A Deterministic Algorithm to Compute Approximate Roots of Polynomial Systems in Polynomial Average Time

We describe a deterministic algorithm that computes an approximate root of  n complex polynomial equations in  n unknowns in average polynomial time with respect to the size of the input, in the Blum–Shub–Smale model with square root. It rests upon a derandomization of an algorithm of Beltrán and Pa...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 17; no. 5; pp. 1265 - 1292
Main Author: Lairez, Pierre
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2017
Springer Nature B.V
Springer Verlag
Subjects:
Algorithms
Applications of Mathematics
Computational Complexity
Computer Science
Economics
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Polynomials
Primary 68Q25
Secondary 65H10
65H20
Polynomial system
Homotopy continuation
65Y20
Smale’s 17th problem
Derandomization
Complexity
homotopy continuation
complexity
derandomization
Smale's 17th problem
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary:We describe a deterministic algorithm that computes an approximate root of  n complex polynomial equations in  n unknowns in average polynomial time with respect to the size of the input, in the Blum–Shub–Smale model with square root. It rests upon a derandomization of an algorithm of Beltrán and Pardo and gives a deterministic affirmative answer to Smale’s 17th problem. The main idea is to make use of the randomness contained in the input itself.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-016-9319-7