Quadratic form representations via generalized continuants

TL;DR

This paper extends Smith's palindromic continuant approach to represent proper binary quadratic forms over various Euclidean rings and introduces new deterministic algorithms for finding these representations.

Contribution

It generalizes Smith's method to broader rings and develops new algorithms for proper quadratic form representations.

Findings

Extended Smith's approach to Euclidean rings of integers and polynomials

Developed deterministic algorithms for quadratic form representations

Validated the methods over rings of odd characteristic fields

Abstract

H. J. S. Smith proved Fermat's two-square theorem using the notion of palindromic continuants. In this paper we extend Smith's approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of integers and rings of polynomials over fields of odd characteristic. Also, we present new deterministic algorithms for finding the corresponding proper representations.