tandard nonperturbative covariant gauge fixing procedure leaves the theory with Gribov copies and on lattice even Neuberger zero-zero problem. Due to this Neuberger problem, BRST and SUSY on lattice are still open and urgent questions to be addressed. I will introduce the problems using Landau gauge for a simple toy model, compact QED on a one dimensional lattice, and propose a modification which completely resolves Gribov-Neuberger problems on this simple toy model and even the higher dimensional generalization. This gauge-fixing term for compact QED on lattice is the classical XY-model Hamiltonian, and in condensed matter terms the problem is to get ALL extrema of this Hamiltonian exactly. To give a full analytical proof for the higher dimensional generalization, I will need to introduce a tailor-made terminology of Algebraic Geometry. I will also go on proposing two algorithms for gauge-fixing on lattice that use sophisticated applied mathematics and give efficient results derived from Numerical Algebraic Geometry.

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