by Carles Bivià-Ausina (Univ. Valencia, Spain)

Given an analytic function germ
,
where
or ,
if
is its
Taylor
expansion, we can consider the problem of determining which is the least
integer *r* such that *f* is topologically equivalent to the *r*-jet
,
where
,
if
.
That is, we want to determine the
least integer *r*such that there exists a germ of homeomorphism
with the condition
that
.
This is called the
*degree of **C*^{0}*-sufficiency of **f* and
we denote it by *s*(*f*). The problem of determining this number is well
known in singularity theory.

In this talk, we show an algorithm to give a sharp estimate for *s*(*f*),
when *f* is an arbitrary analytic function germ. The difference between
the estimate we give and *s*(*f*) is .
Our method is based on the
characterization of *s*(*f*) by Chang-Lu, in the complex case, and
Bochnak-ojasiewicz and Kuo, in the real case. This characterization
allows us to relate *s*(*f*) with the notion of integral closure of an
ideal. But the integral closure of an ideal *I* in a local ring
is related with the multiplicity *e*(*I*), in the
Hilbert-Samuel sense (where *I* is an -primary ideal). Then
we can use the program Singular in order to give an algorithm to
determine *s*(*f*). Hence we use commutative algebra methods to deal with
a problem from real analytic geometry.