On the Moore determinant

Abstract

The existence of certain ${\mathbb{F}}_q$-spaces of differential forms of the projective line over a field $K$ containing ${\mathbb{F}}_q$ 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 ${\mathbb{F}}_q$-vector space of the roots of an ${\mathbb{F}}_q$-linear polynomial and that of the roots of its reversed polynomial.