Rigid continuation paths II. structured polynomial systems

This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomi...

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Veröffentlicht in:Forum of Mathematics, Pi Jg. 11
Hauptverfasser: Bürgisser, Peter, Cucker, Felipe, Lairez, Pierre
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge, UK Cambridge University Press 01.01.2023
Cambridge Univ Press
Schlagworte:
65H10
65H20
65Y20
68Q25
Algebra
Algebraic Geometry
Algorithms
Approximation
Complexity
Computational Mathematics
Computer Science
Mathematics
Numerical Analysis
Polynomials
Symbolic Computation
68Q25
65H10
65Y20
65H20
ISSN:2050-5086, 2050-5086
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Beschreibung
Zusammenfassung:This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomial system given only as black-box evaluation program. Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L. Combining both, we show that we can compute an approximate zero of a random structured polynomial system with n equations of degree at most ${D}$ in n variables with only $\operatorname {poly}(n, {D}) L$ operations with high probability. This exceeds the expectations implicit in Smale’s 17th problem.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:2050-5086
2050-5086
DOI:10.1017/fmp.2023.7