Ehrhart Theory and Unimodular Decompositions of Lattice Polytopes

Polyhedra;Ehrhart Theory;Polytope Decomposition;Valuations;Polytope Algebra;Polyhedral Subdivision;Lattice Polytopes;Combinatorics and Optimization

Tam, Ricci Yik Chi

University of Waterloo

关键词: Polyhedra;    Ehrhart Theory;    Polytope Decomposition;    Valuations;    Polytope Algebra;    Polyhedral Subdivision;    Lattice Polytopes;    Combinatorics and Optimization;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/9105/3/Tam_Ricci.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

Ehrhart theory studies the behaviour of lattice points contained in dilates of lattice polytopes. We provide an introduction to Ehrhart theory. In particular, we prove Ehrhart;;s Theorem, Stanley Non-negativity, and Ehrhart-Macdonald Reciprocity via lattice triangulations. We also introduce the algebra $mathscr{P}(mathbb{R}^d)$ spanned by indicator functions of polyhedra, and valuations (linear functions) on $mathscr{P}(mathbb{R}^d)$. Through this, we derive Brion;;s Theorem, which gives an alternate proof of Ehrhart;;s Theorem. The proof of Brion;;s Theorem makes use of decomposing the lattice polytope in $mathscr{P}(mathbb{R}^d)$ into support cones and other polyhedra. More generally, Betke and Kneser proved that every lattice polytope in $mathscr{P}(mathbb{R}^d)$ (or the sub-algebra $mathscr{P}(mathbb{Z}^d)$, spanned by lattice polytopes) admits a unimodular decomposition; it can be expressed as a formal sum of unimodular simplices. We give a new streamlined proof of this result, as well as some applications and consequences.