On the bilinear structure associated to Bezoutians

TL;DR

This paper explores a twisted form of Bezoutian, called skew Bezoutian, which links polynomial pairs to symplectic and quadratic spaces, revealing their isometry groups and enabling explicit construction of isometries with specified invariants.

Contribution

It introduces the skew Bezoutian, a novel construction connecting reciprocal polynomial pairs to symplectic and quadratic spaces, and analyzes their isometry groups and invariants.

Findings

The isometry group contains a hypergeometric subgroup.

Explicit isometries can be constructed with prescribed invariants.

The skew Bezoutian generalizes classical Bezoutian concepts.

Abstract

This paper is partly a survey of known results on quadratic forms that are hard to find in the literature. Our main focus is a twisted form of a construction due to Bezout. This skew Bezoutian is a symplectic (resp. quadratic) space associated to a pair of reciprocal (or skew reciprocal) coprime polynomials of same degree. The isometry group of this space turns out to contain a certain associated hypergeometric group. Using the skew Bezoutian we construct explicit isometries of bilinear spaces with given invariants (such as the characteristic polynomial or Jordan form and, in the quadratic case, the spinor norm).