Viktor Korotynskiy: Using Monodromy to recover symmetries of polynomial systems

Abstract.

Parametric polynomial systems can be represented as dominant maps between irreducible algebraic varieties of the same dimension where the symmetries correspond to the automorphisms of these maps. Galois/monodromy groups attached to these polynomial systems provide a method for detecting the existence of symmetries in solution sets. Beyond the question of existence, one would like to compute closed-form expressions for these symmetries, towards the eventual goal of solving the systems more efficiently. I will describe one possible approach to this task using numerical homotopy continuation and multivariate rational function interpolation. I will illustrate this approach on practical examples of 3D reconstruction formulated by minimal problems in computer vision. This is joint work with Timothy Duff, Tomas Pajdla and Margaret Regan.