Cyclic resultants

Let k be a field of characteristic zero and let f ∈ k[x]. The m-th cyclic resultant of f is rm = Res(f, x − 1). We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial f of degree d, there are exactly 2 distinct degree d polynomials with the same set of cyclic resultants as f . However, in the generic monic case, degree d polynomials are uniquely determined by their cyclic resultants. Moreover, two reciprocal (“palindromic”) polynomials giving rise to the same set of nonzero rm are equal. The reciprocal case was stated many years ago (for k = ) and has many applications stemming from such disparate fields as dynamics, number theory, and Lagrangian mechanics. In the process, we also prove a unique factorization result in semigroup algebras involving products of binomials.