Splines, lattice points, and arithmetic matroids

2019

Ann. Comb.

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An arithmetic matroid is weakly multiplicative if the multiplicity of at least one of its bases is equal to the product of the multiplicities of its elements. We show that if such an arithmetic matroid can be represented by an integer matrix, then this matrix is uniquely determined. This implies that the integer cohomology ring of a centred toric arrangement whose arithmetic matroid is weakly multiplicative is determined by its poset of layers. This partially answers a question asked by Callegaro-Delucchi.

Representations of Weakly Multiplicative Arithmetic Matroids are Unique

2019

Ann. Comb.

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On powers of Plücker coordinates and representability of arithmetic matroids

Advances in Applied Mathematics

2020

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The first problem we investigate is the following: given k ∈ R ≥0 and a vector v of Plücker coordinates of a point in the real Grassmannian, is the vector obtained by taking the kth power of each entry of v again a vector of Plücker coordinates? For k = 1, this is true if and only if the corresponding matroid is regular. Similar results hold over other fields. We also describe the subvariety of the Grassmannian that consists of all the points that define a regular matroid.The second topic is a related problem for arithmetic matroids. Let A = (E, rk, m) be an arithmetic matroid and let k = 1 be a non-negative integer. We prove that if A is representable and the underlying matroid is non-regular, then A k := (E, rk, m k ) is not representable. This provides a large class of examples of arithmetic matroids that are not representable. On the other hand, if the underlying matroid is regular and an additional condition is satisfied, then A k is representable. Bajo-Burdick-Chmutov have recently discovered that arithmetic matroids of type A 2 arise naturally in the study of colourings and flows on CW complexes. In the last section, we prove a family of necessary conditions for representability of arithmetic matroids.

Zonotopal algebra and forward exchange matroids

Advances in Mathematics

2016

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Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X. It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair (D(X), P(X)), where D(X) denotes the Dahmen-Micchelli space that is spanned by the local pieces of the box spline and P(X) is a space spanned by products of linear forms.The first main result of this paper is the construction of a canonical basis for D(X). We show that it is dual to the canonical basis for P(X) that is already known.The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom.