Abstract:
Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the algorithmic solving of multivariate polynomial equation systems over the complex numbers or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e., polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids "unnecessary" branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.
Registro:
Documento: |
Artículo
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Título: | The hardness of polynomial equation solving |
Autor: | Castro, D.; Giusti, M.; Heintz, J.; Matera, G.; Pardo, L.M. |
Filiación: | Depto. de Ciencias de la Computacion, E.U. Politécnica, Universidad de Alcalá, 28871 Alcalá de Henares, Spain Laboratoire GAGE, Ecole Polytechnique, F-91128 Palaiseau Cedex, France Depto. Matematicas Estadistica Comp., Facultad de Ciencias, Universidad de Cantabria, E-39071 Santander, Spain Departamento de Matemáticas, Fac. de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pabellon I, 1428 Buenos Aires, Argentina CONICET, Argentina Instituto de Desarrollo Humano, Univ. Nacional de General Sarmiento, Campus Universitario, Jose M. Gutierrez 1150, 1613 Los Polvor. Pcia. Buenos Aires, Argentina
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Palabras clave: | Complexity; Continuous data structure; Elimination theory; Holomorphic and continuous encoding; Polynomial equation solving; Arithmetic circuits; Bezout number; Elimination theory; Geometric objects; Algebra; Algorithms; Computational complexity; Data structures; Digital arithmetic; Encoding (symbols); Geometry; Graph theory; Problem solving; Topology; Polynomials |
Año: | 2003
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Volumen: | 3
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Número: | 4
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Página de inicio: | 347
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Página de fin: | 420
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DOI: |
http://dx.doi.org/10.1007/s10208-002-0065-7 |
Título revista: | Foundations of Computational Mathematics
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Título revista abreviado: | Found. Comput. Math.
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ISSN: | 16153375
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16153375_v3_n4_p347_Castro |
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Citas:
---------- APA ----------
Castro, D., Giusti, M., Heintz, J., Matera, G. & Pardo, L.M.
(2003)
. The hardness of polynomial equation solving. Foundations of Computational Mathematics, 3(4), 347-420.
http://dx.doi.org/10.1007/s10208-002-0065-7---------- CHICAGO ----------
Castro, D., Giusti, M., Heintz, J., Matera, G., Pardo, L.M.
"The hardness of polynomial equation solving"
. Foundations of Computational Mathematics 3, no. 4
(2003) : 347-420.
http://dx.doi.org/10.1007/s10208-002-0065-7---------- MLA ----------
Castro, D., Giusti, M., Heintz, J., Matera, G., Pardo, L.M.
"The hardness of polynomial equation solving"
. Foundations of Computational Mathematics, vol. 3, no. 4, 2003, pp. 347-420.
http://dx.doi.org/10.1007/s10208-002-0065-7---------- VANCOUVER ----------
Castro, D., Giusti, M., Heintz, J., Matera, G., Pardo, L.M. The hardness of polynomial equation solving. Found. Comput. Math. 2003;3(4):347-420.
http://dx.doi.org/10.1007/s10208-002-0065-7