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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Pseudozeros of multivariate polynomials


Authors: J. William Hoffman, James J. Madden and Hong Zhang
Journal: Math. Comp. 72 (2003), 975-1002
MSC (2000): Primary 65H10; Secondary 13P99, 14Q99, 14P10
Published electronically: May 15, 2002
MathSciNet review: 1954980
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Abstract: The pseudozero set of a system $f$ of polynomials in $n$ complex variables is the subset of $\mathord{\bf C}^n$ which is the union of the zero-sets of all polynomial systems $g$ that are near to $f$ in a suitable sense. This concept is made precise, and general properties of pseudozero sets are established. In particular it is shown that in many cases of natural interest, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets in coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials.


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Additional Information

J. William Hoffman
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: hoffman@math.lsu.edu

James J. Madden
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: madden@math.lsu.edu

Hong Zhang
Affiliation: Department of Computer Science, Illinois Institute of Technology, Chicago, Illinois 60616
Email: hzhang@mcs.anl.gov

DOI: http://dx.doi.org/10.1090/S0025-5718-02-01429-1
PII: S 0025-5718(02)01429-1
Keywords: Multivariate polynomial, pseudozeros, semialgebraic set, algorithm
Received by editor(s): May 5, 2000
Received by editor(s) in revised form: April 24, 2001
Published electronically: May 15, 2002
Article copyright: © Copyright 2002 American Mathematical Society