Artículo

Castro, D.; Giusti, M.; Heintz, J.; Matera, G.; Pardo, L.M. "The hardness of polynomial equation solving" (2003) Foundations of Computational Mathematics. 3(4):347-420
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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
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
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
Volumen:3
Número:4
Página de inicio:347
Página de fin:420
DOI: http://dx.doi.org/10.1007/s10208-002-0065-7
Título revista:Foundations of Computational Mathematics
Título revista abreviado:Found. Comput. Math.
ISSN:16153375
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