On Abhyankar's irreducibility criterion for quasi-ordinary polynomials

TL;DR

This paper generalizes Abhyankar's irreducibility criterion to higher dimensions, showing that under certain conditions, a polynomial g is irreducible and quasi-ordinary if it interacts with an irreducible quasi-ordinary polynomial f in a specific way.

Contribution

The paper extends Abhyankar's criterion from plane curves to quasi-ordinary polynomials in several variables, providing new conditions for irreducibility and quasi-ordinarity.

Findings

g is irreducible and quasi-ordinary under the given conditions

Resultant monomials have large orders when g is irreducible

Degree of g must be sufficiently small for the criterion to hold

Abstract

Let $f$ and $g$ be Weierstrass polynomials with coefficients in the ring of formal power series over an algebraically closed field of characteristic zero. Assume that $f$ is irreducible and quasi-ordinary. We show that if degree of $g$ is small enough and all monomials appearing in the resultant of $f$ and $g$ have orders big enough, then $g$ is irreducible and quasi-ordinary, generalizing Abhyankar's irreducibility criterion for plane analytic curves.