Definite regular quadratic forms over 𝔽_{𝕢}[𝕋]

Chan, Wai Kiu; Daniels, Joshua

doi:10.1090/S0002-9939-05-08197-9

Definite regular quadratic forms over $\mathbb F_q[T]$
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by Wai Kiu Chan and Joshua Daniels

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

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

Published electronically: June 20, 2005

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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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