Given a system of homogeneous polynomial equations, we study the complexity of solving the system using the Eigenvalue Method. More specifically, fixing a degree $d$, we study the situation where all equations of degree $d$ satisfied by the solution set are available. The complexity of computing these points is governed by the Hilbert function and regularity of the "chopped ideal" generated by the equations. We conjecture values for Hilbert function and regularity, and prove them in many cases. Our conjecture is motivated by an application to symmetric tensor decomposition.

This talk is based on joint work with Fulvio Gesmundo and Simon Telen: https://www.sciencedirect.com/science/article/pii/S0021869324006409.