AbstractWe 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 $$f(x) = c_0 + c_1 x^{d_1} + \cdots + c_k x^{d_k}$$
f
(
x
)
=
c
0
+
c
1
x
d
1
+
⋯
+
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 $$y=x^d$$
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 $$d_k/d$$
d
k
/
d
elements and $${\mathbb {Z}}/d{\mathbb {Z}}$$
Z
/
d
Z
. 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.
Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products
