On the Abhyankar–Moh irreducibility criterion for quasi-ordinary polynomials
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 the degree of $g$ is small enough and all monomials appearing in the resultant of $f$ and $g$ have orders large enough, then $g$ is irreducible and quasi-ordinary. This generalizes the Abhyankar–Moh irreducibility criterion for plane analytic curves.