Definite regular quadratic forms over

Authors:
Wai Kiu Chan and Joshua Daniels

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3121-3131

MSC (2000):
Primary 11E12, 11E20

DOI:
https://doi.org/10.1090/S0002-9939-05-08197-9

Published electronically:
June 20, 2005

MathSciNet review:
2160173

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a power of an odd prime, and be the ring of polynomials over a finite field of elements. A quadratic form over is said to be regular if globally represents all polynomials that are represented by the genus of . In this paper, we study definite regular quadratic forms over . It is shown that for a fixed , there are only finitely many equivalence classes of regular definite primitive quadratic forms over , regardless of the number of variables. Characterizations of those which are universal are also given.

**1.**W. K. Chan and A. G. Earnest,*Discriminant bounds for spinor regular ternary quadratic lattices*, J. London Math. Soc. (2)**69**(2004), 545-561. MR**2048511 (2005b:11042)****2.**W. K. Chan, A. G. Earnest and B.-K. Oh,*Regularity properties of positive definite integral quadratic forms*, Contemporary Math. Amer. Math. Soc.,**344**(2004), 59-71. MR**2058667 (2005c:11043)****3.**W. K. Chan and B.-K. Oh,*Finiteness theorems for positive definite**-regular quadratic forms*, Trans. Amer. Math. Soc.,**355**(2003), 2385-2396. MR**1973994 (2004i:11032)****4.**W. K. Chan and B.-K. Oh,*Positive ternary quadratic forms with finitely many exceptions*, Proc. Amer. Math. Soc.,**132**(2004), 1567-1573. MR**2051115 (2005a:11044)****5.**L. E. Dickson,*Ternary quadratic forms and congruences*, Ann. of Math.**28**(1927), 333-341. MR**1502786****6.**D. Z. Djokovic,*Hermitian matrices over polynomial rings*, J. Algebra**43**(1976), 359-374. MR**0437565 (55:10489)****7.**A. G. Earnest,*An application of character sum inequalities to quadratic forms*, Number Theory, Canadian Math. Soc. Conference Proceedings**15**(1995), 155-158. MR**1353928 (96j:11044)****8.**M. Fried and M. Jarden,*Field Arithmetic*, Springer Verlag, New York, 1986. MR**0868860 (89b:12010)****9.**L. Gerstein,*Definite quadratic forms over*, J. Algebra,**268**(2003), 252-263. MR**2005286 (2004f:11032)****10.**L. Gerstein,*On representation by quadratic**-lattices*, Contemporary Math. Amer. Math. Soc.,**344**(2004), 129-134. MR**2058672****11.**W. Jagy, I. Kaplansky and A. Schiemann,*There are 913 regular ternary forms*, Mathematika,**44**(1997), 332-341. MR**1600553 (99a:11046)****12.**Y. Kitaoka,*Arithmetic of quadratic forms*, Cambridge University Press, Cambridge, 1999. MR**1245266 (95c:11044)****13.**U. Korte,*Class numbers of definite binary quadratic lattices over algebraic function fields*, J. Number Theory**19**(1984), 33-39. MR**0751162 (85m:11023)****14.**M. H. Kim, Y. Wang and F. Xu,*Universal quadratic forms over*, preprint.**15.**O. T. O'Meara,*Introduction to quadratic forms*, Springer Verlag, New York, 1963. MR**0152507 (27:2485)****16.**M. Rosen,*Number theory in function fields*, Springer Verlag, New York, 2001. MR**1876657 (2003d:11171)****17.**G. L. Watson,*Some problems in the theory of numbers*, Ph.D. thesis, University College, London (1953).**18.**G. L. Watson,*The representation of integers by positive ternary quadratic forms*, Mathematika**1**(1954), 104-110. MR**0067162 (16:680c)**

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Additional Information

**Wai Kiu Chan**

Affiliation:
Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459

Email:
wkchan@wesleyan.edu

**Joshua Daniels**

Affiliation:
Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459

Address at time of publication:
2920 Deakin Street #1, Berkeley, California 94705

Email:
jdaniels@wesleyan.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-08197-9

Received by editor(s):
May 21, 2004

Published electronically:
June 20, 2005

Additional Notes:
The research of the first author was partially supported by the National Security Agency and the National Science Foundation.

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.