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Fachgebiet Algorithmische AlgebraProf. Dr. Peter Bürgisser

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Prof. Dr. Peter Bürgisser

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Anschrift
Technische Universität Berlin
Institut für Mathematik
Sekretariat MA 3-2
Straße des 17. Juni 136
10623 Berlin

Büro
Raum MA 317 (3. OG)
Institut für Mathematik

Kontakt

Sekretariat
Beate Nießen
Raum MA 318
Tel.: +49 (0)30 314 - 25771

eMail
peter.buergisser@offmath.tu-berlin.de

Telefon
+49 (0)30 314 - 75902
Faxgerät
+49 (0)30 314 - 25839

Sprechstunde
Während der Vorlesungszeit: Do, 15-16 Uhr.
Während der vorlesungsfreien Zeit: nach Vereinbarung.

Publikationen

Counting Complexity Classes for Numeric Computations II: Algebraic and Semialgebraic Sets
Zitatschlüssel BC-Counting-Complexity-Classes-For-Numeric-Computations-Ii-Algebraic-And-Semialgebraic-Sets-1
Autor Peter Bürgisser and Felipe Cucker
Buchtitel In Proceedings of 36th STOC
Seiten 475-485
Jahr 2004
Adresse Chicago
Zusammenfassung We define counting classes $\#P_R$ and $\#P_C$ in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over $\mathbb R$, or of systems of polynomial equalities over $\mathbb C$, respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over $\mathbb R$) and algebraic sets (over $\mathbb C$). We prove that the problem to compute the Euler characteristic of semialgebraic sets is $FP_R^R}$-complete, and that the problem to compute the geometric degree of complex algebraic sets is $FP_C^C}$-complete. We also define new counting complexity classes in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving , for all $k$ in $N$, the FPSPACE-hardness of the problem of computing the $k$-th Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.
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