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Ehemalige Mitarbeiter

Dr. Peter Scheiblechner


Lucerne University of Applied Sciences and Arts
Technology and Architecture
Technikumstr. 21
CH-6048 Horw
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Publikationen in der Arbeitsgruppe

Differential Forms in Computational Algebraic Geometry
Citation key BS-Differential-Forms-In-Computational-Algebraic-Geometry
Author Peter Bürgisser and Peter Scheiblechner
Pages 61-68
Year 2007
DOI 1277548.1277558
Address Waterloo, Canada
Journal Proc. of ISSAC
Month July 29 - August 1
Abstract We give a uniform method for the two problems $\#CC_C$ and $\#IC_C$ of counting connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of $\mathbb C$. They work efficiently in parallel and can be implemented by algebraic circuits of polynomial depth, i.e., in parallel polynomial time. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Szanto (1997) computing a variant of characteristic sets. The crucial complexity parameter for $\#IC_C$ turns out to be the number of equations. We describe a randomised algorithm solving $\#IC_C$ for a fixed number of rational equations given by straight-line programs (slps), which runs in parallel polylogarithmic time in the length and the degree of the slps.
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