A polynomial analogue of Berggren's theorem on Pythagorean triples

TL;DR

This paper extends Berggren's theorem from integer Pythagorean triples to polynomial rings over fields of characteristic not equal to 2, revealing new structure theorems for the orthogonal group related to the Pythagorean form.

Contribution

It introduces a polynomial analogue of Berggren's theorem, providing a novel framework for understanding Pythagorean triples in polynomial rings and their associated orthogonal groups.

Findings

Established a polynomial version of Berggren's theorem.

Derived structure theorems for the orthogonal group over polynomial rings.

Connected classical Pythagorean triples to polynomial algebra structures.

Abstract

Say that $(x, y, z)$ is a positive primitive integral Pythagorean triple if $x, y, z$ are positive integers without common factors satisfying $x^2 + y^2 = z^2$. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on $(3, 4, 5)$ and $(4, 3, 5)$ generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren's theorem in the context of a one-variable polynomial ring over a field of characteristic $\neq 2$. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.