On Abhyankar's irreducibility criterion for quasi-ordinary polynomials

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.