ∈ Σ satisfy h(z) ≤ h(z * )−ε for allr ∈ K, the proof is just one line. This is all our applications presently require, however the greater generality (although most likely not best possible) may be useful in future work.
Proof: The first step is to prove that for fixedr we may choose {γ c (r) : 0 < |c| < δ} satisfying the conclusion of Lemma 7.1, all supported on a fixed compact set Ξ avoiding Σ. It suffices to avoid Σ because the condition of being supported on V −ε immediately implies separation from z * . The construction in Theorem 6.9 produces a single homotopy for all c, which is then intersected with each V c . It follows that the union of the intersection cycles is contained in a compact set. By the dimension assumption, a small generic perturbation avoids Σ while still being separated from z * .
Having seen that for fixedr the cycles {γ c (r) : 0 < |c| < δ} may be chosen to satisfy the conclusions of Lemma 7.1 and to be supported on a compact set Ξ(r) avoiding Σ, the rest is straightforward. For eacĥ r there is a neighborhood N (r) ⊆ K such thatŝ ∈ N and hr(z) ≤ hr(z * ) − ε imply hŝ(z) ≤ h(z * ) − ε/2. Thus we may choose γ c (ŝ) = γ c (r) to be independent ofŝ over N (r). Choosing a finite cover of K by these neighborhoods, the union of the corresponding sets Ξ(r) supports the cycles γ c (r) for all c andr.
Theorem 6.9 is a rather standard result about pushing the intersection class below height h(z * ) except in a small ball about z * . Our proof of Theorem 6.9 uses an unspecified torus T with polyradius in the descending component B of Definition 3.3, and is therefore not an explicit construction of a chain representing T, but is sufficient to prove Theorem 6.10 and Lemma 7.1 describing the relative homology of the pair (V c , V c (≤ −ε)). Equation (7.1) in Lemma 7.3 is all we need to complete the proof of Theorem 5.3. However, in Section 8 we study the asymptotic contributions of lower critical points, these being the dominant contributions in the lacuna setting, when d is even and greater than 2k. For this purpose we need a more explicit description of a cycle homologous to T at height below the critical point: the quadric approximation of V c is only good in a neighbourhood of the critical point, however finding a torus disjoint from V may require traveling further down. The next lemma finds an explicit cycle homologous to T, having height at most −ε except for an abitrarily small tube around a piece of V ≤0 , in two ways: one when a torus T at height −ε can be chosen disjoint from V and a different way when T intersects V. (i) Suppose that T is disjoint from V, as in the proof of Theorem 6.9. Then there exist ε, c * , c > 0 such that for all |c| < c * , where γ c is the cycle in the conclusion of Lemma 7.1 with c replaced by c .
Proof: For the compact Ξ described in Corollary 7.2, the intersection of V with Ξ is smooth. By Proposition 3.1, there is a neighborhood of Ξ in V * \ Σ that can be parameterized as a 2-dimensional vector bundle over some compact subset Ξ ⊆ V. This bundle is naturally coordinatized by the values of Q so that for some small c * > 0 the tubular vicinity around V Ξ can be identified with D × V Ξ for D := {c ∈ C : |c| < c * }.
We will denote this vicinity as V D Ξ .
Lemma 7.1 implies that for all small enough |c| (which we may assume from now on to be smaller than c * < c * ). The class oγ c can be represented by a small tube around a cycle γ c ∈ V c , which is entirely supported by V D Ξ . Using the product structure V D Ξ ∼ = D × V Ξ we can identify this tube with a product of a small circle (of radius ρ(c) > 0) around c ∈ D and γ * , a cycle in the smooth part of V obtained by projection of γ c . When c * and ρ are sufficiently small, the maximum height of γ * is h * − ε for some ε > 0.
There exists a homeomorphism of the annulus D − D ρ(c) (c) fixing its outer boundary and sending the small circle ∂D ρ(c) (c) around c into the circle of radius c * . Extend this homeomorphism, fiberwise, to all of the tubular vicinity V D Ξ . Furthermore, extend it to the complement of V D Ξ in such a way that it is identity outside of a small vicinity of V D Ξ (and thus near S c and T, T ). Choosing c * smaller if necessary, and taking oγ to be the c * -tube around γ * for all c with |c| < c * , this cycle avoids V c for all c with |c| < c * and has maximum height less than h * − ε where ε is positive once c * has been chosen sufficiently small with respect to ε . This completes the proof of case (i).
For case (ii), let α : [0, 1] → R d be as in Theorem 6.9 parametrizing the line segment from x to some y ∈ B . Let T denote the torus with polyradius y and let T be the torus at height ε which is the slice of the homotopy in Theorem 6.9 for some t ∈ (0, 1). Let y be the corresponding basepoint.
The function H(u, c) is holomorphic in u and bounded on oS 1 uniformly in c. As c → 0, H(u, c) → P (z * )/q(u) k and the conclusion (5.3) follows. 2 8 Application to the GRZ function with critical parameter Having established the exponential drop, this section extends Theorem 2.3 to obtain more precise asymptotics for a r . Most of what follows concentrates on the GRZ example, however we first state a result holding more generally in the presence of a lacuna. A critical point at infinity, formally defined in [BMP22], can be viewed as a sequence of singularities going off to infinity in such a way that the limit of the differential of the height function at the points approaches zero. Here, we note only that there is an effective test for critical points at infinity [BMP22, Algorithm 1] and that our GRZ example does not have any. . Then for any ε > 0, any chain Γ of maximum height at most b can be homotopically deformed into a chain Γ whose maximum height is at most a + ε.
Proof of Theorem 8.1: Apply Proposition 8.2 with a = c 2 and b = c 1 , resulting in the chain Γ . Applying Theorem 2.3 and the homotopy equivalence of Γ and Γ in M, where R decreases super-exponentially, and in the polynomial case is in fact zero for all but finitely many r. The height condition on Γ implies that this integral is bounded above by the volume of Γ , multiplied by the maximum value of |F | on Γ , multiplied by e (c2+ε)|r| . 2 In the remainder of this section, as in Example 2.4, we let Fixr to be the diagonal direction. We will prove Theorem 2.5 by first computing an estimate for a r up to an unknown integer factor m. We then use the theory of D-finite functions and rigorous numerical bounds to find the value of m. Lastly, we indicate how the value of m could possibly be determined by topological methods. In order to discuss the sets V(ε) relative to different critical heights, we extend the notation in (2.2) via V ≤t := V ∩ {z : hr(z) < t} .
• The solutions of a D-finite equation form a C-vector space, here equal to dimension three.
• A solution of (8.3) can only have a singularity when the leading polynomial coefficient z 2 (81z 2 +14z+1) vanishes. Here the roots are 0, ζ 4 , and its algebraic conjugate ζ 4 , where ζ is the complex number appearing in the coordinates of the critical point c 2 .
• Equation (8.3) is a Fuchsian differential equation, meaning its solutions have only regular singular points, and its indicial equation has rational roots. Because of this, at any point ω ∈ C, including potentially singularities, any solution of (8.3) has an expansion of the form in a disk centered at ω with a line from ω to the boundary of the disk removed, where α is rational and each g j are analytic. At any algebraic point z = ω there are effective algorithms to determine initial terms of the expansion (8.4) for a basis of the vector space of solutions of (8.3).
• If g(z) = n≥0 c n z n is a solution of (8.3) which has no singularity in some disk |z| < ρ except at a point z = ω, and g(z) has an expansion (8.4) in a slit disk near ω (a disk centered at ω minus a ray from the center to account for a branch cut) then asymptotics of c n are determined by adding asymptotic contributions of the terms in (8.4). In particular, a term of the form C(1 − z/ω) α log r (1 − z/ω) with α / ∈ N gives an asymptotic contribution of ω −n n −α−1 log r (n) C Γ(−α) to c n . Furthermore, if g(z) has a finite number of singularities in a disk and each has the above form, then one can simply add the asymptotic contributions coming from each point in the disk to determine asymptotics of c n .
Problem 1. Give a direct demonstration of these coefficients being 3.
Our best explanation at present is this. If W is a smooth algebraic hypersurface, Morse theory gives us a basis for H d−1 (W ) consisting of the unsable manifolds for downward gradient flow at each critical point.
The stable manifolds at each critical point are an upper tringular dual to this via the intersection pairing. The original torus of integration is a tube over a torus T 0 in V. If V were smooth, we would be trying to show that the stable manifold at w in V * intersects T 0 with signed multiplicity ±3, where w = (ζ, ζ, ζ, ζ). This is probably not true in the smooth varieties V c . However, as c → 0, part of the stable manifold at w gets drawn toward z * = (1/3, 1/3, 1/3, 1/3). Therefore, in the limit, we need to check how many total signed paths in the gradient field ascend from w to z * .
By the symmetry, we expect to find these paths along the three partial diagonals: {x = y, z = w}, {x = z, y = w} and {x = w, y = z}. Solving for gradient ascents on any one of these yields three that go to z rather than to the coordinate planes. If these all had the same sign, the multiplicity would be 9 rather than 3, therefore, in any one partial diagonal, the three paths are two of one sign and one of the other. It remains to show that the signs are as predicted, that these are the only paths going from w to z * , and to rigorize passage from the smooth case to the limit as c → 0.

Computational Morse theory
One of the central problems in ACSV is effective computation of coefficients in integer homology. Specifically, the class [T ] ∈ H d (M) must be resolved as an integer combination of classes oσ where σ ∈ H d−1 (V * ) projects to a homology generator for one of the attachment pairs H d−1 (V ≤c , V ≤c−ε ) near a critical point with critical value c. What is known is nonconstructive. There is a highest critical value c where [T ] has nonzero homology in the attachment pair. The projection of [T ] to H d−1 (V ≤c , V ≤c−ε ) is well defined. If this relative homology element is the projection of an absolute homology element σ ∈ H d−1 (V ≤c \ V ≤c−ε ) then there is no Stokes phenomenon, meaning one can replace [T ] by [T ] − σ and continue to the next lower attachment pair where [T ] − σ projects to a nonzero homology element.
The data for this problem is algebraic. Therefore, one might hope for an algebraic solution, which can be found via computer algebra without resorting to numerical methods, rigorous or otherwise. At present, however, we have only heuristic geometric arguments.
Problem 2. Given an integer polynomial and rationalr, algebraically compute the highest critical points z for which the projection of [T ] to the attachment pair is nonzero. Then compute these integer coefficients.