Representations of Definite Binary Quadratic Forms over F_q[t]

TL;DR

This paper investigates binary definite quadratic forms over polynomial rings over finite fields, showing they are determined by low-degree polynomial representations and classifying forms with class number one for large q.

Contribution

It proves that such forms are uniquely determined by their low-degree represented polynomials and characterizes forms with class number one for q > 13.

Findings

Forms are determined by polynomials up to degree 3m-2

Classification of forms with class number one for q > 13

Complete characterization of definite binary quadratic forms over F_q[t]

Abstract

In this paper, we prove that a binary definite quadratic form over F_q[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m-2, where m is the degree of its discriminant. We also characterize, when q>13, all the definite binary forms over F_q[t] that have class number one.