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

Abstract

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 $f(x) = c_0 + c_1 x^{d_1} + \ldots + c_k x^{d_k}$ by varying its coefficients. If the GCD of the exponents is $d$, then the polynomial admits the change of variable $y=x^d$, 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 $d_k/d$ elements and $\mathbb{Z}/d\mathbb{Z}$. The aim of this paper is to prove this equality and study its multidimensional generalization: we show that 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.