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`Algebraic Geometry and Lattice Landau Gauge Fixing'

Abstract: 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. It should be noted that some of the methods are indeed being
used in string phenomenology areas (to find the string vacua of a
given model, for example), however, I will be presenting a numerical
method which could go beyond the existing methods quite efficiently. 
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