Theory of linear and integer programming
Theory of linear and integer programming
Symmetric polynomials and Hall's theorem
Discrete Mathematics
On uniquely 3-colorable graphs
Discrete Mathematics
Proceedings of the first Malta conference on Graphs and combinatorics
Triangle-free four-chromatic graphs
Discrete Mathematics
Lower bounds for the polynomial calculus and the Gröbner basis algorithm
Computational Complexity
On Markov chains for independent sets
Journal of Algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Good Degree Lower Bounds on Nullstellensatz Refutations of the Induction Principle
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
The Hilbert zonotope and a polynomial time algorithm for universal Gröbner bases
Advances in Applied Mathematics
The division algorithm and the hilbert scheme
The division algorithm and the hilbert scheme
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series A
Semidefinite representations for finite varieties
Mathematical Programming: Series A and B
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Reduction of symmetric semidefinite programs using the regular $$\ast$$-representation
Mathematical Programming: Series A and B
Algebraic characterization of uniquely vertex colorable graphs
Journal of Combinatorial Theory Series B
Computational Commutative Algebra 1
Computational Commutative Algebra 1
Combinatorics, Probability and Computing
Computing infeasibility certificates for combinatorial problems through Hilbert's Nullstellensatz
Journal of Symbolic Computation
An algebraic characterization of rainbow connectivity
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
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Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows. In the first part of the paper, we show that the minimum degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree. In the second part of this paper, for the purpose of computation, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colourable subgraph. We include some applications to graph theory.