Selections of bounded variation for roots of smooth polynomials

• Adam Parusinski ^[1] ; Armin Rainer ^[2]
  1. [1] Université Côte d'Azur

     Université Côte d'Azur

     Arrondissement de Grasse, Francia

     
  2. [2] University of Vienna

     University of Vienna

     Innere Stadt, Austria

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 1, 2020
• Idioma: inglés
• DOI: 10.1007/s00029-020-0538-z
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• Resumen

  • We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain Ω in Rm admit a parameterization by functions of bounded variation uniformly with respect to the coefficients. This result is best possible in the sense that discontinuities of the roots are in general unavoidable due to monodromy. We show that the discontinuity set can be chosen to be a finite union of smooth hypersurfaces. On its complement the parameterization of the roots is of optimal Sobolev class W1,p for all 1≤p