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Explicit estimates for polynomial systems defining irreducible smooth complete intersections

Volume 188 / 2019

Joachim von zur Gathen, Guillermo Matera Acta Arithmetica 188 (2019), 209-240 MSC: Primary 14M10; Secondary 11G25, 14N05. DOI: 10.4064/aa8387-8-2018 Published online: 8 March 2019

Abstract

This paper deals with properties of the algebraic variety defined as the set of zeros of a “typical” sequence of polynomials. We consider various types of “nice” varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero “genericity” polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Here, the number of polynomials and their degrees are fixed. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.

Authors

  • Joachim von zur GathenB-IT
    Universität Bonn
    D-53113 Bonn, Germany
    e-mail
  • Guillermo MateraInstituto del Desarrollo Humano
    Universidad Nacional de General Sarmiento
    J. M. Gutiérrez 1150
    (B1613GSX) Los Polvorines, Argentina
    and
    National Council of Science and Technology (CONICET)
    Argentina
    e-mail

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