Applications of tropical combinatorics and monomial modules
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We study three aspects of tropical combinatorics and monomial modules. In the fi rst, we consider the tropical geometry specifi cally arising from convergent Puiseux series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is the study of families of ordinary convex polytopes depending on more than one parameter through tropical geometry. In the second, we consider matching fi elds and their connections to combinatorial geometry. We show that the Chow covectors of a linkage matching fi eld defi ne a bijection of lattice points, resolving two open questions from Sturmfels & Zelevinsky. We use a similar method to prove that, given a triangulation of a product of two simplices encoded by a set of bipartite trees, the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fi elds and characterise the flip graph of a linkage matching fi eld in terms of its prodsimplicial flag complex. In the third, we study commutative algebra arising from generalised Frobenius numbers. We defi ne generalised lattice modules, a class of monomial modules whose Castelnuovo{ Mumford regularity captures the k-th Frobenius number. We study the fi ltration of generalised lattice modules providing an explicit characterisation of their minimal generators, and show that there are only finitely many isomorphism classes of generalised lattice modules. As a consequence of our commutative algebraic approach, we prove structural results on the sequence of generalised Frobenius numbers and also construct an algorithm to compute them.
Authors
Smith, BCollections
- Theses [4248]