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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Polynomial recurrences and cyclic resultants


Authors: Christopher J. Hillar and Lionel Levine
Journal: Proc. Amer. Math. Soc. 135 (2007), 1607-1618
MSC (2000): Primary 11B37, 14Q99; Secondary 15A15, 20M25
Published electronically: December 29, 2006
MathSciNet review: 2286068
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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

$\displaystyle r_m =$   Res$\displaystyle (f,x^m-1).$

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.


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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
Email: levine@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08672-2
PII: S 0002-9939(06)08672-2
Keywords: Cyclic resultants, linear recurrence, polynomial recurrence, semigroup algebra, Toeplitz determinant, topological dynamics, Vandermonde determinant
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
Article copyright: © Copyright 2006 American Mathematical Society