Solving Parameterized Polynomial Systems with Decomposable Projections

The Galois group of a parameterized polynomial system of equations encodes the structure of the solutions. This monodromy group acts on the set of solutions for a general set of parameters, that is, on the fiber of a projection from the incidence variety of parameters and solutions onto the space of parameters. When this projection is decomposable, the Galois group is imprimitive, and we show that the structure can be exploited for computational improvements. Furthermore, we develop a new algorithm for solving these systems based on a suitable trace test. We illustrate our method on examples in statistics, kinematics, and benchmark problems in computational algebra. In particular, we resolve a conjecture on the number of solutions of the moment system associated to a mixture of Gaussian distributions.