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Factorization of bivariate sparse polynomials

Volume 191 / 2019

Francesco Amoroso, Martín Sombra Acta Arithmetica 191 (2019), 361-381 MSC: Primary 13P05; Secondary 12Y05. DOI: 10.4064/aa171219-18-12 Published online: 19 September 2019

Abstract

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with a fixed set of complex coefficients and varying exponents. Roughly speaking, this result shows that the truly bivariate irreducible factors of these sparse Laurent polynomials are also sparse. The proofs are based on a variant of the toric Bertini theorem due to Zannier and to Fuchs, Mantova and Zannier.

Authors

  • Francesco AmorosoLaboratoire de mathématiques Nicolas Oresme
    CNRS UMR 6139
    Université de Caen
    BP 5186, 14032 Caen Cedex, France
    e-mail
  • Martín SombraInstitució Catalana de Recerca
    i Estudis Avançats (ICREA)
    Passeig Lluís Companys~23
    08010 Barcelona, Spain
    and
    Departament de Matemàtiques i Informàtica
    Universitat de Barcelona (UB)
    Gran Via 585
    08007 Barcelona, Spain
    e-mail

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