2014/11/19 by Akihiro Higashitani, Higashitani, Akihiro
Computer Science · Mathematics · #52B20 #Advanced Combinatorial Mathematics #Analytic and geometric function theory #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:52B20 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1411.5250
17 pages
openalex publication_date 2014/11/19 · arxiv created 2015/04/16 · arxiv updated 2015/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate two properties concerning the unimodality of the δ-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes P of dimension d, we prove that the dilated lattice polytopes nP have strictly log-concave and strictly alternatingly increasing δ-vectors if n > max\s,d+1-s\, where s is the degree of the δ-polynomial of P. The bound max\s,d+1-s\ for n is reasonable. We also provide several kinds of unimodal (or non-unimodal) δ-vectors. Concretely, we give examples of lattice polytoeps whose δ-vectors are not unimodal, unimodal but neither log-concave nor alternatingly increasing, alternatingly increasing but not log-concave, and log-concave but not alternatingly increasing, respectively.