Bounded reduction of orthogonal matrices over polynomial rings

TL;DR

This paper proves that matrices in the split orthogonal group over polynomial rings can be reduced to smaller matrices using a bounded number of elementary transformations, providing an explicit bound and extending classical group results.

Contribution

It offers an effective reduction method for orthogonal matrices over polynomial rings, completing the understanding for split classical groups.

Findings

Bounded reduction of orthogonal matrices established

Explicit bounds for transformations provided

Extends classical results to orthogonal groups

Abstract

We prove that a matrix from the split orthogonal group over a polynomial ring with coefficients in a small-dimensional ring can be reduced to a smaller matrix by a bounded number of elementary orthogonal transformations. The bound is given explicitly. This result is an effective version of the early stabilisation of the orthogonal K1 functor proven by Suslin and Kopeiko. Since the similar effective results for special linear and symplectic groups were obtained by Vaserstein, the present paper closes the problem for split classical groups.