2004/11/18 by Christopher J. Hillar, Lionel Levine, Hillar, Christopher J. +1
Mathematics · #11B37 #14Q99 (primary) #15A15 #20M25 (secondary) #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Algebra over a field #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Mathematical analysis #Mathematics #Polynomial #Pure mathematics #math.AG #math.CO #msc:11B37 #msc:14Q99 #msc:15A15 #msc:20M25
paper · pdf · doi:10.48550/arxiv.math/0411414
Proceedings of the AMS
openalex publication_date 2004/11/18 · arxiv created 2006/11/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let K be an algebraically closed field of characteristic zero and let f ∈ K[x]. The m-th \it cyclic resultant of f is rm = Res(f,xm-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 2d+1 cyclic resultants and that a generic monic reciprocal polynomial of even degree d is determined by its first 2⋅ 3d/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.