  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.
