Using monodromy and representation theory to recover symmetries of polynomial systems
Parametric polynomial systems can be represented as dominant maps between irreducible algebraic varieties of the same dimension, where symmetries correspond to automorphisms of these maps. Galois, or monodromy, groups provide a numerical tool for detecting the existence of such symmetries in solution sets. A central computational challenge, however, is to recover explicit formulas for these automorphisms in order to better understand and more efficiently solve the systems. We combine numerical homotopy continuation with multivariate rational function interpolation to compute candidate symmetries, implemented in the Julia package DecomposingPolynomialSystems.jl.
For structured systems with many variables, such as minimal problems in computer vision, the resulting Vandermonde-like interpolation matrices become prohibitively large, leading to expensive nullspace computations and numerical instability. We address this by exploiting equivariance of minimal problems with respect to matrix Lie group actions. The interpolation space of bounded-degree polynomials decomposes into isotypic components as a representation of the Lie symmetry group, allowing a substantial reduction of the problem size. This representation-theoretic decomposition is implemented in the Julia package DecomposingGroupRepresentations.jl. Together, these tools provide a scalable Julia-based framework that integrates monodromy and computational representation theory to recover closed-form symmetries of polynomial systems.