arXiv subjects

Zafeirakis Zafeirakopoulos

Publications and source records attributed to Zafeirakis Zafeirakopoulos.

6 recordsLinked to original sources

arXiv2026

Resultant multiplicity via projective degrees and applications to tensor eigenvalues

Given a system $\mathbf{f}=(f_1,\ldots,f_n)$ of $n$ homogeneous forms in $n$ variables of the same degree, Macaulay's resultant vanishes precisely when the polynomials have a common projective zero. Its order of vanishing measures the singularity of the resultant hypersurface at $\mathbf{f}$. In this paper, we study how this multiplicity reflects the geometry of the projective zero scheme defined by $\mathbf{f}$. We give an exact formula for the multiplicity, expressed in terms of the projective degrees of the rational map defined by $\mathbf{f}$. As a consequence, we obtain a geometric lower bound involving the degrees, dimensions, and multiplicities of the irreducible components of the projective zero scheme. This extends the multiplicity estimates of Roy and Ghidelli from zero-dimensional schemes to schemes of arbitrary dimension. Finally, we apply this geometric estimate to tensor eigenvalues. It translates directly into a lower bound for the algebraic multiplicity of a tensor eigenvalue in terms of the geometry of its eigenscheme. This settles a conjecture by Canino et al. and consequently settles earlier conjectures of Qi and of Hu and Ye concerning the relationship between algebraic, geometric, and span multiplicities of tensor eigenvalues.

math.AG↗

arXiv2026

Small Resultant Systems via Linear Combinations

For a system of $s$ homogeneous polynomials of degree $d$ in $n$ variables, say ${\bf{f}} = 0$, we consider the problem of constructing resultant systems. A resultant system is a finite set of polynomials in the coefficients of the input polynomials, the vanishing of which characterizes the systems $\bf{f}$ with a common non-zero solution. The classical approaches for constructing resultant systems rely either on maximal minors of large coefficient matrices or on the coefficients of a resultant of generic linear combinations of the input polynomials. Typically, they produce resultant systems containing a very large number of polynomials. We develop new constructions based on taking resultants of linear combinations of the input polynomials; this results in resultant systems of small cardinality. Our main results are: 1) We prove that a resultant system with ${d+n-1 \choose n-1} s-n^2+1$ polynomials exists; each polynomial is the resultant of $n$ linear combinations of the input polynomials. This improves the previously known upper bounds, even for systems of bivariate homogeneous polynomials. 2) Under the assumption that the input polynomials are non-zero, we construct explicit resultant systems with cardinality $\mathrm{poly}(s,d)$, when $n$ is fixed.

math.AC↗

arXiv2017

Generating functions and triangulations for lecture hall cones

We investigate the arithmetic-geometric structure of the lecture hall cone \[ L_n \ := \ \left\{λ\in \mathbb{R}^n: \, 0\leq \frac{λ_1}{1}\leq \frac{λ_2}{2}\leq \frac{λ_3}{3}\leq \cdots \leq \frac{λ_n}{n}\right\} . \] We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial is given by the $(n-1)$st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $L_n$, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of $L_n$, including connections between enumerative and algebraic properties of $L_n$ and cones over unit cubes.

math.CO↗

arXiv2015

On Computing the Elimination Ideal Using Resultants with Applications to Gröbner Bases

Resultants and Gröbner bases are crucial tools in studying polynomial elimination theory. We investigate relations between the variety of the resultant of two polynomials and the variety of the ideal they generate. Then we focus on the bivariate case, in which the elimination ideal is principal. We study - by means of elementary tools - the difference between the multiplicity of the factors of the generator of the elimination ideal and the multiplicity of the factors of the resultant.

math.AC↗

arXiv2015

Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems

Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.

math.CO↗

arXiv2013

s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones

In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$. We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ for at most finitely many n. We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of h*-vectors for s-lecture hall polytopes. We end with open questions and directions for further research.

math.CO↗