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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Bibliographic Details
Published inForum of mathematics. Pi Vol. 11
Main Authors Bürgisser, Peter, Cucker, Felipe, Lairez, Pierre
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.01.2023
Cambridge Univ Press
Subjects
65H10
65H20
65Y20
68Q25
Algebra
Algebraic Geometry
Algorithms
Approximation
Complexity
Computational Mathematics
Computer Science
Mathematics
Numerical Analysis
Polynomials
Symbolic Computation
68Q25
65H10
65Y20
65H20
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Summary: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.
ISSN:2050-5086
2050-5086
DOI:10.1017/fmp.2023.7