Solving Parameterized Polynomial Systems with Decomposable Projections
arXiv:1612.08807
Abstract
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.
arXiv comment18 pages, 2 figures, 2 tables
References in corpus (5)
- Moment Varieties of Gaussian Mixtures
- Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
- Cyclic p-roots of prime lengths p and related complex Hadamard matrices
- Investigating the Maximum Number of Real Solutions to the Power Flow Equations: Analysis of Lossless Four-Bus Systems
- HomotopyContinuation.jl: A package for homotopy continuation in Julia