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Toric ideals of lattice path matroids and polymatroids. (English) Zbl 1230.13028

Let \(M\) be the set of all monomials in a subset of variables \(\underline X\) which are basis of a discrete polymatroidal \(\Gamma \), let \(K[m\in M]\subset K[\underline X]\) be the toric ring associated to the discrete polymatroid \(\Gamma \). It was conjectured by White that the binomial ideal \(I\) defining \(K[m\in M]\) as a quotient \(K[\underline Y]/I\), is generated by quadric binomials corresponding to symmetric basis exchanges. In the paper under review the author proves this conjecture for a special class of discrete polymatroids, namely lattice paths polymatroids, for which the set of quadric binomials corresponding to symmetric basis exchanges is a Gröbner basis of \(I\).

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
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References:

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