Polynomials that Vanish on Distinct $n$th Roots of Unity

Abstract

Let ${\bf C}$ denote the field of complex numbers and $\Omega_n$ the set of $n$th roots of unity. For $t = 0,\ldots,n-1$, define the ideal $\Im(n,t+1) \subset {\bf C}[x_0,\ldots,x_{t}]$ consisting of those polynomials in $t+1$ variables that vanish on distinct $n$th roots of unity; that is, $f \in \Im(n,t+1)$ if and only if $f(\omega_0,\ldots,\omega_{t}) = 0$ for all $(\omega_0,\ldots,\omega_{t}) \in \Omega_n^{t+1}$ satisfying $\omega_i \neq \omega_j$, for $0 \le i < j \le t$.

In this paper we apply Gröbner basis methods to give a Combinatorial Nullstellensatz characterization of the ideal $\Im(n,t+1)$. In particular, if $f \in {\bf C}[x_0,\ldots,x_{t}]$, then we give a necessary and sufficient condition on the coefficients of $f$ for membership in $\Im(n,t+1)$.