# Higher order polars of quasi-ordinary singularities

## TL;DR

This paper investigates higher order derivatives of quasi-ordinary polynomials, providing new factorizations and generalizations of existing results, extending to irreducible cases and quasi-ordinary power series.

## Contribution

It generalizes the factorization of higher order polars for quasi-ordinary polynomials, including irreducible cases and power series, advancing the theoretical understanding of these singularities.

## Key findings

- Derived factorizations of higher order polars.
- Extended results to irreducible quasi-ordinary polynomials.
- Generalized factorization to quasi-ordinary power series.

## Abstract

A quasi-ordinary polynomial is a monic polynomial with coefficients in the power series ring such that its discriminant equals a monomial up to unit. In this paper we study higher derivatives of quasi-ordinary polynomials, also called higher order polars. We find factorizations of these polars. Our research in this paper goes in two directions. We generalize the results of Casas-Alvero and our previous results on higher order polars in the plane to irreducible quasi-ordinary polynomials. We also generalize the factorization of the first polar of a quasi-ordinary polynomial (not necessary irreducible) given by the first-named author and Gonz\'alez-P\'erez to higher order polars. This is a new result even in the plane case. Our results remain true when we replace quasi-ordinary polynomials by quasi-ordinary power series.