Stationary Points at Infinity for Analytic Combinatorics

Abstract

On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates both 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. This allows for simplified arguments in the field of analytic combinatorics in several variables, and forms the basis for new methods applying to problems beyond the reach of previous techniques.

A Appendix

We now give an abstract answer to [41, Conjecture 2.11], in a manner suggested to us by Justin Hilburn and Roberta Guadagni. Let $H(\mathbf{z}) := \mathbf{z}^\mathbf{m}$ be the monomial function on $C_*^d$ and $G\subset C_*^d \times \mathbb {C}^*$ its graph. An easy case of toric resolution of singularities (see, e.g., [28]) implies the following result.

Theorem 6

There exists a compact toric manifold K such that $C_*^d$ embeds into it as an open dense stratum and the function H extends from this stratum to a smooth $\mathbb {P}^1$-valued function on K.

Proof

The graph G is the zero set of the polynomial $P_\mathbf{m}:=h-\mathbf{z}^\mathbf{m}$ on $C_*^d\times \mathbb {C}^*$, where h is the coordinate on the second factor. Theorem 2 in [28] implies that a compactification of $C_*^d\times \mathbb {C}^*$ in which the closure of G is smooth exists if the restrictions of the polynomial $P_\mathbf{m}$ to any facet of the Newton polyhedron of $P_\mathbf{m}$ is nondegenerate (defines a nonsingular manifold in the corresponding subtorus). In our case, the Newton polytope is a segment, connecting the points $(\mathbf{m},0)$ and $(\mathbf{0},1)$, and this condition follows immediately. Hence, the closure of G in the compactification of $C_*^d\times \mathbb {C}$ is a compact manifold K. We notice that the projection to $C_*^d$ is an isomorphism on G, and therefore K compactifies $C_*^d$ in such a way that H lifts to a smooth function on K.

Lifting the variety ${\mathcal {V}}_*$ to $G\subset K$ and taking the closure produces the desired result: a compactification of ${\mathcal {V}}_*$ in a compact manifold K on which H is smooth. $\square$

A practical realization of the embedding requires construction of a simple fan (partition of $\mathbb {R}^{d+1}$ into simplicial cones with unimodular generators) which subdivides the fan dual to the Newton polytope of $h-\mathbf{z}^\mathbf{m}$. While this is algorithmically doable (and implementations exist, for example in macaulay2), the resulting fans depend strongly on $\mathbf{m}$, and the resulting compactifications K are hard to work with.

Definition 9

(compactified stationary point) Define a compactified stationary point of H, with respect to a compactification of $C_*^d$ to which H extends smoothly, as a point $\mathbf{x}$ in the closure of ${\mathcal {V}}$ such that dH vanishes at $\mathbf{x}$ on the stratum ${\overline{S}}(\mathbf{x})$, and $H(\mathbf{x})$ is not zero or infinite.

Applying basic results of stratified Morse theory [23] to ${\mathcal {K}}$ directly yields the following consequence.

Corollary 4

(no compactified stationary point implies Morse results) (i) If there are no stationary points or compactified stationary points with heights in [a, b], then ${\mathcal {V}}_{\le b}$ is homotopy equivalent to ${\mathcal {V}}_{\le a}$ via the downward gradient flow.

(ii) If there is a single stationary point x with critical value in [a, b], and there is no compactified stationary point with height in [a, b], then the homotopy type of the pair $\left( {\mathcal {M}}_{\le b} \, , \, {\mathcal {M}}_{\le b} \right)$ is determined by a neighborhood of x, with an explicit description following from results in [23].