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

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.