A polynomial analogue of Berggren's theorem on Pythagorean triples

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 $\ne 2$. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.

arXiv commentA version of this paper will appear in the Proceedings of the American Mathematical Society