arXiv subjects

Zachary Flores

Publications and source records attributed to Zachary Flores.

8 recordsLinked to original sources

arXiv2024

Symmetry, Unimodality, and Lefschetz Properties for Graded Modules

We investigate the Weak Lefschetz Properties for modules whose minimal free resolutions are given by generalized Kosuzl complexes in dimension three through a careful study of their Betti numbers and the symmetry and unimodality of their Hilbert functions. We also study the non-Lefschetz locus for finite length modules in arbitrary dimension, and are able to generalize several previous results on the non-Lefschetz locus in this setting. Along the way, we find several connections with a Gorenstein analogue for finite length modules and Artin level modules that are both interesting and useful throughout this paper.

math.AC↗

arXiv2024

A Construction of Optimal Quasi-cyclic Locally Recoverable Codes using Constituent Codes

A locally recoverable code of locality $r$ over $\mathbb{F}_{q}$ is a code where every coordinate of a codeword can be recovered using the values of at most $r$ other coordinates of that codeword. Locally recoverable codes are efficient at restoring corrupted messages and data which make them highly applicable to distributed storage systems. Quasi-cyclic codes of length $n=m\ell$ and index $\ell$ are linear codes that are invariant under cyclic shifts by $\ell$ places. %Quasi-cyclic codes are generalizations of cyclic codes and are isomorphic to $\mathbb{F}_{q} [x]/ \langle x^m-1 \rangle$-submodules of $\mathbb{F}_{q^\ell} [x] / \langle x^m-1 \rangle$. In this paper, we decompose quasi-cyclic locally recoverable codes into a sum of constituent codes where each constituent code is a linear code over a field extension of $\mathbb{F}_q$. Using these constituent codes with set parameters, we propose conditions which ensure the existence of almost optimal and optimal quasi-cyclic locally recoverable codes with increased dimension and code length.

cs.IT↗

arXiv2023

A Formalization of Operads in Coq

What provides the highest level of assurance for correctness of execution within a programming language? One answer, and our solution in particular, to this problem is to provide a formalization for, if it exists, the denotational semantics of a programming language. Achieving such a formalization provides a gold standard for ensuring a programming language is correct-by-construction. In our effort on the DARPA V-SPELLS program, we worked to provide a foundation for the denotational semantics of a meta-language using a mathematical object known as an operad. This object has compositional properties which are vital to building languages from smaller pieces. In this paper, we discuss our formalization of an operad in the proof assistant Coq. Moreover, our definition within Coq is capable of providing proofs that objects specified within Coq are operads. This work within Coq provides a formal mathematical basis for our meta-language development within V-SPELLS. Our work also provides, to our knowledge, the first known formalization of operads within a proof assistant that has significant automation, as well as a model that can be replicated without knowledge of Homotopy Type Theory.

math.CT↗

arXiv2023

A Formal Algebraic Framework for DSL Composition

We discuss a formal framework for using algebraic structures to model a meta-language that can write, compose, and provide interoperability between abstractions of DSLs. The purpose of this formal framework is to provide a verification of compositional properties of the meta-language. Throughout our paper we discuss the construction of this formal framework, as well its relation to our team's work on the DARPA V-SPELLS program via the pipeline we have developed for completing our verification tasking on V-SPELLS. We aim to give a broad overview of this verification pipeline in our paper. The pipeline can be split into four main components: the first is providing a formal model of the meta-language in Coq; the second is to give a specification in Coq of our chosen algebraic structures; third, we need to implement specific instances of our algebraic structures in Coq, as well as give a proof in Coq that this implementation is an algebraic structure according to our specification in the second step; and lastly, we need to give a proof in Coq that the formal model for the meta-language in the first step is an instance of the implementation in the third step.

math.CT↗

arXiv2020

The apolar algebra of a product of linear forms

Apolarity is an important tool in commutative algebra and algebraic geometry which studies a form, $f$, by the action of polynomial differential operators on $f$. The quotient of all polynomial differential operators by those which annihilate $f$ is called the apolar algebra of $f$. In general, the apolar algebra of a form is useful for determining its Waring rank, which can be seen as the problem of decomposing the supersymmetric tensor, associated to the form, minimally as a sum of rank one supersymmetric tensors. In this article we study the apolar algebra of a product of linear forms, which generalizes the case of monomials and connects to the geometry of hyperplane arrangements. In the first part of the article we provide a bound on the Waring rank of a product of linear forms under certain genericity assumptions; for this we use the defining equations of so-called star configurations due to Geramita, Harbourne, and Migliore. In the second part of the article we use the computer algebra system Bertini, which operates by homotopy continuation methods, to solve certain rank equations for catalecticant matrices. Our computations suggest that, up to a change of variables, there are exactly six homogeneous polynomials of degree six in three variables which factor completely as a product of linear forms defining an irreducible multi-arrangement and whose apolar algebras have dimension six in degree three. As a consequence of these calculations, we find six cases of such forms with cactus rank six, five of which also have Waring rank six. Among these are products defining subarrangements of the braid and Hessian arrangements.

math.AC↗

arXiv2019

A generalization of Wilf's conjecture for Generalized Numerical Semigroups

A numerical semigroup is a submonoid of $\mathbb N$ with finite complement in $\mathbb N$. A generalized numerical semigroup is a submonoid of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. In the context of numerical semigroups, Wilf's conjecture is a long standing open problem whose study has led to new mathematics and new ways of thinking about monoids. A natural extension of Wilf's conjecture, to the class of $\mathcal C$-semigroups, was proposed by García-García, Marín-Aragón, and Vigneron-Tenorio. In this paper, we propose a different generalization of Wilf's conjecture, to the setting of generalized numerical semigroups, and prove the conjecture for several large families including the irreducible, symmetric, and monomial case. We also discuss the relationship of our conjecture to the extension proposed by García-García, Marín-Aragón, and Vigneron-Tenorio.

math.CO↗

arXiv2019

G-groups of Cohen-Macaulay Rings with $n$-Cluster Tilting Objects

Let $(R, \mathfrak{m}, k)$ denote a local Cohen-Macaulay ring such that the category of maximal Cohen-Macaulay $R$-modules $\textbf{mcm}\ R$ contains an $n$-cluster tilting object $L$. In this paper, we compute $G_1(R) := K_1(\textbf{mod}\ R)$ explicitly as a direct sum of a free group and a specified quotient of $\text{aut}_R(L)_{\text{ab}}$ when $R$ is a $k$-algebra and $k$ is algebraically closed (and $\text{char}(k)\neq 2$). Moreover, we give some explicit computations of $\text{aut}_R(L)_{\text{ab}}$ and $G_1(R)$ for certain hypersurface singularities.

math.AC↗

arXiv2018

On the Weak Lefschetz Property for Vector Bundles on $\mathbb P^2$

Let $R=\mathbb K[x,y,z]$ be a standard graded polynomial ring where $\mathbb K$ is an algebraically closed field of characteristic zero. Let $M = \oplus_j M_j$ be a finite length graded $R$-module. We say that $M$ has the Weak Lefschetz Property if there is a homogeneous element $L$ of degree one in $R$ such that the multiplication map $\times L : M_j \rightarrow M_{j+1}$ has maximal rank for every $j$. The main result of this paper is to show that if $\mathcal E$ is a locally free sheaf of rank 2 on $\mathbb P^2$ then the first cohomology module of $\mathcal E$, $H^1_*(\mathbb P^2, \mathcal E)$, has the Weak Lefschetz Property.

math.AG↗