Experimental Mathematics

Universal Gröbner Bases of Colored Partition Identities

Tristram Bogart, Ray Hemmecke, and Sonja Petrovíc

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Associated to any toric ideal are two special generating sets: the universal Gröbner basis and the Graver basis, which encode polyhedral and combinatorial properties of the ideal, or equivalently, its defining matrix. If the two sets coincide, then the complexity of the Graver bases of the higher Lawrence liftings of the toric matrices is bounded.

While a general classification of all matrices for which both sets agree is far from known, we identify all such matrices within two families of nonunimodular matrices, namely, those defining rational normal scrolls and those encoding homogeneous primitive colored partition identities. This also allows us to show that higher Lawrence liftings of matrices with fixed Gröbner and Graver complexities do not preserve equality of the two bases.

The proof of our classification combines computations with the theoretical tool of Graver complexity of a pair of matrices.

Article information

Experiment. Math., Volume 21, Issue 4 (2012), 395-401.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E40: Combinatorial aspects of commutative algebra 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 11P84: Partition identities; identities of Rogers-Ramanujan type 90C27: Combinatorial optimization

Graver bases universal Gröbner bases partition identities colored partitions rational normal scrolls state polytope toric ideal


Bogart, Tristram; Hemmecke, Ray; Petrovíc, Sonja. Universal Gröbner Bases of Colored Partition Identities. Experiment. Math. 21 (2012), no. 4, 395--401. https://projecteuclid.org/euclid.em/1356038822

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