Polynomial recurrences and cyclic resultants
| dc.creator | Hillar, Christopher J. | |
| dc.creator | Levine, Lionel | |
| dc.date | 2004-11-18 | |
| dc.date | 2006-11-07 | |
| dc.date.accessioned | 2026-07-07T06:39:02Z | |
| dc.date.available | 2026-07-07T06:39:02Z | |
| dc.description | Let $K$ be an algebraically closed field of characteristic zero and let $f \in K[x]$. The $m$-th {\it cyclic resultant} of $f$ is \[r_m = \text{Res}(f,x^m-1).\] 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 $d$ 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 $2\cdot 3^{d/2}$ 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. | |
| dc.description | Proceedings of the AMS | |
| dc.identifier | https://arxiv.org/abs/math/0411414 | |
| dc.identifier | http://arxiv.org/abs/math/0411414 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100945 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Combinatorics | |
| dc.subject | 11B37, 14Q99 (primary), 15A15, 20M25 (secondary) | |
| dc.title | Polynomial recurrences and cyclic resultants | |
| dc.type | text |