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Eric Katz

Publications and source records attributed to Eric Katz.

At least 19 recordsLinked to original sources
arXiv2026

Frobenius identities for the volume map on Cohen--Macaulay rings

We study the volume map on Artinian quotients of Cohen-Macaulay algebras in characteristic $p$, and the interaction between it and the action of Frobenius on resolutions. This allows us to provide a general, conceptual way to understand Parseval-Rayleigh identities, curious inhomogeneous identities on the volume map which were developed for the proof of the Ohsugi-Hibi conjecture. This general perspective gives a new approach to generic Lefschetz theory. We use this perspective to do the following: we give sufficient conditions for anisotropy and the Hard Lefschetz property for generic Artinian reductions of graded Gorenstein rings; we study the codimension-$3$ Gorenstein quotient of a polynomial ring by the ideal generated by Pfaffians, proving a Parseval-Rayleigh identity and deriving anisotropy and Hard Lefschetz in characteristic $2$; we deduce the $g$-theorem for simplicial spheres and the Ohsugi-Hibi conjecture following previous work of Adiprasito, Papadakis, and Petrotou; and we provide further examples of Parseval-Rayleigh identities for Gorenstein rings.

math.AC↗
arXiv2024

The Unipotent Tropical Fundamental Group

We define the unipotent tropical fundamental group of a polyhedral complex in $\mathbb{R}^n$ as the Tannakian fundamental group of the category of unipotent tropical vector bundles with integrable connection. We show that it is computable in that it satisfies a Seifert--Van Kampen theorem and has a description for fans in terms of a bar complex. We then review an analogous classical object, the unipotent de Rham fundamental group of a sch\"{o}n subvariety of a toric variety. Our main result is a correspondence theorem between classical and tropical unipotent fundamental groups: there is an isomorphism between the unipotent completion of the fundamental group of a generic fiber of a tropically smooth family over a disc and the tropical unipotent fundamental group of the family's tropicalization. This theorem is established using Kato--Nakayama spaces and a descent argument. It requires a slight enlargement of the relevant categories, making use of enriched structures and partial compactifications.

math.AG↗
arXiv2023

Matroidal Mixed Eulerian Numbers

We make a systematic study of matroidal mixed Eulerian numbers which are certain intersection numbers in the matroid Chow ring generalizing the mixed Eulerian numbers introduced by Postnikov. These numbers are shown to be valuative and obey a log-concavity relation. We establish recursion formulas and use them to relate matroidal mixed Eulerian numbers to the characteristic and Tutte polynomials, reproving results of Huh-Katz and Berget-Spink-Tseng. Generalizing Postnikov, we show that these numbers are equal to certain weighted counts of binary trees. Lastly, we study these numbers for perfect matroid designs, proving that they generalize the remixed Eulerian numbers of Nadeau-Tewari.

math.CO↗
arXiv2022

p-adic iterated integration on semistable curves

We reformulate the theory of p-adic iterated integrals on semistable curves using the unipotent log rigid fundamental group. This fundamental group carries Frobenius and monodromy operators whose basic properties are established. By identifying the Frobenius-invariant subgroup of the fundamental group with the fundamental group of the dual graph, we characterize Berkovich--Coleman integration, which is path-dependent, as integration along the Frobenius-invariant lift of a path in the dual graph. Vologodsky's path-independent integration theory which was previously described using a monodromy condition can now be identified as Berkovich--Coleman integration along a combinatorial canonical path arising from the theory of combinatorial iterated integration as developed by the first-named author and Cheng.

math.AG↗
arXiv2020

p-adic integration on bad reduction hyperelliptic curves

In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser for regular and meromorphic 1-forms on good reduction curves, respectively. We then employ tropical geometric techniques due to the first-named author with Rabinoff and Zureick-Brown to convert the Berkovich-Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish.

math.NT↗
arXiv2020

Poset subdivisions and the mixed cd-index

The cd-index is an invariant of Eulerian posets expressed as a polynomial in noncommuting variables c and d. It determines another invariant, the h-polynomial. In this paper, we study the relative setting, that of subdivisions of posets. We introduce the mixed cd-index, an invariant of strong formal subdivisions of posets, which determines the mixed h-polynomial introduced by the second author with Stapledon. The mixed cd-index is a polynomial in noncommuting variables c',d',c,d, and e and is defined in terms of the local cd-index of Karu. Here, use is made of the decomposition theorem for the cd-index. We extend the proof of the decomposition theorem, originally due to Ehrenborg-Karu, to the class of strong formal subdivisions. We also compute the mixed cd-index in a number of examples.

math.CO↗
arXiv2019

Multivariate polynomials for generalized permutohedra

Using the notion of Mahonian statistic on acyclic posets, we introduce a $q$-analogue of the $h$-polynomial of a simple generalized permutohedron. We focus primarily on the case of nestohedra and on explicit computations for many interesting examples, such as $S_n$-invariant nestohedra, graph associahedra, and Stanley--Pitman polytopes. For the usual (Stasheff) associahedron, our generalization yields an alternative $q$-analogue to the well-studied Narayana numbers.

math.CO↗
arXiv2017

Total $p$-differentials on schemes over $Z/p^2$

For a scheme $X$ defined over the length $2$ $p$-typical Witt vectors $W_2(k)$ of a characteristic $p$ field, we introduce total $p$-differentials which interpolate between Frobenius-twisted differentials and Buium's $p$-differentials. They form a sheaf over the reduction $X_0$, and behave as if they were the sheaf of differentials of $X$ over a deeper base below $W_2(k)$. This allows us to construct the analogues of Gauss-Manin connections and Kodaira-Spencer classes as in the Katz-Oda formalism. We make connections to Frobenius lifts, Borger-Weiland's biring formalism, and Deligne--Illusie classes.

math.AG↗
arXiv2017

Combinatorial Iterated Integrals and the Harmonic Volume of Graphs

Let $\Gamma$ be a connected bridgeless metric graph, and fix a point $v$ of $\Gamma$. We define combinatorial iterated integrals on $\Gamma$ along closed paths at $v$, a unipotent generalization of the usual cycle pairing and the combinatorial analogue of Chen's iterated integrals on Riemann surfaces. These descend to a bilinear pairing between the group algebra of the fundamental group of $\Gamma$ at $v$ and the tensor algebra on the first homology of $\Gamma$, $\int\colon \mathbf{Z}\pi_1(\Gamma,v) \times T\mathrm{H}_1(\Gamma,\mathbf{R}) \to \mathbf{R}$. We show that this pairing on the two-step unipotent quotient of the group algebra allows one to recover the base-point $v$ up to well-understood finite ambiguity. We encode the data of this structure as the combinatorial harmonic volume which is valued in the tropical intermediate Jacobian. We also give a potential-theoretic characterization for hyperelliptiicity for graphs.

math.CO↗
arXiv2016

Newton-Okounkov Bodies over Discrete Valuation Rings and Linear Systems on Graphs

The theory of Newton-Okounkov bodies attaches a convex body to a line bundle on a variety equipped with flag of subvarieties. This convex body encodes the asymptotic properties of sections of powers of the line bundle. In this paper, we study Newton-Okounkov bodies for schemes defined over discrete valuation rings. We give the basic properties and then focus on the case of toric schemes and semistable curves. We provide a description of the Newton-Okounkov bodies for semistable curves in terms of the Baker--Norine theory of linear systems on graphs, finding a connection with tropical geometry. We do this by introducing an intermediate object, the Newton-Okounkov linear system of a divisor on a curve. We prove that it is equal to the set of effective elements of the real Baker-Norine linear system of the specialization of that divisor on the dual graph of the curve. As a bonus, we obtain an asymptotic algebraic geometric description of the Baker-Norine linear system.

math.AG↗
arXiv2016

Diophantine and tropical geometry, and uniformity of rational points on curves

We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.

math.NT↗
arXiv2015

Hodge Theory for Combinatorial Geometries

We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the f-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.

math.CO↗
arXiv2015

Uniform bounds for the number of rational points on curves of small Mordell--Weil rank

Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by [CHM97] to follow from the Bombieri--Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{{\rm tors},\dagger}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel--Jacobi map. For fixed $X$ this quantity was conjectured to be finite by Manin--Mumford, and was proved to be so by Raynaud [Ray83]. We give an explicit uniform bound on $\#X(F)$ when $X$ has Mordell--Weil rank $r\leq g-3$. This generalizes recent work of Stoll on uniform bounds on hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of $F$-rational torsion points of $J$ lying on the image of $X$ under an Abel--Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of $J$ lying on $X$ when the reduction type of $X$ is highly degenerate. Our methods combine Chabauty--Coleman's $p$-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.

math.NT↗
arXiv2014

Local $h$-polynomials, invariants of subdivisions, and mixed Ehrhart theory

There are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and $h^*$-polynomials, respectively. In this paper, we study their generalization to subdivisions and lattice subdivisions of polytopes. By abstracting constructions in mixed Hodge theory, we introduce multivariable polynomials which specialize to the $h$-, $h^*$- polynomials. These polynomials, the mixed $h$-polynomial and the (refined) limit mixed $h^*$-polynomial have rich symmetry, non-negativity, and unimodality properties, which both refine known properties of the classical polynomials, and reveal new structure. For example, we prove a lower bound theorem for a related invariant called the local $h^*$-polynomial. We introduce our polynomials by developing a very general formalism for studying subdivisions of Eulerian posets that extends the work of Stanley, Brenti and Athanasiadis on local $h$-vectors. In particular, we prove a conjecture of Nill and Schepers, and answer a question of Athanasiadis.

math.CO↗
arXiv2014

Matroid theory for algebraic geometers

This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be representable. Still, one may apply linear algebraic constructions to non-representable matroids. There are a number of different definitions of matroids, a phenomenon known as cryptomorphism. In this survey, we begin by reviewing the classical definitions of matroids, develop operations in matroid theory, summarize some results in representability, and construct polynomial invariants of matroids. Afterwards, we focus on matroid polytopes, introduced by Gelfand-Goresky-MacPherson-Serganova, which give a cryptomorphic definition of matroids. We explain certain locally closed subsets of the Grassmannian, thin Schubert cells, which are labeled by matroids, and which have applications to representability, moduli problems, and invariants of matroids following Fink-Speyer. We explain how matroids can be thought of as cohomology classes in a particular toric variety, the permutohedral variety, by means of Bergman fans, and apply this description to give an exposition of the proof of log-concavity of the characteristic polynomial of representable matroids due to the author with Huh.

math.AG↗
arXiv2014

Tropical geometry, the motivic nearby fiber and limit mixed Hodge numbers of hypersurfaces

The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure associated to the degeneration which is encoded as a limit mixed Hodge structure. However, this invariant is difficult to compute in practice. Using the techniques of tropical geometry we present a new formula for the motivic nearby fiber. Moreover, since there is a range of available software implementing the main algorithms in tropical geometry, our formula can be computed in practice. We specialize to the case of families of sch\"{o}n complex hypersurfaces of tori where we provide explicit formulas describing the action of the unipotent part of monodromy on the graded pieces (with respect to the Deligne weight filtration) of the cohomology with compact supports. These families are described combinatorially by a polyhedral subdivision of the associated Newton polytope. We develop new mixed Hodge theory-inspired combinatorial invariants of such subdivisions, among them the `refined limit mixed $h^*$-polynomial'. These invariants are related to Stanley's combinatorial study of subdivisions: in a companion combinatorial paper whose results are applied here, we situate our invariants in Stanley's theory where they become multi-variable extensions of his invariants. Our results generalize work of Danilov and Khovanski{\u\i} and Batyrev and Borisov on the Hodge numbers of hypersurfaces. We also present analogous formulas describing the action of the unipotent part of monodromy on the intersection cohomology groups of a family of sch\"on hypersurfaces of a projective toric variety.

math.AG↗