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

DOI:
https://doi.org/10.1090/S0002-9939-06-08672-2

Published electronically:
December 29, 2006

MathSciNet review:
2286068

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an algebraically closed field of characteristic zero and let . The -th *cyclic resultant* of is

**1.**J. H. Conway and R. Guy,*The Book of Numbers*, Springer-Verlag, 1996. MR**1411676 (98g:00004)****2.**D. Cox, J. Little, and D. O'Shea,*Ideals, varieties, and algorithms*, second ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997.MR**1417938 (97h:13024)****3.**J.J. Duistermaat and V. Guillemin,*The spectrum of positive elliptic operators and periodic bicharacteristics*, Inv. Math. 25 (1975) 39-79.MR**0405514 (53:9307)****4.**S. Elaydi,*An Introduction to Difference Equations*, Springer, 1999.MR**1711587 (2001g:39001)****5.**G. Everest and T. Ward. Heights of Polynomials and Entropy in Algebraic Dynamics. Springer-Verlag London, Ltd., London, 1999. MR**1700272 (2000e:11087)****6.**D. Fried,*Cyclic resultants of reciprocal polynomials*, in Holomorphic Dynamics (Mexico 1986), Lecture Notes in Math. 1345, Springer-Verlag, 1988, 124-128. MR**0980956 (90h:57004)****7.**V. Guillemin,*Wave trace invariants*, Duke Math. J. 83 (1996), 287-352.MR**1390650 (97f:58131)****8.**C. Hillar,*Cyclic resultants*, J. Symb. Comp. 39 (2005), 653-669; erratum 40 (2005), 1126-1127.MR**2167674****9.**A. Iantchenko, J. Sjöstrand, and M. Zworski,*Birkhoff normal forms in semi-classical inverse problems*, Math. Res. Lett. 9 (2002), 337-362.MR**1909649 (2003f:35284)****10.**K. Kedlaya, Computational Complexity 15 (2006), 1-19. MR**2226067****11.**C. Krattenthaler, ``Advanced Determinant Calculus,''*Sem. Lothar. Combin.*42 (1999), Art. B42q. MR**1701596 (2002i:05013)****12.**E. Miller and B. Sturmfels,*Combinatorial Commutative Algebra*, Springer, 2004.MR**2110098 (2006d:13001)****13.**K. Purbhoo,*A Nullstellensatz For Amoebas*, preprint.**14.**W. H. Stevens,*Recursion formulas for some abelian knot invariants*, Journal of Knot Theory and Its Ramifications, Vol. 9, No. 3 (2000) 413-422.**15.**D. Zeilberger,*Dodgson's determinant-evaluation rule proved by two-timing men and women*, Elec. J. Comb. 4(2), 1997.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11B37,
14Q99,
15A15,
20M25

Retrieve articles in all journals with MSC (2000): 11B37, 14Q99, 15A15, 20M25

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:
https://doi.org/10.1090/S0002-9939-06-08672-2

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