The Computational Complexity of the Chow Form
We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is chara...
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2004
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paper:paper_16153375_v4_n1_p41_Jeronimo2023-06-08T16:25:23Z The Computational Complexity of the Chow Form Jeronimo, Gabriela Tali Krick, Teresa Elena Genoveva Sabia, Juan Vicente Rafael Sombra, Martín Chow form Equidimensional decomposition of algebraic varieties Overdetermined polynomial equation system Sparse resultant Symbolic Newton algorithm Algorithms Computational methods Partial differential equations Polynomials Probability Set theory Bounded probability algorithm Chow form Computational complexity We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system. Fil:Jeronimo, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Krick, T. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Sabia, J. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Sombra, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2004 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16153375_v4_n1_p41_Jeronimo http://hdl.handle.net/20.500.12110/paper_16153375_v4_n1_p41_Jeronimo |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Chow form Equidimensional decomposition of algebraic varieties Overdetermined polynomial equation system Sparse resultant Symbolic Newton algorithm Algorithms Computational methods Partial differential equations Polynomials Probability Set theory Bounded probability algorithm Chow form Computational complexity |
| spellingShingle |
Chow form Equidimensional decomposition of algebraic varieties Overdetermined polynomial equation system Sparse resultant Symbolic Newton algorithm Algorithms Computational methods Partial differential equations Polynomials Probability Set theory Bounded probability algorithm Chow form Computational complexity Jeronimo, Gabriela Tali Krick, Teresa Elena Genoveva Sabia, Juan Vicente Rafael Sombra, Martín The Computational Complexity of the Chow Form |
| topic_facet |
Chow form Equidimensional decomposition of algebraic varieties Overdetermined polynomial equation system Sparse resultant Symbolic Newton algorithm Algorithms Computational methods Partial differential equations Polynomials Probability Set theory Bounded probability algorithm Chow form Computational complexity |
| description |
We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system. |
| author |
Jeronimo, Gabriela Tali Krick, Teresa Elena Genoveva Sabia, Juan Vicente Rafael Sombra, Martín |
| author_facet |
Jeronimo, Gabriela Tali Krick, Teresa Elena Genoveva Sabia, Juan Vicente Rafael Sombra, Martín |
| author_sort |
Jeronimo, Gabriela Tali |
| title |
The Computational Complexity of the Chow Form |
| title_short |
The Computational Complexity of the Chow Form |
| title_full |
The Computational Complexity of the Chow Form |
| title_fullStr |
The Computational Complexity of the Chow Form |
| title_full_unstemmed |
The Computational Complexity of the Chow Form |
| title_sort |
computational complexity of the chow form |
| publishDate |
2004 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16153375_v4_n1_p41_Jeronimo http://hdl.handle.net/20.500.12110/paper_16153375_v4_n1_p41_Jeronimo |
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