Polynomials that Vanish on Distinct nth Roots of Unity

Let C denote the ̄eld of complex numbers and n the set of nth roots of unity. For t = 0; : : : ; n ¡ 1, de ̄ne the ideal =(n; t +1) 1⁄2 C [x0; : : : ; xt] consisting of those polynomials in t + 1 variables that vanish on distinct nth roots of unity; that is, f 2=(n; t+ 1) if and only if f (!0 ; : : : ; !t) = 0 for all (!0 ; : : : ; !t) 2 t+1 n satisfying !i 6= !j, for 0 · i < j · t. In this paper we apply GrÄ obner basis methods to give a Combinatorial Nullstellensatz characterization of the ideal =(n; t+1). In particular, if f 2 C [x0; : : : ; xt], then we give a necessary and su±cient condition on the coe±cients of f for membership in =(n; t +1).