A nullstellensatz for sequences over F_p

TL;DR

This paper explores the structure of solutions to linear equations over finite fields, establishing new bounds and characterizations for sequences of length at least the prime p, using algebraic and combinatorial methods.

Contribution

It proves that for sequences of length at least p, the solution set characterizes the sequence up to a scalar, and refines Olson's theorem for the case when length equals p.

Findings

For l >= p, solution sets characterize A up to scalar.

When l = p and A is nonconstant, at least p-1 minimal solutions exist.

Detailed analysis of the case l = p - 1.

Abstract

Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.