Computational Aspects of the Combinatorial Nullstellensatz Method

We discuss here some computational aspects of the Combinatorial Nullstellensatz argument. Our main result shows that the order of magnitude of the symmetry group associated with permutations of the variables in algebraic constraints, determines the performance of algorithms naturally deduced from Alon's Combinatorial Nullstellensatz arguments. Finally we present a primal-dual polynomial constructions for certifying the existence or the non-existence of solutions to combinatorial problems.