Ehrhart Polynomials of Dilated Polytopes
summary
Proves that dilating a lattice polytope by any integer R larger than the degree of its H-polynomial makes the H-polynomial real‑rooted (strengthening the earlier dimension bound and giving a sharp uniform threshold), and proves zonotope H-polynomials are real‑rooted with coefficient counts coming from a half-open dissection into half-open parallelepipeds. Method combines Stanley’s nonnegativity, an operator picking coefficients congruent mod R on generating series, and interlacing recursions (Savage–Visontai style) producing interlacing sequences* built from Eulerian and J‑Eulerian polynomials.
AI-written summary · about
chapters (21)
- 0:00Intro
- 0:43Ehrhart-Theory
- 4:07Ehrhart series and h*-polynomial
- 5:18Inequalities for the h*-polynomial
- 6:56Unimodality and real-rootedness
- 7:41Dilated lattice polytopes
- 9:44Lattice parallelepipeds and lattice zonotopes
- 11:06h-polynomials of IDP-polytopes
- 14:06Outline
- 14:32Interlacing polynomials
- 15:35Interlacing sequences of polynomials
- 16:20Constructing interlacing sequences
- 18:26Decomposing
- 23:49ht-polynomials of half-open unimodular simplices
- 26:41Half-open unit cubes
- 27:43Refined Eulerian numbers
- 29:18Refined Eulerian polynomials
- 31:49Dilation operator
- 33:43h-polynomials of dilated polytopes
- 36:37Putting the pieces together...
- 37:59Concluding remarks
related talks
43mApplied Hodge Theory
56mEntropy, Capacity, and Counting
01h 00mAverage-case Complexity for Polynomials, and All That
01h 00mOn Approximating the Covering Radius and Finding Dense Lattice Subspaces
59mLP/SDP Hierarchies and Sum of Squares Proofs 3
01h 07m(Hidden) Symmetries of Computational Problems
01h 03mHigh Dimensional Geometry and Concentration II
43mGeometric Complexity Theory and Tensor Rank