APPROXIMATE ROOTS, TORIC RESOLUTIONS AND DEFORMATIONS OF A PLANE BRANCH

Pedro Daniel Gonzalez Perez (1)

1 Departamento de Álgebra [Madrid]

We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial $f$ defining a plane branch $(C,0)$, in the light of the toric embedded resolution of the branch. This leads to the definition of a class of (non equisingular) deformations of a plane branch $(C,0)$ supported on certain monomials in the approximate roots of $f$. As a consequence we find out a Kouchnirenko type formula for the Milnor number $(C,0)$. Our results provide a geometrical approach to Abhyankar's straight line conditions and its consequences. As an application we give an equisingularity criterion for a family of plane curves to be equisingular to a plane branch and we express it algorithmically.

Mots clés: approximate roots; deformations of a plane curve; equisingularity criterion

Domaines: Géométrie algébrique [math.AG]

Pedro Daniel Gonzalez Perez. APPROXIMATE ROOTS, TORIC RESOLUTIONS AND DEFORMATIONS OF A PLANE BRANCH. 2008. ⟨hal-00308807v2⟩