B\'ezoutians and injectivity of polynomial maps

TL;DR

This paper establishes a connection between the constancy of Bézoutians and the injectivity of polynomial maps on rational points, providing new criteria for injectivity based on algebraic properties.

Contribution

It introduces conditions involving Bézoutians that determine when polynomial maps are injective on rational points, extending understanding of polynomial map injectivity criteria.

Findings

Injectivity on rational points linked to constant Bézoutian

Injectivity at a point characterized by reduced Bézoutian

Invertible Jacobian implies equivalence of injectivity and Bézoutian constancy

Abstract

We prove that an endomorphism $f$ of affine space is injective on rational points if its B\'ezoutian is constant. Similarly, $f$ is injective at a given rational point if its reduced B\'ezoutian is constant. We also show that if the Jacobian determinant of $f$ is invertible, then $f$ is injective at a given rational point if and only if its reduced B\'ezoutian is constant.