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 necessarily
irreducible) given by the first-named author and González-Pérez 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.
