Algebraic Geometry and Landau Gauge Fixing on the Lattice

On the lattice, the standard way to fix Landau gauge is to minimizing the so-called lattice Landau gauge (LLG)-fixing functional numerically. Minimizing a multivariate function efficiently is one of the fundamental problems in many branches of theoretical physics. The conventional numerical minimization methods such as Simulated Annealing and Over-relaxation are known to fail in obtaining the global minimum. We observe that the extremizing equations for the LLG-fixing functional and, in general, for multivariate functions arising in many physical phenomena have 'polynomial-like' non-linearity. After explaining how one can transform the extremizing equations for the LLG, for the compact U(1) case as an example, to a system of multivariate polynomial equations, we propose a few methods to solve these equations and obtain all the extrema of the LLG-fixing functional. I will demonstrate our preliminary results from both these methods. If time permits, I will also present the results for the modified lattice Landau gauge which was proposed recently, in order to establish the BRST symmetry on the lattice.