A polynomial analogue of Berggren’s theorem on Pythagorean triples

Abstract

Say that ( x , y , z ) (x, y, z) is a positive primitive integral Pythagorean triple if x , y , z x, y, z are positive integers without common factors satisfying x 2 + y 2 = z 2 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 ) (3, 4, 5) and ( 4 , 3 , 5 ) (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 ≠ 2 \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.