Polynomial recurrences and cyclic resultants

Let K be an algebraically closed field of characteristic zero and let f is an element of 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 center dot 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.