Approximate roots of quasi-ordinary polynomials
Volume 572 / 2022
Abstract
The work is devoted to approximate roots of quasi-ordinary polynomials. This research topic was initiated by Abyankar and Moh in the 1970s. They considered characteristic approximate roots of Weierstrass polynomials over the power series ring in one variable.
In 2003 González Pérez extended the Abhyankar–Moh theorem to power series ring in several variables and in 2011 Brzostowski extended the Abhyankar–Moh result to non-characteristic approximate roots, but still for power series ring in one variable.
In this work we prove a result similar to González Pérez but for quasi-ordinary Weierstrass polynomials that could be reducible (González Pérez assumed irreducibility). We also generalize the Brzostowski theorem to the power series ring in several variables and to polynomials that need not be irreducible. The main tools we use are generalized Puiseux series and monomial substitutions.
