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 Gonzalez-Perez 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.

中文翻译：

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准普通奇点的高阶极坐标

拟普通多项式是一个单项多项式，其系数在幂级数环中，这样它的判别式就等于一个单项式到单位。在本文中，我们研究了拟普通多项式的高阶导数，也称为高阶极坐标。我们找到了这些极坐标的因式分解。我们在本文中的研究有两个方向。我们将 Casas-Alvero 的结果和我们之前在平面中高阶极坐标上的结果推广到不可约的拟普通多项式。我们还将第一作者和 Gonzalez-Perez 给出的拟普通多项式（不一定不可约）的第一极的因式分解推广到更高阶极。即使在平面情况下，这也是一个新结果。当我们用拟常幂级数代替拟常多项式时，我们的结果仍然成立。