Polynomial recurrences and cyclic resultants

Christopher J. Hillar, Lionel Levine
2006 Proceedings of the American Mathematical Society  
Let K be an algebraically closed field of characteristic zero and let f ∈ K [x]. The m-th cyclic resultant of f is 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
more » ... m. 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. Let Per m (T ) = {x ∈ T d : T m (x) = x} be the set of points on the torus fixed by the map T m . Under the ergodicity condition that no eigenvalue of A is a root of unity, it follows (see [5] ) that #Per m (T ) = | det(A m − Id)| = |r m (f )|, 1991 Mathematics Subject Classification. Primary 11B37, 14Q99; Secondary 15A15, 20M25.
doi:10.1090/s0002-9939-06-08672-2 fatcat:navbx5cxjfemddbjeq4zd3wfsq