A nullstellensatz for sequences over

Let p be a prime and let A = ( a 1 ,..., a ℓ ) be a sequence of nonzero elements in . In this paper, we study the set of all 0–1 solutions to the equation 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.