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## Continuity properties for flat families of polynomials (I)
Continuous parametrizations

pdf file.

with Galligo A., González-Vega L.

Journal of Pure and Applied Algebra,
**184**, (2003), 77-103.
** Abstract :**
Inspired by classical results in algebraic geometry, we study the
continuity with respect to the coefficients, of the zero set of a
system of complex homogeneous polynomials with a given pattern and
when the Hilbert polynomial of the generated ideal is fixed.
In this work we prove topological
properties of some classifying spaces, e.g. the space of systems with
given pattern, fixed Hilbert polynomial is locally compact,
and we establish continuous parametrizations of Nullstellensatz formulae.
In the general case we get local rational results but
in the complex case we get global results using rational
polynomials in the real and imaginary parts of the coefficients.
In a second companion paper, we shall treat the continuity of zero sets
for the Hausdorff distance, i.e., from a metric point of view.