Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products
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 + c1 x^d1 +. . .+ ck x^dk
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 dk/d elements and Z/dZ.
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.