Polynomials with restricted support and maximal zeros on a finite Cartesian set

2026-10-05

AI Summary

This study addresses the problem of determining the maximum number of zeros of polynomials with restricted support over finite Cartesian sets, revealing that the existence of non-canonical extremal polynomials is more constrained than in the classical Prouhet-Tarry-Escott problem. By integrating algebraic combinatorics, coordinate decomposition, and elementary symmetric function analysis, this work rigorously proves that only canonical solutions arise under a single maximal element, whereas non-canonical solutions for two maximal elements require stricter symmetry conditions. Furthermore, it precisely quantifies the number of triples corresponding to quadratic non-canonical extremal polynomials and establishes their intrinsic connection to minimum-weight codewords in coding theory, thereby providing a novel paradigm for constructing such codewords.

Abstract

Given a finite Cartesian set $S=X \times Y$ and a decreasing set of monomials $\mathcal M$, we call extremal polynomials for $\mathcal M$ over $S$ those that have the maximum number of zeros in $S$ and whose support belongs to $\mathcal M$. Coordinate factorizations give a family of extremal polynomials; we call them canonical. If $\max(\mathcal M)$, taken with respect to divisibility, is a single monomial, all extremal polynomials are canonical. If $|\max(\mathcal M)|=2$, either all extremal polynomials are canonical, or the problem reduces to the case where $\max(\mathcal M)=\{x^{d_x},y^{d_y}\}$. In the latter case, we prove that the existence of noncanonical extremal polynomials depends on finding families of subsets whose elementary symmetric functions agree. This condition is more restrictive than the classical Prouhet--Tarry--Escott problem, which asks for two sets whose elementary symmetric functions agree. We determine the number of triples $(X, Y, h)$, where $h$ is a quadratic noncanonical extremal polynomial. We apply extremal polynomials to coding theory via minimum-weight codewords.

Problem

• extremal polynomials
• restricted support
• Cartesian set
• canonical factorization
• Prouhet-Tarry-Escott problem

Innovation

• Extremal polynomials
• Finite Cartesian set
• Coordinate factorization
• Elementary symmetric functions
• Coding theory