| dc.description.abstract | To an enumerative problem, one may associate a Galois group which encodes symmetries of the solutions to the problem. Galois groups of enumerative problems were first defined and studied by Jordan who considered them in the context of several classical enumerative problems. Recently, Galois groups of enumerative problems have been exploited for fast and efficient solving of polynomial systems. As such, determining Galois groups of enumerative problems and understanding how they may be used in numerical computations is of great importance.
We detail the mathematical background needed to define Galois groups of enumerative problems and then describe tools from numerical and computational algebraic geometry used to compute and exploit Galois groups. We then give the algebraic definition of the Galois group originally used by Jordan, as well as a geometric definition. We prove these definitions are equivalent and explore Galois groups for two classes of enumerative problems, sparse polynomial systems and Fano problems.
A sparse polynomial system is a polynomial systems such that the monomials appearing in each polynomial have been chosen a priori. Esterov observed two classes of sparse polynomial systems whose Galois group is an imprimitive permutation group and determined that the Galois group is the symmetric group for all other sparse polynomial systems. In special cases, there are results which determine the Galois group when it is imprimitive, though the answer is not known in general. We detail a computational method used to decompose systems into simpler systems when the Galois group is imprimitive. This approach has shown to increase speed and accuracy in solving polynomial systems when the Galois group is imprimitive.
Fano problems, problems of enumerating linear spaces on a variety, were among those enumerative problems considered by Jordan in his study of Galois theory. Recently, Galois groups of Fano problems were nearly classified by Hashimoto and Kadets. All Galois groups of Fano problems which are unknown are either the alternating group or the symmetric group. We present a computational technique which may be used to prove the existence of a transposition in the Galois group. This method was recently used to prove that several Galois groups of Fano problems are symmetric groups, which was previously unknown. | |
