On M-O.Ore determinants

TL;DR

This paper explores determinants related to Moore matrices and their connection to differential forms on the projective line over a field containing Fq, revealing new identities and interpretations of Elkies pairing.

Contribution

It introduces a novel identity linking Moore matrix determinants with cofactors, and provides a new interpretation of Elkies pairing via residues of differential forms.

Findings

Established a determinant identity for Moore matrices and cofactors.

Linked Fq-spaces of differential forms to Elkies pairing.

Provided a duality interpretation for roots of Fq-linear polynomials.

Abstract

The existence of certain Fq-spaces of differential forms of the projective line over a field K containing Fq leads us to prove an identity linking the determinant of the Moore matrix of n indeterminates with the determinant of the Moore matrix of the cofactors of its first row. These same spaces give an interpretation of Elkies pairing in terms of residues of differential forms. This pairing puts in duality the Fq-vector space of the roots of a Fq-linear polynomial and that of the roots of its reversed polynomial.