Polynomial recurrences and cyclic resultants
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- by Christopher J. Hillar and Lionel Levine PDF
- Proc. Amer. Math. Soc. 135 (2007), 1607-1618 Request permission
Abstract:
Let $K$ be an algebraically closed field of characteristic zero and let $f \in K[x]$. The $m$-th 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.References
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Additional Information
- Christopher J. Hillar
- Affiliation: Department of Mathematics, Texas A & M University, College Station, TX 77843
- Email: chillar@math.tamu.edu
- Lionel Levine
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- MR Author ID: 654666
- Email: levine@math.berkeley.edu
- Received by editor(s): November 23, 2004
- Received by editor(s) in revised form: February 8, 2006
- Published electronically: December 29, 2006
- Additional Notes: Both authors were supported under a NSF Graduate Research Fellowship.
- Communicated by: Bernd Ulrich
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 1607-1618
- MSC (2000): Primary 11B37, 14Q99; Secondary 15A15, 20M25
- DOI: https://doi.org/10.1090/S0002-9939-06-08672-2
- MathSciNet review: 2286068