Ehrhart Theory and Unimodular Decompositions of Lattice Polytopes
| dc.contributor.author | Tam, Ricci Yik Chi | |
| dc.date.accessioned | 2015-01-20T18:12:31Z | |
| dc.date.available | 2015-01-20T18:12:31Z | |
| dc.date.issued | 2015-01-20 | |
| dc.date.submitted | 2015-01-16 | |
| dc.description.abstract | 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. | en |
| dc.identifier.uri | http://hdl.handle.net/10012/9105 | |
| dc.language.iso | en | en |
| dc.pending | false | |
| dc.publisher | University of Waterloo | en |
| dc.subject | Polyhedra | en |
| dc.subject | Ehrhart Theory | en |
| dc.subject | Polytope Decomposition | en |
| dc.subject | Valuations | en |
| dc.subject | Polytope Algebra | en |
| dc.subject | Polyhedral Subdivision | en |
| dc.subject | Lattice Polytopes | en |
| dc.subject.program | Combinatorics and Optimization | en |
| dc.title | Ehrhart Theory and Unimodular Decompositions of Lattice Polytopes | en |
| dc.type | Master Thesis | en |
| uws-etd.degree | Master of Mathematics | en |
| uws-etd.degree.department | Combinatorics and Optimization | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |