Journal of the ACM (JACM)
Using the Groebner basis algorithm to find proofs of unsatisfiability
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Proof complexity in algebraic systems and bounded depth Frege systems with modular counting
Computational Complexity
Short proofs are narrow—resolution made simple
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Good Degree Lower Bounds on Nullstellensatz Refutations of the Induction Principle
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
Separation of the monotone NC hierarchy
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Exponential Separations between Restricted Resolution and Cutting Planes Proof Systems
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
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In standard implementations of the Gröbner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the proof complexity of refutations of polynomials derived from Boolean formulas in both the Polynomial Calculus (PC) and Nullstellensatz systems. We show that the PC refutations of homogenized formulas give crucial information about the complexity of the original formulas. The minimum PC refutation degree of homogenized formulas is equal to the Nullstellensatz refutation degree of the original formulas, whereas the size of the homogenized PC refutation is equal to the size of the PC refutation for the originals. Using this relationship, we prove nearly linear (Ω(n/logn) vs. O(1)) separations between Nullstellensatz and PC degree, for a family of explicitly constructed contradictory 3CNF formulas. Previously, a Ω(n1/2) separation had been proved for equations that did not correspond to any CNF formulas, and a log n separation for equations derived from kCNF formulas.