Name: Félix Delgado de la Mata

title:  Poincaré series for curve singularities and Alexander polynomial.

Abstracts:

documentstyle[12pt]{article}
\input amssym.def
\input amssym.tex
\begin{document}
Let $C=\bigcup\limits_{i=1}^r C_i$ be a reduced curve singularity
(defined over an algebraically closed feld); $C_1,\ldots , C_r$
being the branches of $C$ and let $v_1, \ldots, v_r$ be its
corresponding discrete valuations. Let
$\underline{v} : {\cal O}_C^* \to {\Bbb Z}^r$ be the map defined
by $\underline{v}(g)=(v_1(g), \ldots,v_r(g))$ for $g$ in the set
of non-zero divisors ${\cal O}_C^*$ of the local ring
${\cal O}_C$ of $C$.
The natural order in ${\Bbb Z}^r$ for this situation is the
product one:
$$\underline{n}=(n_1,\ldots,n_r)\le
\underline{m}=(m_1,\ldots,m_r) \iff n_i\le m_i\ (i=1,
\ldots,r)$$
The fact that this ordering
is not a total one dificults
the study of the ``filtrations" given by the valuation ideals
$J(\underline{v}) = \{g\in {\cal O}_C : \underline{v}(g)\ge
\underline{n}\}$ for
$\underline{n}\in {\Bbb Z}^r$.
In a join work with A. Campillo and S.M. Gusein-Zade we study the
Hilbert function defined by
$$
\underline{n}\to \dim_{\Bbb C}
J(\underline{n})/J(\underline{n}+(1,\ldots,1))
$$
showing that a suitable Poincar\'e series coincides (for complex
plane germs) with the Alexander polynomial of the algebraic link
defined by $C$. As a consequence deep topological invariants of
$C$ (zeta functions, Alexander polynomials,...) could be defined
and computed in a pure algebraic way; also provides a (possible)
way to extend such invariants for cases in which the
corresponding topological ones does not exist.

\end{document}



Felix Delgado de la Mata
Dto. de Algebra, Geometria y Topologia
Facultad de Ciencias.
Universidad de Valladolid
email: <fdelgado@agt.uva.es>
Phone: 983423050