Abstract 

In numerical algebraic geometry, we commonly face the problem of deciding if a set of solutions is complete. In other words, we wish to determine whether a given set of solutions to a system of polynomial equations consists of all the solutions to that system. The most common numerical heuristic for answering this question is called the trace test. In the trace test, we use the local behavior of the solutions to a parametric family of systems to provide evidence for the completeness of the set of solutions. The theory behind the trace test is based on properties of the branched cover of the solution-parameter incidence variety for this parametric family. In this setup, the monodromy/Galois group corresponds to possible permutations of the zeros induced by changing the parameters. In this talk, I will motivate the talk with problems from numerical algebraic geometry, and then, I will transition to discussing the theoretical underpinnings of the numerical computations. I will provide both an overview of what is known about these monodromy/Galois groups and end with new results concerning their behavior for sparse polynomial systems.