Euclidean algorithm and polynomial equations after Labatie

We recall Labatie's effective method of solving polynomial equations with two unknowns by using the Euclidean algorithm

Higher order polars of quasi-ordinary singularities

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

Invariants des singularités de courbes planes et courbure des fibres de Milnor

La curvatura de la fibra de Milnor asociada a un germen de curva analítica plana reducida compleja presenta una concentración asintótica en las zonas evanescentes a varias escalas diferentes, escalas que sólo dependen de la topología de la curva de partida. Un procedimiento explícito permite comparar el diagrama de Eggers con el grafo de intersección de las componentes del divisor excepcional de una resolución sumergida minimal de la singularidad del germen. Este diagrama de Eggers, que describe el contacto y los exponentes característicos de las diferentes ramas de la curva, permite estudiar el contacto de dicha curva con las distintas ramas de la polar. Se demuestra un teorema de descomposición de la curva polar en paquetes de componentes irreducibles

Decomposition in bunches of the critical locus of a quasi-ordinary map

A polar hypersurface P of a complex analytic hypersurface germ f = 0 can be investigated by analyzing the invariance of certain Newton polyhedra associated with the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated with f. We develop this general principle of Teissier when f = 0 is a quasi-ordinary hypersurface germ and P is the polar hypersurface associated with any quasi-ordinary projection of f = 0. We show a decomposition of P into bunches of branches which characterizes the embedded topological types of the irreducible components of f = 0. This decomposition is also characterized by some properties of the strict transform of P by the toric embedded resolution of 0 given by the second author. In the plane curve case this result provides a simple algebraic proof of a theorem of Le et al

On Lê’s formula in arbitrary characteristic

In this note we extend, to arbitrary characteristic, Lˆe’s formula (Calculation of Milnor number of isolated singularity of complete intersection. Funct. Anal. Appl. 8 (1974), 127–131)