Polynomials with restricted support and maximal zeros on a finite Cartesian set
Abstract
Given a finite Cartesian set S=X × Y and a decreasing set of monomials ℳ, we call extremal polynomials for ℳ over S those that have the maximum number of zeros in S and whose support belongs to ℳ. Coordinate factorizations give a family of extremal polynomials; we call them canonical. If max(ℳ), taken with respect to divisibility, is a single monomial, all extremal polynomials are canonical. If |max(ℳ)|=2, either all extremal polynomials are canonical, or the problem reduces to the case where max(ℳ)={xdx,ydy}. 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.
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BibTeX
@misc{bhunia2026polynomials,
title = {{Polynomials with restricted support and maximal zeros on a finite Cartesian set}},
author = {Dipak K. Bhunia and Eduardo Camps-Moreno and Ignacio GarcÍa-Marco and Hiram H. López and Irene Márquez-Corbella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05813},
eprint = {2610.05813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05813},
}
RIS
TY - UNPB
TI - Polynomials with restricted support and maximal zeros on a finite Cartesian set
AU - Dipak K. Bhunia
AU - Eduardo Camps-Moreno
AU - Ignacio GarcÍa-Marco
AU - Hiram H. López
AU - Irene Márquez-Corbella
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05813
UR - https://arxiv.org/abs/2610.05813
AB - Given a finite Cartesian set S=X × Y and a decreasing set of monomials ℳ, we call extremal polynomials for ℳ over S those that have the maximum number of zeros in S and whose support belongs to ℳ. Coordinate factorizations give a family of extremal polynomials; we call them canonical. If max(ℳ), taken with respect to divisibility, is a single monomial, all extremal polynomials are canonical. If |max(ℳ)|=2, either all extremal polynomials are canonical, or the problem reduces to the case where max(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.
ER -