# Galois groups of lacunary sparse polynomial systems

The Galois group of a polynomial system is a group of symmetries of the zeros of the system that reflects its intrinsic structure. These groups were initially studied algebraically by Jordan, and much later Harris interpreted them as geometric monodromy groups. This geometric view allows one to study Galois groups through analytic means and to approximate them via numerical methods. It has been shown that knowledge of the Galois group may be used to reduce computation in solving polynomial systems, and this has been used in computer vision, for example.

We will consider Galois groups of sparse polynomial systems, systems whose coefficients are general and whose monomial support is fixed. There are two special structures that occur in sparse systems: lacunary systems are those that have been precomposed with a non-invertible monomial map, and triangular systems are those that contain a proper subsystem. Galois groups of lacunary systems and triangular systems act imprimitively on the zeros of the system--there is a non-trivial partition of the zeros which is preserved. This implies that the Galois group of a lacunary or triangular system is a subgroup of a certain wreath product, and it is expected that the Galois group is equal to this wreath product in most cases. However, a classification of these Galois groups remains open.

We determine the Galois group of a pure lacunary polynomial system. A pure lacunary polynomial system is a sparse polynomial system which is lacunary and not triangular. We use analytic methods akin to those of Harris to show that the Galois group of a pure lacunary system is determined by the automorphism group of a certain variety. Further, using a resultant product formula of D'Andrea, this variety is defined by binomial equations and its automorphism group is a group of roots of unity acting by coordinate-wise multiplication. We use this to compute the Galois group of some pure lacunary systems and demonstrate that the Galois group may be strictly smaller than the expected wreath product in many cases.