A numerical algorithm for zero counting, I: Complexity and accuracy

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f . The algorithm performs O ( log ( n D κ ( f ) ) ) iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials...

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Published in:	Journal of Complexity Vol. 24; no. 5; pp. 582 - 605
Main Authors:	Cucker, Felipe, Krick, Teresa, Malajovich, Gregorio, Wschebor, Mario
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01.10.2008
Subjects:	Counting algorithms Finite precision Polynomial systems Polynomial systems Finite precision Counting algorithms
ISSN:	0885-064X, 1090-2708
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Abstract	We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f . The algorithm performs O ( log ( n D κ ( f ) ) ) iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials’ degree, and κ ( f ) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature of our results is a bound for the precision required to ensure that the returned output is correct which is polynomial in n and D and logarithmic in κ ( f ) . The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time in n , log D and log ( κ ( f ) ) .
AbstractList	We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f . The algorithm performs O ( log ( n D κ ( f ) ) ) iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials’ degree, and κ ( f ) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature of our results is a bound for the precision required to ensure that the returned output is correct which is polynomial in n and D and logarithmic in κ ( f ) . The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time in n , log D and log ( κ ( f ) ) .
Author	Malajovich, Gregorio Krick, Teresa Wschebor, Mario Cucker, Felipe
Author_xml	– sequence: 1 givenname: Felipe surname: Cucker fullname: Cucker, Felipe email: macucker@cityu.edu.hk organization: Department of Mathematics, City University of Hong Kong, Hong Kong – sequence: 2 givenname: Teresa surname: Krick fullname: Krick, Teresa email: krick@dm.uba.ar organization: Departamento de Matemática, Univ. de Buenos Aires, Argentina – sequence: 3 givenname: Gregorio surname: Malajovich fullname: Malajovich, Gregorio email: gregorio@ufrj.br organization: Depto. de Matemática Aplicada, Univ. Federal do Rio de Janeiro, Brazil – sequence: 4 givenname: Mario surname: Wschebor fullname: Wschebor, Mario email: wschebor@cmat.edu.uy organization: Centro de Matemática, Universidad de la República, Uruguay
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Title	A numerical algorithm for zero counting, I: Complexity and accuracy
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