Polynomial recurrences and cyclic resultants

Abstract

Let K be an algebraically closed field of characteristic zero and let f ∈ K[x]. The m-th cyclic resultant of f is r m = Res(f , x m - 1). A generic monic polynomial is determined by its full sequence of cyclic resultants; however, the known techniques proving this result give no effective computational bounds. We prove that a generic monic polynomial of degree d is determined by its first 2 d+1 cyclic resultants and that a generic monic reciprocal polynomial of even degree d is determined by its first 2 3 d/2 of them. In addition, we show that cyclic resultants satisfy a polynomial recurrence of length d+ 1. This result gives evidence supporting the conjecture of Sturmfels and Zworski that d + 1 resultants determine f. In the process, we establish two general results of independent interest: we show that certain Toeplitz determinants are sufficient to determine whether a sequence is linearly recurrent, and we give conditions under which a linearly recurrent sequence satisfies a polynomial recurrence of shorter length.