We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
  arXiv:1201.434

Heintz, Joos

Kuijpers, Bart

Paredes, Andres Rojas

English

arXiv

One may represent polynomials not only by their coefficients but also by arithmetic circuits which evaluate them. This idea allowed in the past fifteen years considerable complexity progress in effective polynomial equation solving. We present a circuit based computation model which captures all known symbolic elimination algorithms in effective Algebraic Geometry and exhibit a class of simple elimination problems which require exponential size circuits to be solved in this model. This implies that the known, circuit based elimination algorithms are already optimal.FWO G.0344.05Software engineering; Complexity; Algebraic Geometry; Database

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Software Engineering and complexity in effective Algebraic Geometry

One may represent polynomials not only by their coefficients but also by arithmetic circuits which evaluate them. This idea allowed in the past fifteen years considerable complexity progress in effective polynomial equation solving. We present a circuit based computation model which captures all known symbolic elimination algorithms in effective Algebraic Geometry and exhibit a class of simple elimination problems which require exponential size circuits to be solved in this model. This implies that the known, circuit based elimination algorithms are already optimal.Fil: Heintz, Joos Ulrich. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Universidad de Cantabria; España. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Kuijpers, Bart. Hasselt University; BélgicaFil: Rojas Paredes, Andres Avelino. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Software Engineering and Complexity in Effective Algebraic Geometry

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