```Bernd Schober (Université de Versailles) Título "Characteristic Polyhedra of Singularities without
completion"

Fecha y Lugar "05.06.2014, 12:30. Seminario A125 de la Facultad de Ciencias de la Universidad de Valladolid Resumen: Let \$ R \$ be a Noetherian regular local ring, \$  (u ) = ( u_1, \ldots, u_e ) \$ a system of regular elements in \$ R \$, and \$ J \subset R \$ a non-zero ideal.
For these data Hironaka introduced the characteristic polyhedron \$ \Delta ( J;u) \$ which yields crucial information on the singularity defined by \$ J \$.

In his original work Hironaka also gave a procedure how to compute \$ \Delta ( J;u) \$ starting with a given set of generators \$ ( f ) = ( f_1, \ldots, f_m ) \$ and a system of elements \$ ( y) = ( y_1, \ldots, y_r ) \$ which extends \$ ( u ) \$ to a regular system of parameters for \$ R \$.
But there exist examples where this process is not finite.
Recently V.~Cossart and O.~Piltant have shown that one can achieve \$ \Delta ( J;u) \$ with a different procedure in finitely many steps under the assumption that \$ R \$ is a \$ G \$-ring, \$ J \$ is a principal ideal (\$ m = 1 \$), and \$ r = 1 \$.
In this talk I show that their process is finite if \$ J \$ is any ideal, \$ R \$ is excellent, and the reduced ridge of \$ J \$ coincides with its directrix.

First, I recall Hironaka's definition of the characteristic polyhedron of a singularity.
In particular I explain the procedures of normalization and vertex solving with whom one can achieve the characteristic polyhedra in the completion of \$ R \$.
After that I show that the normalization process is always finite and then I reduce the proof for the finiteness of solving vertices to the case of an empty characteristic polyhedron.
All the previous steps are valid in any excellent regular local ring \$ R \$ without the assumption on the ridge.
Finally, I discuss the key case of an empty characteristic polyhedron, \$ \Delta (J;u) = \emptyset \$, and under which assumptions one can attain it without going to the completion.

(This is joint work with V.~Cossart)```