Evelia R. García Barroso

Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility

In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two variables and necessary conditions which a complex curve has to satisfy in order to be the discriminant of a complex plane branch.

On the Łojasiewicz numbers

Let f be a holomorphic function of two complex variables with an isolated critical point at 0∈C2. We give some necessary conditions for a rational number to be the smallest θ>0 in the Łojasiewicz inequality |gradf(z)|⩾C|z|θ for z near 0∈C2. To cite this article: E. Garcia Barroso, A. Ploski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

A discriminant criterion of irreducibility

In this paper we give a criterion of irreducibility for a complex power series in two variables, using the notion of jacobian Newton diagrams, defined with respect to any direction. Moreover we study the singularity at infinity of a plane affine curve with one point at infinity for which the global counterpart of our main result holds.

Ultrametric Spaces of Branches on Arborescent Singularities

Let S be a normal complex analytic surface singularity. We say that S is arborescent if the dual graph of any good resolution of it is a tree. Whenever A, B are distinct branches on S, we denote by A ⋅ B their intersection number in the sense of Mumford. If L is a fixed branch, we define UL(A, B) = (L ⋅ A)(L ⋅ B)(A ⋅ B)−1 when A ≠ B and UL(A, A) = 0 otherwise. We generalize a theorem of Ploski concerning smooth germs of surfaces, by proving that whenever S is arborescent, then UL is an ultrametric on the set of branches of S different from L. We compute the maximum of UL, which gives an analog of a theorem of Teissier. We show that UL encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both S and L are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of S. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.

On the Abhyankar-Moh inequality

Abhyankar and Moh in their fundamental paper on the embeddings of the line in the plane proved an important inequality which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. In this note we study the semigroups of integers satisfying the Abhyankar-Moh inequality and give a simple proof of the Abhyankar-Moh embedding theorem.

Quasi-Ordinary Singularities: Tree Model, Discriminant, and Irreducibility

Let

On the Milnor Formula in Arbitrary Characteristic

f(Y)\in K[[X_1,\dots,X_d]][Y]

Decompositions of the higher order polars of plane branches

be a quasi-ordinary Weierstrass polynomial with coefficients in the ring of formal power series over an algebraically closed field of characteristic zero. In this paper we study the discriminant

Euclidean algorithm and polynomial equations after Labatie

D_f

On the approximate Jacobian Newton diagrams of an irreducible plane curve

of