Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

• A. Esterov ^[1] ; L. Lang ^[2]
  1. [1] HSE University, Rusia
  2. [2] University of Gävle, Suecia
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 2, 2022
• Idioma: inglés
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• Resumen

  • We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial f(x)=c0+c1xd1+⋯+ckxdk by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable y=xd, and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over dk/d elements and Z/dZ. We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.