Ehemalige Mitarbeiter

Dr. Peter Scheiblechner

Kontakt
Lucerne University of Applied Sciences and Arts
Technology and Architecture
Technikumstr. 21
CH-6048 Horw
Switzerland

Persönliche Homepage
www.scheiblechner.ch

Differential Forms in Computational Algebraic Geometry
Zitatschlüssel	BS-Differential-Forms-In-Computational-Algebraic-Geometry
Autor	Peter Bürgisser and Peter Scheiblechner
Seiten	61-68
Jahr	2007
DOI	1277548.1277558
Adresse	Waterloo, Canada
Journal	Proc. of ISSAC
Monat	July 29 - August 1
Zusammenfassung	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.