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Definite regular quadratic forms over $\mathbb F_q[T]$


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


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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.

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