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TU Berlin

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Leitung

Prof. Dr. Peter Bürgisser

Lupe

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

Rigid continuation paths II. Structured polynomial systems
Citation key BCL-RigidContinuationPathsII-StructuredPolynomialSystems
Author Peter Bürgisser and Felipe Cucker and Pierre Lairez
Year 2020
Abstract We design a probabilistic algorithm that, given ϵ>0 and a polynomial system F given by black-box evaluation functions, outputs an approximate zero of F, in the sense of Smale, with probability at least 1−ϵ. When applying this algorithm to u⋅F, where u is uniformly random in the product of unitary groups, the algorithm performs poly(n,δ)⋅L(F)⋅(Γ(F)logΓ(F)+log log ϵ^(−1)) operations on average. Here n is the number of variables, δ the maximum degree, L(F) denotes the evaluation cost of F, and Γ(F) reflects an aspect of the numerical condition of F. Moreover, we prove that for inputs given by random Gaussian algebraic branching programs of size poly(n,δ), the algorithm runs on average in time polynomial in n and δ. Our result may be interpreted as an affirmative answer to a refined version of Smale's 17th question, concerned with systems of \emphstructured polynomial equations.
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