= Zero-sum sequences as differences of permutations of a prime field
Every length-$p$ sequence in the <prime field> $\mathbb F_p$ with sum zero, including sequences with repetitions, is the coordinatewise difference of two enumerations of that field. Apply the <Combinatorial Nullstellensatz> on $(\mathbb F_p^*)^{p-1}$ to the product of the <Vandermonde determinant> in variables $x_i$ and its translate by the first $p-1$ sequence terms. The <Dyson constant-term identity> gives a nonzero top coefficient, and the zero-sum condition supplies the final missing field element. The argument includes $p=2$.
Back to article page
