Critical points at infinity for analytic combinatorics

On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates the description and computation via Morse theory of key topological invariants. Here we establish checkable conditions under which the behavior at infinity may be ignored, and the usual theorems of classical and stratified Morse theory may be applied.

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1 Introduction

Recent studies of multivariate generating functions motivate several problems in the topology of complex algebraic varieties. Even when they fall within the scope of well-established theories, these questions are often outside of the realm of existing literature. Here we study one such problem, which lies at the heart of effective methods for the field of analytic combinatorics in several variables (ACSV), and give its solution using tools from deformation theory, Morse theory and algebraic geometry. We begin by describing the purely topological problem; its combinatorial origin is discussed below.

Fix a real Laurent polynomial

and a nonzero real vector

with normalization , where . Let denote the affine algebraic variety defined by , and with .

Furthermore, define the function by

and assume that is Morse on (i.e. the Hessian of at each critical point has full rank).

Problem.

Use standard methods of (stratified) Morse theory to give a topological decomposition of in terms of relative cycles near critical points of the potentially non-proper height function .

The key word here is ‘non-proper’: most basic results of standard and stratified Morse theory require that the height function be a proper map (see [Mil63] and [GM88], respectively). Although stratified Morse theory contains results for certain types of non-proper height functions, they apply primarily in the case where the topological space in question fails to be complete because a point was removed [GM88, Chapter 10]. In our situation the function , fixed by the underlying combinatorics, fails to be proper not only near points where one of the coordinates is zero but also due to points at infinity (for certain vectors there might be ‘critical point at infinity’ which stay at finite height). For this reason, books and papers on analytic combinatorics in several variables [PW08, PW13]

often use Morse-theoretic heuristics to motivate certain constructions, but cannot use Morse theory outright to prove general results.

One solution to the problem of non-properness is to compactify. Suppose we can embed in a compact space such that may be extended smoothly to the closure of in . Then Morse theory gives a decomposition of the topology of in terms of cycles local to (stratified) critical points. Chains in may be decomposed into sums of relative homology cycles near these critical points, some of which are critical points for on and some of which may be critical points at infinity. Conjecture 2.11 of [Pem10] posits that a compacitification may always be found on which extends smoothly. In Appendix A of this paper we include a one-page argument relying on standard results in toric geometry which resolves the conjecture. Unfortunately, this solution works only when is an integer vector and the structure of the compactification depends on . This is not ideal for the underlying combinatorial applications, as it hinders a complete analysis of multivariate generating functions and hides strong links between the different sequences they encode.

Furthermore, for generic parameter , there are no critical points at infinity, and no need to embark on a cumbersome computational exercise. For this reason, we were motivated to find a solution better adapted to use in the combinatorial setting. Instead of trying to construct a compactification of compatible with , we give instead a computationally effective criteria under which the requisite Morse decomposition on remains confined to a compact region.

Our main results are the following.

  1. A definition of critical points at infinity for a polynomial with respect to a direction (Definition 2.3).

  2. An algorithm for detecting critical points at infinity (Section 3), including a Maple implementation.

  3. Theorem 2.4, which states that in the absence of critical points at infinity of the height function the usual Morse theoretic decomposition of the topology of applies.

In the remainder of this section we discuss the combinatorial origins of the problem.

Section 2 sets the notation for the study of stratified spaces and critical points, formulates the definition of critical points at infinity and states the main result. Section 3 shows how to determine all critical points at infinity using a computer algebra system. Some examples are given in Section 4. Section 5 constructs a Morse deformation in the absence of critical points at infinity, and Section 6 concludes by proving Theorem 2.4.

Motivation from ACSV

Analytic combinatorics in several variables (ACSV) studies coefficients of multivariate generating functions via analytic methods; see, for example, [PW13]. The most developed part of the theory is the asymptotic determination of coefficients of multivariate series , wherein the coefficients are defined by the multivariate Cauchy integral

(1.1)

with an appropriate chain of integration (a certain torus defined by for ). In many applications, is rational function with a power series expansion whose coefficients are indexed by , an integer vector. More generally, one often looks for asymptotics in the directions that vary over a convex cone .

Given with we define the functions

where is taken coordinate-wise and denotes the real part of complex .

In order to estimate

for and , one isolates the exponential part of the Cauchy integral (1.1) by writing

One then deforms to a chain on which the maximum value of is minimized; the maximum value on will occur at some (stratified) critical point of on . Let denote the domain of holomorphy of the integrand. The value of the Cauchy integral depends on only via its homology class . In the absence of the non-properness issue discussed above, and for smooth , Morse theory would allow us to find a basis for composed of tubes around unstable manifolds of the critical points of on (the critical points are necessarily of of index ). Resolving the class into this basis results in the decomposition where are integers and are tubes around certain cycles corresponding to the critical points .

Taking residues results in a sum of integrals

which are generally well understood asymptotically, computed either as stationary phase integrals or by more difficult singularity theory in the case where is a singular point of the variety .

2 Definitions and results

2.1 Stratified spaces and critical points

Let and denote , referred to as “log space”. Log space is diffeomorphic to via the map defined by

(2.1)

We use a tilde to denote the result of pulling back to the log space via . Most of our constructions will take place on the log space, with the analogous constructions on defined as the image under . The reason for keeping both and around is that the geometric constructions are more transparent in but polynomial computations via computer algebra are carried out in .

A Whitney stratification of a complex algebraic variety is a decomposition of into a disjoint union of complex manifolds of dimensions between and with so-called Whitney conditions on how the tangent planes of the different manifold interact (see, for example, [PW13, Definition 5.4.1]). Such a stratification of always exists and may be chosen to be compatible with one for . If is a Whitney stratification of , we let denote the stratification of , where is the inverse image of under .

We recall the stratified definition of an affine critical point of . Note that critical points are only defined relative to stratifications; a stratification is assumed but does not appear in the notation.

Definition 2.1 (critical points).

The function is said to have a critical point (in the stratified sense) at if and only if for the unique stratum containing (the projection of the differential of onto the tangent space of the stratum is zero). The set of critical points is denoted ; it depends on because depends on . Later we will call these affine critical points to distinguish them from critical points at infinity.

Proposition 2.2 (complex versus real).

The following conditions are equivalent:

  • The point is a stratified critical point for on

  • The point is a stratified critical point for on

  • The point is a stratified critical point for on

  • The point is a stratified critical point for on

We summarize in figure 1.

Figure 1: Critical points are the same in and , and for the complex valued height or its real part.

Proof: By functoriality, critical points of for the stratification pull back to critical points of for the stratification and likewise for and . It remains to see that and have the same critical points. This is equivalent to showing

Clearly implies on any subspace. On the other hand, if for some tangent vector , then because we require stratifications to have complex structure, the vector is also a tangent vector to , and either or will be nonzero.

Except in degenerate cases, the set of critical points in direction is a zero-dimensional ideal, easily computed in any computer algebra system; see, for example, [Mel17, Chapter 8].

2.2 Critical points at infinity and main deformation result

The logspace is an Abelian group acting on itself, and therefore one can introduce a shift-invariant Hermitian structure on it. In what follows, when we refer to orthogonal projections or angles between tangent vectors in the logspace, it is this shift-invariant structure that we mean.

Definition 2.3 (critical point at infinity).

Given , , and a stratification , let be the corresponding stratification in the log-space. A critical point at infinity is a sequence of points in some stratum such that the pullbacks approach infinity and the length of the projection of to the tangent space of at goes to zero. The set of heights of a critical point at infinity (possibly empty) is the limit set of values . If is the limit of a sequence for a critical point at infinity , we say witnesses a critical point at infinity at height . When there is no critical point at infinity in the direction then there is none for directions in some neighborhood of .

For any space with height function and any real , we denote by the set . Our main result is the following theorem, together with an algorithm in Section 3 which shows that critical points at infinity are easily computed.

Theorem 2.4 (no critical point at infinity implies Morse results).

  1. Suppose there are no critical points at infinity with heights in , nor any ordinary critical points with heights in . Then is homotopy equivalent to via the downward gradient flow.

  2. Suppose there is a single critical point with critical value , and there is no critical point at infinity with height in . Then for any compact cycle supported on there are such that the downward gradient flow run for time takes to a cycle supported on , where denotes the ball of radius around . In other words, every cycle can be pushed down so it is supported on the union of a neighborhood of and the part of below height .

This immediately implies the following corollary. Because the cycle can be pushed down at least until hitting the first critical point corresponding to direction , the magnitude of coefficients in this direction is bounded above by the Cauchy integral over a contour at this height. This is given only as a conjecture in [PW13] because it was not known under what conditions could be pushed down to the critical height.

Corollary 2.5.

Fix a Laurent polynomial and in the cone supporting the Laurent expansion . Let be the maximal height of an affine critical point. Assume there are no critical points at infinity with height in . Then

In the remainder of Section 2 we outline how Theorem 2.4 is used to obtain Morse theoretic representations of integrals. These results, already quoted in several preprints, can be skipped if one is only interested in examples, proofs and computations of critical points at infinity.

2.3 Intersection classes

It is useful to be able to transfer between and : topologically this is the Thom isomorphism and, when computing integrals, this corresponds to taking a single residue. We outline this construction, which goes back at least to Griffiths [Gri69]. Assume does not vanish on , so that is smooth. Then the well known Collar Lemma [MS74, Theorem 11.1] states111See [Lan02] for a full proof. that an open tubular vicinity of is diffeomorphic to the space of the normal bundle to . We require a stratified version: if intersects a manifold transversely, then an open tubular neighborhood of the pair is diffeomorphic to the product where is a two-dimensional disk (because the normal bundle to is ). This statement follows from the version of Thom’s first isotopy lemma given in [GM88, Section 1.5]; the proof is only sketched, but the integral curve argument there mirrors what is given explicitly in [PW13, Section A.4].

In particular, for any -chain in , one can define a -chain , obtained by taking the boundary of the union of small disks in the fibers of the normal bundle. The radii of these disk should be small enough to fit into the domain of the collar map, but can (continuously) vary with the point on the base. Different choices of the radii matching over the boundary of the chain lead to homologous tubes. We will be referring to informally as the tube around . Similarly, the symbol denotes the product with the solid disk. The elementary rules for boundaries of products imply

(2.2)

Because commutes with , cycles map to cycles, boundaries map to boundaries, and the map on the singular chain complex of induces a map on homology ; we also denote this map on homology by to simplify notation.

Proposition 2.6 (intersection classes).

Suppose vanishes on a smooth variety and let and be two cycles in that are homologous in . Then there exists a class such that

The class is represented by the cycle for any cobordism between and in that intersects transversely, all such intersections yielding the same class in .

Accordingly, we may define the intersection class of and by

which, by Proposition 2.6, is well defined.

Proof: Let be any -chain in . If intersects transversely, let denote the intersection of with . We claim that induces a map . This follows from taking to be on any representing cycle transverse to , provided that maps cycles to cycles and boundaries to boundaries. To see that this is the case, observe that by transversality and the product formula, and implies .

The Thom-Gysin long exact sequence implies exactness in the following diagram,

(2.3)

This may be found in [Gor75, page 127], taking , though in the particular situation at hand it goes back to Leray [Ler50].

Consider now two -cycles and , homologous in and both avoiding . We define the intersection class as follows. Let be any cobordism in between and , generically perturbed if necessary so as to be transverse to . Let , therefore

(2.4)

If is another such cobordism then is a -cycle in , hence null-homologous, so and in .

2.4 Relative homology

Morse theory decomposes the topology of a manifold into the direct sum of relative homologies of attachment pairs. In the absence of critical points at infinity, we can harness this for the particular Morse functions of interest. Given , suppose the affine critical values, listed in decreasing order, are . Assume that there is precisely one critical point at each affine critical height.

Even without ruling out critical points at infinity, one has the following decomposition which works at the level of filtered spaces. Fix real numbers interleaving the critical values, so that , and for define . Without ambiguity we may let denote any of the inclusions of into and denote any of the projections of a pair to a pair . A simple diagram chase proves the following result.

Proposition 2.7 ([Pw13, Lemma B.2.1]).

Let be any class in . The least such that is not in the image of is equal to the least such that in . Denoting this by , there is a class such that in . The class is not unique but the projection to is unique.

The interpretation is that the relative homology classes are the possible obstructions, and that may be pushed down until the first obstruction, which is a well defined relative homology element. Taking , we see that so we may iterate, arriving at where is a class supported on .

When there are no critical points at infinity, Theorem 2.4 allows us to upgrade this to the following result. Informally, until you hit a critical value at infinity, you can push down to each critical point, subtract the obstruction, and continue downward. The obstructions are the homology groups of the attachment pairs.

Proposition 2.8 (the Morse filtration).

Fix and suppose there are no critical points at infinity with height in . Let be any class. Then for sufficiently small and all , the pair is naturally homotopy equivalent to the pair which is further homotopic to for any ball about the critical point that reaches past . It follows that there are such that each is well defined given and projects to zero in .

Proof: The first homotopy equivalence follows from part  of Theorem 2.4 and the second from part . The remainder follows from Proposition 2.7 and the homotopy equivalences.

Combining this representation of as with the construction of intersection classes in Proposition 2.6 immediately proves a similar result directly in terms of the homology of .

Theorem 2.9 (filtration on ).

Fix and suppose there are no critical points at infinity with height in . Let be any class that is homologous in to some cycle in . Then for all there are classes such that

The least for which and the projection of to are uniquely determined. Each is either zero in or nonvanishing in .

2.5 Integration

Integrals of holomorphic forms on a space are well defined on homology classes in . Relative homology is useful for us because it defines integrals up to terms of small order. Throughout the remainder of the paper, denotes a quotient of polynomials except when a more general numerator is explicitly noted. Let denote the amoeba associated to the polynomial , where and are taken coordinatewise. Components of the complement of are open convex sets and are in correspondence with Laurent expansions , each Laurent expansion being convergent when and determined by the Cauchy integral (1.1) over the torus for any . The support set will be contained in the dual cone to the recession cone of ; see [BP11, Section 2.2] for details.

Definition 2.10 ( and the pair ).

Let denote the infimum of heights of critical points, including both affine critical points and critical points at infinity. Denote by the homology of the pair for any . By part  of Theorem 2.4, these pairs are all naturally homotopy equivalent.

For functions of , let denote the relation of differing by a quantity decaying more rapidly than any exponential function of . Homology relative to and equivalence up to superexponentially decaying functions are related by the following result.

Theorem 2.11.

Let with rational and holomorphic. Suppose that . For -cycles in , the equivalence class of the integral

depends only on the relative homology class when projected to .

Proof: Fix any . Suppose in . From the exactness of

observing that projects to zero in , it follows that is homologous in to some cycle . Homology in determines the integral exactly. Therefore, it suffices to show that .

As a consequence of the homotopy equivalence in part  of Theorem 2.4, for any there is a cycle supported on and homologous to in . Fix such a collection of cycles . Let and let denote the volume of . Observe that on . It follows that

and is thus seen to be smaller than any exponential function of .

When is rational we may strengthen determination up to to exact equality. The Newton polytope, denoted , is defined as the convex hull of degrees of monomials in . It is known (see, e.g. [FPT00]) that the components of map injectively into the integer points in , and that to each extreme point corresponds a non-empty component. Moreover, the recession cone of these components (collection of directions of rays contained in the component) correspond to the dual cones of the vertices. Hence the linear function is bounded on a component if and only if the vector is within the dual to the tangent cone of the Newton polytope at the corresponding integer point. Fix the component corresponding to the Laurent expansion and integer point in the Newton polytope, with not in the dual to the tangent cone of the Newton polytope at . Then there exists another component of the amoeba complement with in the dual to the tangent cone of the Newton polytope at the corresponding integer point (this integer point lying on the opposite side of the Newton polytope from ).

Proposition 2.12.

If is rational and , then

for all but finitely many .

Proof: Observe that there is a continuous path moving to infinity within . On the corresponding tori, the (constant) value of approaches . Let denote such a torus supported on . Because the tori are all homotopic in , the value of the integral

(2.5)

cannot change. On the other hand, with and as in the proof of the first part, both and are bounded by polynomials in , the common polyradius of points in . Once any coordinate is great enough so that the product of the volume and the maximum grows more slowly than , the integral for that fixed goes to zero as , hence is identically zero.

The utility of this is to represent the Cauchy integral precisely as a tube integral. Let for , the component of defining the Laurent expansion, and choose for as in Proposition 2.12. By Proposition 2.6, if denotes the intersection class , we have in . The integral over vanishes by Proposition 2.12, yielding

Corollary 2.13.

If is rational and there are no critical points at infinity, then

2.6 Residues

Having transferred homology from to , we transfer integration there as well. The point of this is that the minimax height cycles live on , not on where the minimax height is never achieved. Thus we reduce to saddle-point integrals on whose asymptotics can be approximated. In what follows, denotes the holomorphic de Rham complex, whose -cochains are holomorphic forms. The following duality between residues and tubes is well known.

Proposition 2.14 (residue theorem).

There is a functor such that for any class ,

(2.6)

The residue functor is defined locally and, when is squarefree, it commutes with products by any locally holomorphic scalar function. If, furthermore, is rational, there is an implicit formula

For higher order poles, the residue can be computed by choosing coordinates: if , and locally defines a graph of a function, , then

(2.7)

Proof: Restrict to a neighborhood of the support of the cycle in the smooth variety coordinatized so that the last coordinate is . The result follows by applying the (one variable) residue theorem, taking the residue in the last variable.

Applying this to intersection classes and using homology relative to to simplify integrals yields the following representation.

Theorem 2.15.

Let be the quotient of Laurent polynomials with Laurent series converging on when , for some component of . Assume the minimal critical value is finite. Let denote the low (with respect to ) component of as in Proposition 2.12. Then for any and ,

for all but finitely many . If is replaced by any holomorphic function, the same representation of holds up to a function decreasing super-exponentially in .

Proof: If is polynomial, then

The first line is Cauchy’s integral formula, the second is Proposition 2.6, the third is (2.6) and the last is Proposition 2.12. If is not polynomial, use Theorem 2.11 in place of Proposition 2.12 in the last line.

Combining Theorems 2.9 and 2.15 yields the most useful form of the result: a representation of the coefficients in terms of integrals over relative homology generators produced by the stratified Morse decomposition. Let enumerate the critical points of in weakly decreasing order of height . For each , denote the relevant homology pair by

(2.8)

where is a sufficiently small ball around in . Let and let denote cycles in that project to a basis for with integer coefficients. In the case where is a smooth point of , stratified Morse theory [GM88] implies that and is a cycle agreeing locally with the unstable manifold for the downward gradient flow on .

Theorem 2.16 (Stratified Morse homology decomposition).

Let be rational. Let the critical points for on be enumerated as above and assume there are no critical points at infinity. Then there are integers such that

(2.9)

For each smooth critical point , the cycle agrees locally with the unstable manifold at for the downward gradient flow on .

3 Computation of Critical Points at Infinity

We begin by recalling some background about stratifications and affine critical points.

Computing a stratification

To compute critical points at all requires a stratification. In practice there is nearly always an obvious stratification. Generically, in fact, the variety is smooth and the trivial stratification suffices222Formally one must join with the stratification generated by the coordinate planes, but no affine critical point can be on a coordinate plane, so in the end one works only with the single stratum .. In non-generic cases, however, one must produce a stratification of before proceeding with the search for affine critical points as well as those at infinity.

There are two relevant facts to producing a stratification. One is that there is a coarsest possible Whitney stratification, called the canonical Whitney stratification of . It is shown in [Tei82, Proposition VI.3.2] that there are algebraic sets such that the set of all connected components of for all forms a Whitney stratification of and such that every Whitney stratification of is a refinement of this stratification. This canonical stratification is effectively computable; that is, an algorithm exists, given (in general, given the generators of any radical ideal), to produce where is a prime decomposition of the radical ideal corresponding to the Zariski closed set . A bound on the computation time, doubly exponential in , is given in [MR91, page 282].

Computing the affine critical points

Assume now that a Whitney stratification is given, meaning the index set is stored, along with, for each , a collection of polynomial generators for the radical ideal . The set is the algebraic set minus the union of varieties of higher codimension. Potentially by replacing with its prime components, we may assume that the tangent space of at any smooth point has constant codimension . Note that although the gradients of generators of will generate the cotangent space of locally at each , the number of generators, , may be greater than the codimension .

Critical points for the height function are determined by orthogonality of to in the log space. In particular, Definition 2.1 is satisfied for if and only if the vector is orthogonal to ; equivalently, if is in the span of the normal vectors . Define, for ,

Let be the matrix whose rows are the vector together with the vectors for . We then have the following computational definition (see, e.g., [Mum76, Sec. 1A]).

Proposition 3.1.

Fix a stratum and a point . Then if and only if all the minors of vanish. Consequently, the critical points are found by taking the union over of the solutions to the polynomial equalities and non-equalities saying that: these minors all vanish; that (in other words, for all , ); and that for any with .

Condition  is polynomial in both and . For later use, we let denote the minors of with replaced by a variable vector ; note that these polynomials will be homogeneous in the variables. Condition  is of course defined by polynomials in the variables. Homogenizing the and in the variables gives a set of polynomials that together define the Zariski closure of the graph of the relation in . The actual graph of the relation is obtained by removing with for some where (a lower dimensional stratum) and removing points when the homogenizing coordinate vanishes (points at infinity).

Computing critical points at infinity

To determine whether there exist critical points at infinity we use ideal quotients, corresponding to the difference of algebraic varieties. Recall that the variety defined by the saturation of two ideals and is the Zariski closure of the set difference (see [