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Smale's 17th problem: Average polynomial time to compute affine and projective solutions


Authors: Carlos Beltrán and Luis Miguel Pardo
Journal: J. Amer. Math. Soc. 22 (2009), 363-385
MSC (2000): Primary 65H20, 14Q20; Secondary 12Y05, 90C30
DOI: https://doi.org/10.1090/S0894-0347-08-00630-9
Published electronically: November 6, 2008
MathSciNet review: 2476778
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Abstract: Smale's 17th problem asks: ``Can a zero of $ n$ complex polynomial equations in $ n$ unknowns be found approximately, on the average, in polynomial time with a uniform algorithm?'' We give a positive answer to this question. Namely, we describe a uniform probabilistic algorithm that computes an approximate zero of systems of polynomial equations $ f:\mathbb{C}^n\longrightarrow\mathbb{C}^n$, performing a number of arithmetic operations which is polynomial in the size of the input, on the average.


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Additional Information

Carlos Beltrán
Affiliation: Departmento de Matemáticas, Estadística y Computación. Fac. de Ciencias. Avda. Los Castros s/n. 39005 Santander, Spain
Email: beltranc@unican.es

Luis Miguel Pardo
Affiliation: Departmento de Matemáticas, Estadística y Computación. Fac. de Ciencias. Avda. Los Castros s/n. 39005 Santander, Spain
Email: luis.pardo@unican.es

DOI: https://doi.org/10.1090/S0894-0347-08-00630-9
Keywords: Approximate zero, homotopy methods, complexity of methods in algebraic geometry
Received by editor(s): November 2, 2006
Published electronically: November 6, 2008
Additional Notes: The authors’ research was partially supported by FECYT, Spanish Ministry of Science, MTM2007-62799, and an NSERC grant.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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