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