A nullstellensatz for sequences over $\mathbb{F}_p $.

Let p be a prime and let A = ( a,..., a) be a sequence of nonzero elements in $\mathbb{F}_p $. In this paper, we study the set of all 0-1 solutions to the equation $$a_1 x_1 + \cdots + a_\ell x_\ell = 0$$ 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. [ABSTRACT FROM AUTHOR]