Work

Zero testing and equation solving for sparse polynomials on rectangular domains

Year: 2024

Type: article

Abstract: We consider sparse polynomials in N variables over a finite field, and ask whether they vanish on a set SN, where S is a set of nonzero elements of the field. We see that if for a polynomial f, there is with , then there is such a in every ball (with respect to the Hamming distance) inside SN, where the radius of the ball is bounded by a multiple of the logarithm of the number of monomials that appear in f. A similar result holds for the solutions of the equations f1=⋯=fr=0 inside SN. When 0∉S, this provides algorithms for testing whether a given polynomial f vanishes on SN and for checking whether f1=⋯=fr=0 has a solution in SN that are of time complexity in O(nclog⁡(n)), where c>0 and n is the size of the input polynomials given in sparse representation.

Source: Finite Fields and Their Applications

DOI: 10.1016/j.ffa.2024.102379

Authors: Erhard Aichinger, Simon Grünbacher, Paul Hametner

Institution: Johannes Kepler University of Linz

Language: English

Topic: Polynomial and algebraic computation

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,640

Funder: Austrian Science Fund