Polynomial recurrences and cyclic resultants
Christopher J. Hillar, Lionel Levine · Proceedings of the American Mathematical Society · 2006
Let K K be an algebraically closed field of characteristic zero and let f ∈ K [ x ] f \in K[x] . The m m -th cyclic resultant of f f is \[ r m = Res ( f , x m − 1 ) . r_m = \text {Res}(f,x^m-1). \] 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 d is determined by its first 2 d + 1 2^{d+1} cyclic resultants and that a generic monic reciprocal polynomial of even degree d d is determined by its first 2 ⋅ 3 d / 2 2\cdot 3^{d/2} of them. In addition, we show that cyclic resultants satisfy a polynomial recurrence of length d + 1 d+1 . This result gives evidence supporting the conjecture of Sturmfels and Zworski that d + 1 d+1 resultants determine f 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.