On the Abhyankar–Moh irreducibility criterion for quasi-ordinary polynomials
Volume 164 / 2021
Colloquium Mathematicum 164 (2021), 149-160 MSC: Primary 32S55; Secondary 14H20. DOI: 10.4064/cm8116-2-2020 Published online: 18 September 2020
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.