Deformation Techniques for Sparse Systems

  • Authors:

  • Gabriela Jeronimo;Guillermo Matera;Pablo Solernó;Ariel Waissbein

  • Affiliations:

  • Universidad de Buenos Aires, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón I, 1428, Buenos Aires, Argentina and National Counci ...;National Council of Sci. and Technol. (CONICET), Buenos Aires, Argentina and Universidad Nacional de General Sarmiento, Instituto de Desarrollo Humano, J.M. Gutiérrez 1150, 1613 Los Polvori ...;Universidad de Buenos Aires, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón I, 1428, Buenos Aires, Argentina and National Counci ...;CORE Security Technologies, CoreLabs, Humboldt 1967, C1414CTU, Buenos Aires, Argentina and Instituto Tecnológico de Buenos Aires, Doctorado en Ingeniería, Av. Eduardo Madero 399, C11 ...

  • Venue:
  • Foundations of Computational Mathematics
  • Year:
  • 2009

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Abstract

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is cubic in the size of the combinatorial structure of the input system. This size is mainly represented by the cardinality and mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration.