High probability analysis of the condition number of sparse polynomial systems

  • Authors:

  • Gregorio Malajovich;J. Maurice Rojas

  • Affiliations:

  • Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brazil;Department of Mathematics, Texas A&M University, TAMU 3368, College Station, TX

  • Venue:
  • Theoretical Computer Science - Algebraic and numerical algorithm
  • Year:
  • 2004

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Abstract

Let f := (f1 ..... fn) be a random polynomial system with fixed n-tuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/ε. The bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the Newton polytopes (and underlying variances) are all identical.We also consider polynomials with real coefficients and give bounds for the expected number of real roots and (restricted) condition number. Using a Kähler geometric framework throughout, we also express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form, thus recovering the classical mixed volume when U = (C*)n.