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Systems of polynomial equations over an algebraically-closed field K can beused to concisely model many combinatorial problems. In this way, acombinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian,etc.) if and only if a related system of polynomial equations has a solutionover K. In this paper, we investigate an algorithm aimed at provingcombinatorial infeasibility based on the observed low degree of Hilbert'sNullstellensatz certificates for polynomial systems arising in combinatoricsand on large-scale linear-algebra computations over K. We report on experimentsbased on the problem of proving the non-3-colorability of graphs. Wesuccessfully solved graph problem instances having thousands of nodes and tensof thousands of edges.

Hilbert's Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility