2 research outputs found
A Note on Lower Bounds for Colourful Simplicial Depth
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14
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Semigroups of Polyhedral Lattice Points: Convexity, Combinatorics, and Algebra
This dissertation explores problems of convexity, combinatorics, and algebra associated with semigroups of polyhedral lattice points.In \Cref{ChapterColored}, we first attempt to generalize and extend three well-known convexity theorems, including Helly theorem, Tverberg theorem, and Colorful Carath\'eodory theorem, to affine semigroups. We define a novel notion, chromatic representations of semigroup elements, this is in contrast to the colorful theory developed by B\'ar\'any et al. Later, we focus on one-dimensional affine semigroups, numerical semigroups, and study the number of chromatic solutions in numerical semigroups.In \Cref{ChapterWeighted}, we generalize the classical Hilbert functions and Hilbert series of a semigroup algebra to have weightings. We list three ways to add weightings, -weighting, -weighting, and -weighting, and study their relationships. We find that the -weighting can derive other weightings. Later, we specialize to the special family of semigroup algebras, the Ehrhart rings. We study and extend the properties of -nonnegativity and Ehrhart–Macdonald reciprocity for the Ehrhart series under these three weightings.In \Cref{ChapterEhrhart}, we focus on the Ehrhart functions under the -weighting and give a practical method to evaluate the -weighted Ehrhart function. Specifically, we construct a new polytope, the weight-lifting polytope, and build a connection between the -weighted Ehrhart function and the classical Ehrhart function. Later, we present several applications and experiments of our method in combinatorial representation theory and number theory.In \Cref{ChapterKakeya}, we discuss a long-standing conjecture, Kakeya's conjecture, which brings a surprising connection between numerical semigroups and symmetric polynomials. We give partial results, prove the conjecture for two variables, and outline a general computer proof for an arbitrary number of variables