Ignacio Garcia Marco

Universidad de La Laguna | ULL · Matemáticas, Estadística e Investigación Operativa

In this paper we study sums of powers of affine functions in (mostly) one variable. Although quite simple, this model is a generalization of two well-studied models: Waring decomposition and sparsest shift. For these three models there are natural extensions to several variables, but this paper is mostly focused on univariate polynomials. We presen...

A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code [F...

A numerical semigroup is said to be universally free if it is free for any possible arrangement of its minimal generating set. In this work, we establish that toric ideals associated with universally free numerical semigroups can be generated by their set of circuits. Additionally, we provide a characterization of universally free numerical semigro...

In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight...

A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a commutative algebra an...

In this work, we explore the relationship between the graded free resolution of some monomial ideals and the Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure that is smaller than the set of codewords of minimal support that provides us with some information about the GHWs. We prove that the first and secon...

Huang proved that every set of more than half the vertices of the d-dimensional hypercube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley gra...

A toric ideal is called robust if its universal Gr\"obner basis is a minimal set of generators, and is called generalized robust if its universal Gr\"obner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a Commutative Algebr...

In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight...

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron -- both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs...

In chomp on graphs, two players alternatingly pick an edge or a vertex from a graph. The player that cannot move any more loses. The questions one wants to answer for a given graph are: Which player has a winning strategy? Can a explicit strategy be devised? We answer these questions (and determine the Nim-value) for the class of generalized Kneser...

Few decoding algorithms for hyperbolic codes are known in the literature, this article tries to fill this gap. The first part of this work compares hyperbolic codes and Reed-Muller codes. In particular, we determine when a Reed-Muller code is a hyperbolic code. As a byproduct, we state when a hyperbolic code has greater dimension than a Reed-Muller...

Let \({{\mathcal {S}}}\subseteq {{\mathbb {Z}}}^m \oplus T\) be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in \({\mathcal {S}}\) having at least two factorizations of the same length, namely the ideal \({\mathcal {L}}_{{\mathcal {S}}}\). To this end, we work with a certain (latt...

We introduce Cayley posets as posets arising naturally from pairs S<T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S<T$$\end{document} of semigroups, much in the same...

Huang proved that every set of more than half the vertices of the $d$-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$, which is tight by a result of Chung, F\"uredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite famili...

Given a linear code C, its square code C(2) is the span of all component-wise products of two elements of C. Motivated by applications in multi-party computation, our purpose with this work is to answer the following question: which families of affine variety codes have simultaneously high dimension k(C) and high minimum distance of C(2), d(C(2))?...

Let $\mathcal S \subseteq \mathbb Z^m \oplus T$ be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in $\mathcal S$ having at least two factorizations of the same length, namely the ideal $\mathcal L_{\mathcal S}$. To this end, we work with a certain (lattice) ideal associated to the...

We introduce Cayley posets as posets arising naturally from pairs $S<T$ of semigroups, much in the same way that Cayley graph arises from a (semi)group and a subset. We show that Cayley posets are a common generalization of several known classes of posets, e.g. posets of numerical semigroups (with torsion) and more generally affine semigroups. Furt...

Given a linear code $\mathcal{C}$, its square code $\mathcal{C}^{(2)}$ is the span of all component-wise products of two elements of $\mathcal{C}$. Motivated by applications in multi-party computation, our purpose with this work is to answer the following question: which families of affine variety codes have simultaneously high dimension $k(\mathca...

Let R := K[x1,...,xn] be a polynomial ring over an infinite field K, and let I ⊂ R be a homogeneous ideal with respect to a weight vector ω = (ω1, ..., ωn) ∈ (Z⁺)ⁿ such that dim(R/I) = d. We consider the minimal graded free resolution of R/I as A-module, that we call the Noether resolution of R/I, whenever A := K[xn-d+1,...,xn] is a Noether normali...

A sum of affine powers is an expression of the form [f(x1,...,xn) = ∑i=1s αi li(x1,...,xn)ei] where li is an affine form. We propose polynomial time black-box algorithms that find the decomposition with the smallest value of s for an input polynomial f . Our algorithms work in situations where s is small enough compared to the number of variables o...

A sum of affine powers is an expression of the form f(x) = s∑/i=1 αi (x - ai)ei. Although quite simple, this model is a generalization of two well-studied models: Waring decomposition and Sparsest Shift. For these three models there are natural extensions to several variables, but this paper is mostly focused on univariate polynomials. We propose a...

Let $K$ be an infinite field and let $m_1,\ldots,m_n$ be a generalized arithmetic sequence of positive integers, i.e., there exist $h, d, m_1 \in\mathbb{Z}^+$ such that $m_i = h m_1 + (i-1)d$ for all $i \in \{2,\ldots,n\}$. We consider the projective monomial curve $\mathcal C\subset \mathbb{P}^{n}_{K}$ parametrically defined by $$x_1=s^{m_1}t^{m_n...

We consider the two-player game chomp on posets associated to numerical semigroups and show that the analysis of strategies for chomp is strongly related to classical properties of semigroups. We characterize, which player has a winning-strategy for symmetric semigroups, semigroups of maximal embedding dimension and several families of numerical se...

We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to give a lower bound on the dimension of the space of polynomials spanned by $F$. In particular, we give simple...

Let $R:= K[x_1,\ldots,x_{n}]$ be a polynomial ring over an infinite field $K$, and let $I \subset R$ be a homogeneous ideal with respect to a weight vector $\omega = (\omega_1,\ldots,\omega_n) \in (\mathbb{Z}^+)^n$ such that $\dim(R/I) = d$. In this paper we study the minimal graded free resolution of $R/I$ as $A$-module, that we call the Noether r...

A graph has strong convex dimension 2 if it admits a straight-line drawing in the plane such that its vertices form a convex set and the midpoints of its edges also constitute a convex set. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension 2 are planar and therefore have at most 3n - 6 edges. We prove that all such graphs...

The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero mono...

This paper is a continuation of the paper “Numerical Semigroups: Apéry Sets and Hilbert Series". We consider the general numerical AA-semigroup, i.e., semigroups consisting of all non-negative integer linear combinations of relatively prime positive integers of the form $a, a+d, a+2d,\ldots,a+kd, c$. We first prove that, in contrast to arbitrary nu...

In this paper, we investigate the Möbius function $\mu_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. We introduce and develop a new approach to study $\mu_{\mathcal{S}}$ by using the Hilbert series of $\mathcal{S}$. The latter enables us to provide formulas for $\mu_{\mathcal{S}}$ wh...

A graph has strong convex dimension $2$, if it admits a straight-line drawing in the plane such that its vertices are in convex position and the midpoints of its edges are also in convex position. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension $2$ are planar and therefore have at most $3n-6$ edges. We prove that all su...

In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a representation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower...

One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \sum_{i = 0}^d a_i X^i \in \mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition $a_i^2 > \tau a_{i-1}a_{i+1}$ for every $i \in \{1,\ldots,d-1\},$ where $\tau > 0$. Whene...

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph $G$, checks whether its toric ideal $P_G$ is a complete intersection or not. Whenever $P_G$ is a complete intersection, the...

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators...

Let $k$ be an arbitrary field, the purpose of this work is to provide families of positive integers $\mathcal{A} = \{d_1,\ldots,d_n\}$ such that either the toric ideal $I_{\mathcal A}$ of the affine monomial curve $\mathcal C = \{(t^{d_1},\ldots,\,t^{d_n}) \ | \ t \in k\} \subset \mathbb{A}_k^n$ or the toric ideal $I_{\mathcal A^{\star}}$ of its pr...

Given a set $\mathcal A = \{a_1,...,a_n\} \subset \N^m$ of nonzero vectors defining a simplicial toric ideal $I_{\mathcal A} \subset k[x_1,...,x_n]$, where $k$ is an arbitrary field, we provide an algorithm for checking whether $I_{\mathcal A}$ is a complete intersection or not. This algorithm does not require the explicit computation of a minimal...

Some changes of the traditional scheme for finding rational solutions of linear differential, difference and q-difference homogeneous equations with rational coefficients are proposed. In many cases these changes allow one to predict the absence ...

Let K be an arbitrary field and {d1, . . . , dn} a set of all-different positive integers. The aim of this work is to propose and evaluate an algo- rithm for checking whether or not the toric ideal of the affine monomial curve {(td 1 , . . . , td n ) | t 2 K} � AnK is a complete intersection. The algorithm is based on new results regarding the tori...