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

Let p be a prime and let A = (a 1,...,a l ) 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