Equiprojectable decomposition of zero-dimensional varieties
Marc Moreno Maza, University of Western Ontario
Tuesday, December 14, 2004, at U. of Waterloo.
Abstract:
Equidimensional decompositions of algebraic varieties, such as triangular
decompositions, are used for many situations. However, even a
zero-dimensional variety V may have several triangular decompositions. The
a priori canonical choice, namely the irreducible decomposition of V, does
not have good specialization properties.
For a variable ordering, we introduce the equiprojectable decomposition of
V. This is a canonical equidimensional decomposition of V with good
computational properties. We show how to compute the equiprojectable
decomposition of V from any triangular decomposition or primitive element
representation of V.
Given a zero-dimensional polynomial system F over Q, we show that there
exists an integer A which height is softly in the order of the square of
the Bezout number of F such that any prime number not dividing A is a
good prime for specializing the equiprojectable decomposition of F.
Using Hensel lifting techniques, we deduce a modular algorithm for computing
the equiprojectable decomposition of zero-dimensional varieties over Q. We
have realized a preliminary implementation with the Triade library developed
in Maple by F. Lemaire. Our theoretical results are comforted by these
experiments.
This is joint work with Xavier Dahan, Eric Schost, Wenyuan Wu and Yuzhen Xie.
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