A nullstellensatz for sequences over \(\mathbb{F}_p\) (Q2341924)

Let \(p\) be a prime and \(A=(a_1,a_2,\ldots,a_{\ell})\) be a sequence of nonzero elements in \(\mathbb{F}_p\). Let \(S_A\) be the set of all \(0-1\) solutions to the equation \[ a_1x_1+\cdots+a_{\ell} x_{\ell}=0. \] Let \(\langle S_A\rangle\) be the span of \(S_A\) in \(\mathbb{F}_p^{\ell}\) and let the dimension of this subspace be denoted by \(\dim(A)\). The authors show that \(\dim(A)=\ell-1\) whenever \(\ell>p\). In particular, \(S_A\) characterizes \(A\) up to scalar multiples. In the case \(\ell=p\), there are exceptional sequences \(A\) such that \(\dim(A)<p-1\), but the authors characterize them completely. Namely, they either have \(\dim(A)=1\) and then \(A\) is a scalar multiple of \((1,1,\ldots,1)\), or \(\dim(A)=p-2\), in which case, up to a permutation of the indices, \(A\) is a scalar multiple of \((1,1,\ldots,1,-1,-1,\ldots,-1,-(t+1),-(t+1))\), where we have \(t\) copies of \(1\) for some \(t\in [1,p-3]\). They deduce that \(S_A\) determines \(A\) up to a scalar multiple in case \(\ell=p\) as well. They also study the case \(\ell=p-1\) in detail. The proofs use a Combinatorial Nullstellensatz due to Alon, the classical Cauchy-Davenport theorem, as well as a couple of Vosper type results.