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

  • Authors:

  • André E. Kézdy;Hunter S. Snevily

  • Affiliations:

  • Department of Mathematics, University of Louisville, Louisville, Kentucky 40292, USA (e-mail: kezdy@louisville.edu);Department of Mathematics, University of Idaho, Moscow, Idaho 83844, USA (e-mail: snevily@uidaho.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2004

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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