arXiv subjects

Lorenzo Venturello

Publications and source records attributed to Lorenzo Venturello.

At least 19 recordsLinked to original sources

arXiv2026

The number of edges of a symmetric edge polytope

The symmetric edge polytope of a simple graph is a lattice polytope defined as the convex hull of a subset of the type A roots corresponding to the edges of the graph. In this article we prove a sharp lower bound for the number of edges of the symmetric edge polytope of a graph as a function of elementary graph invariants. Moreover, we characterize graphs attaining this bound. We highlight a connection with the h*-polynomial of such polytopes and, motivated by a conjecture of Ohsugi and Tsuchiya, we investigate the behaviour of such polynomial under edge-deletion in the graph.

math.CO↗

arXiv2025

Preservation of log-concavity on gamma polynomials

Every symmetric polynomial $h(x)$ with center of symmetry $n/2$ can be expressed as a linear combination in the basis $x^i(1+x)^{n-2i}$. The $γ$-polynomial of $h(x)$, which we denote $γ_h(x)$, records the coefficients of this linear combination. Two decades ago, Brändén and Gal independently showed that if $γ_h(x)$ has nonpositive real roots only, then so does $h(x)$. More recently, Brändén, Ferroni, and Jochemko proved using Lorentzian polynomials that if $γ_h(x)$ is ultra log-concave, then so is $h(x)$, and they raised the question of whether a similar statement can be proved for the usual notion of log-concavity. The purpose of this article is to show that the answer to the question of Brändén, Ferroni, and Jochemko is affirmative. One of the crucial ingredients of the proof is an inequality involving binomial numbers that we establish via a path-counting argument.

math.CO↗

arXiv2025

Simplicial complexes and matroids with vanishing $T^2$

We investigate quotients by radical monomial ideals for which $T^2$, the second cotangent cohomology module, vanishes. The dimension of the graded components of $T^2$, and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete characterization and a full list of one dimensional complexes with $T^2=0$. We characterize the graded components of $T^2$ when the simplicial complex is a uniform matroid. Finally, we show that $T^2$ vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing $T^2$ are of corank at most two.

math.CO↗

arXiv2024

The $cd$-index of semi-Eulerian posets

We generalize the definition of the $cd$-index of an Eulerian poset to the class of semi-Eulerian posets. For simplicial semi-Eulerian Buchsbaum posets, we show that all coefficients of the $cd$-index are non-negative. This proves a conjecture of Novik for odd dimensional manifolds and extends it to the even dimensional case.

math.CO↗

arXiv2023

Characteristic polynomials and eigenvalues of tensors

We lay the geometric foundations for the study of the characteristic polynomial of tensors. For symmetric tensors of order $d \geq 3$ and dimension $2$ and symmetric tensors of order $3$ and dimension $3$, we prove that only finitely many tensors share any given characteristic polynomial, unlike the case of symmetric matrices and the case of non-symmetric tensors. We propose precise conjectures for the dimension of the variety of tensors sharing the same characteristic polynomial, in the symmetric and in the non-symmetric setting.

math.AG↗

arXiv2023

Triangulations of cosmological polytopes

A cosmological polytope is defined for a given Feynman diagram, and its canonical form may be used to compute the contribution of the Feynman diagram to the wavefunction of certain cosmological models. Given a subdivision of a polytope, its canonical form is obtained as a sum of the canonical forms of the facets of the subdivision. In this paper, we identify such formulas for the canonical form via algebraic techniques. It is shown that the toric ideal of every cosmological polytope admits a Gröbner basis with a squarefree initial ideal, yielding a regular unimodular triangulation of the polytope. In specific instances, including trees and cycles, we recover graphical characterizations of the facets of such triangulations that may be used to compute the desired canonical form. For paths and cycles, these characterizations admit simple enumeration. Hence, we obtain formulas for the normalized volume of these polytopes, extending previous observations of Kühne and Monin.

math.CO↗

arXiv2022

On the gamma-vector of symmetric edge polytopes

We study $γ$-vectors associated with $h^*$-vectors of symmetric edge polytopes both from a deterministic and a probabilistic point of view. On the deterministic side, we prove nonnegativity of $γ_2$ for any graph and completely characterize the case when $γ_2 = 0$. The latter also confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side, we show that the $γ$-vectors of symmetric edge polytopes of most Erdős-Rényi random graphs are asymptotically almost surely nonnegative up to any fixed entry. This proves that Gal's conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting.

math.CO↗

arXiv2022

Voronoi diagrams of algebraic varieties under polyhedral norms

We study Voronoi diagrams of manifolds and varieties with respect to polyhedral norms. We provide upper and lower bounds on the dimensions of Voronoi cells. For algebraic varieties, we count their full-dimensional Voronoi cells. As an application, we consider the polyhedral Wasserstein distance between discrete probability distributions.

math.AG↗

arXiv2022

Equations of tensor eigenschemes

We study schemes of tensor eigenvectors from an algebraic and geometric viewpoint. We characterize determinantal defining equations of such eigenschemes via linear equations in their coefficients, both in the general and in the symmetric case. We give a geometric necessary condition for a 0-dimensional scheme to be an eigenscheme.

math.AG↗

arXiv2022

On flag spheres with few equators

In this note we construct a flag simplicial $3$-sphere $Δ$ with the following properties: - $Δ$ is not a suspension; - $Δ$ has no edge that can be contracted to obtain another flag sphere; - The only equators (induced subcomplexes which are spheres of codimension $1$) of $Δ$ are vertex links. Our construction has $12$ vertices, the minimum number of vertices such a simplicial complex can have. This answers a question posed by Chudnovsky and Nevo.

math.CO↗

arXiv2021

Koszul Gorenstein algebras from Cohen-Macaulay simplicial complexes

We associate with every pure flag simplicial complex $Δ$ a standard graded Gorenstein $\mathbb{F}$-algebra $R_Δ$ whose homological features are largely dictated by the combinatorics and topology of $Δ$. As our main result, we prove that the residue field $\mathbb{F}$ has a $k$-step linear $R_Δ$-resolution if and only if $Δ$ satisfies Serre's condition $(S_k)$ over $\mathbb{F}$, and that $R_Δ$ is Koszul if and only if $Δ$ is Cohen-Macaulay over $\mathbb{F}$. Moreover, we show that $R_Δ$ has a quadratic Gröbner basis if and only if $Δ$ is shellable. We give two applications: first, we construct quadratic Gorenstein $\mathbb{F}$-algebras which are Koszul if and only if the characteristic of $\mathbb{F}$ is not in any prescribed set of primes. Finally, we prove that whenever $R_Δ$ is Koszul the coefficients of its $γ$-vector alternate in sign, settling in the negative an algebraic generalization of a conjecture by Charney and Davis.

math.AC↗

arXiv2021

Spectral theory of weighted hypergraphs via tensors

One way to study an hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eingenvalues with some techniques from numerical algebraic geometry.

math.CO↗

arXiv2021

Minimality of tensors of fixed multilinear rank

We discover a geometric property of the space of tensors of fixed multilinear (Tucker) rank. Namely, it is shown that real tensors of fixed multilinear rank form a minimal submanifold of the Euclidean space of tensors endowed with the Frobenius inner product. We also establish the absence of local extrema for linear functionals restricted to the submanifold of rank-one tensors, finding application in statistics.

math.DG↗

arXiv2021

Nonlinear Algebra and Applications

We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.

math.AG↗

arXiv2021

The leading coefficient of Lascoux polynomials

Lascoux polynomials have been recently introduced to prove polynomiality of the maximum-likelihood degree of linear concentration models. We find the leading coefficient of the Lascoux polynomials (type C) and their generalizations to the case of general matrices (type A) and skew symmetric matrices (type D). In particular, we determine the degrees of such polynomials. As an application, we find the degree of the polynomial $δ(m,n,n-s)$ of the algebraic degree of semidefinite programming, and when $s=1$ we find its leading coefficient for types C, A and D.

math.AG↗

arXiv2021

Eigenschemes of Ternary Tensors

We study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors and we compute its dimension. Moreover, we characterize the locus of triples of homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor. Our results allow us to implement algorithms to check whether a given set of points is the eigenscheme of a symmetric tensor, and to reconstruct the tensor. Finally, we give a geometric characterization of all reduced zero-dimensional eigenschemes. The techniques we use rely both on classical and modern complex projective algebraic geometry.

math.AG↗

arXiv2020

Wasserstein Distance to Independence Models

An independence model for discrete random variables is a Segre-Veronese variety in a probability simplex. Any metric on the set of joint states of the random variables induces a Wasserstein metric on the probability simplex. The unit ball of this polyhedral norm is dual to the Lipschitz polytope. Given any data distribution, we seek to minimize its Wasserstein distance to a fixed independence model. The solution to this optimization problem is a piecewise algebraic function of the data. We compute this function explicitly in small instances, we examine its combinatorial structure and algebraic degrees in the general case, and we present some experimental case studies.

math.OC↗

arXiv2020

A new family of triangulations of $\mathbb{R}P^d$

We construct a family of PL triangulations of the $d$-dimensional real projective space $\mathbb{R}P^d$ on $Θ((\frac{1+\sqrt{5}}{2})^{d+1})$ vertices for every $d\geq 1$. This improves a construction due to Kühnel on $2^{d+1}-1$ vertices.

math.CO↗