The Cassowary linear arithmetic constraint solving algorithm
ACM Transactions on Computer-Human Interaction (TOCHI)
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Fast congruence closure and extensions
Information and Computation
Superfluous S-polynomials in Strategy-Independent Groebner Bases
SYNASC '09 Proceedings of the 2009 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
An algebraic approach for the unsatisfiability of nonlinear constraints
CSL'05 Proceedings of the 19th international conference on Computer Science Logic
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Hilbert's weak Nullstellensatz guarantees the existence of algebraic proof objects certifying the unsatisfiability of systems of polynomial equations not satisfiable over any algebraically closed field. Such proof objects take the form of ideal membership identities and can be found algorithmically using Gröbner bases and cofactor-based linear algebra techniques. However, these proof objects may contain redundant information: a proper subset of the equational assumptions used in these proofs may be sufficient to derive the unsatisfiability of the original polynomial system. For using Nullstellensatz techniques in SMT-based decision methods, a minimal proof object is often desired. With this in mind, we introduce a notion of locally minimal Nullstellensatz proofs and give ideal-theoretic algorithms for their construction.