Tale of Two Homographies

This is a talk about symmetries in polynomial systems. Once we know what they look like, we can exploit them in various ways (e.g. inside of parameter homotopies, monodromy, etc.) How do we know if they exist? One reasonable answer comes via the monodromy group itself -- viewed as a subgroup of the symmetric group, the centralizer of the monodromy group is isomorphic to a group of rational deck transformations. These deck transformations are typically the symmetries we seek. In general, the equations defining them might be complicated. A notable example of such a deck transformation is the "twisted pair" map which appears when estimating the relative pose between two cameras. Two solutions to this problem which differ up to a twisted pair map to the same essential matrix. The ideal of polynomials that vanish on all essential matrices is minimally generated by ten cubic equations. I will introduce an analogous problem involving 3 cameras and the interesting equations we discovered. (Joint work with Viktor Korotynskiy, Tomas Pajdla, and Maggie Regan.)