Corpus ID: 119600570

New irrational polygons with Ehrhart-theoretic period collapse.

@article{Le2018NewIP,
  title={New irrational polygons with Ehrhart-theoretic period collapse.},
  author={Quang-Nhat Le},
  journal={arXiv: Combinatorics},
  year={2018}
}
  • Quang-Nhat Le
  • Published 1 August 2018
  • Mathematics
  • arXiv: Combinatorics
In a recent paper, Cristofaro-Gardiner--Li--Stanley [CGLS15] constructed examples of irrational triangles whose Ehrhart functions (i.e. lattice-point count) are polynomials when restricted to positive integer dilation factors. This is very surprising because the Ehrhart functions of rational polygons are usually only quasi-polynomials. We demonstrate that most of their triangles can also be obtained by a simple cut-and-paste procedure that allows us to build new examples with more sides. Our… Expand

References

SHOWING 1-10 OF 10 REFERENCES
New examples of period collapse
"Period collapse" refers to any situation where the period of the Ehrhart function of a polytope is less than the denominator of that polytope. We study several interesting situations where thisExpand
The ghost stairs stabilize to sharp symplectic embedding obstructions
In determining when a four-dimensional ellipsoid can be symplectically embedded into a ball, McDuff and Schlenk found an infinite sequence of "ghost" obstructions that generate an infinite "ghostExpand
Ehrhart functions and symplectic embeddings of ellipsoids
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a ball, and found that part of the answer is given by an infinite "Fibonacci staircase." Similarly,Expand
The minimum period of the Ehrhart quasi-polynomial of a rational polytope
TLDR
It is shown that for any D, there is a 2-dimensional triangle P such that D(P) = D but such that the minimum period of ip(n) is 1, that is, ip( n) is a polynomial in n. Expand
Maximal periods of (Ehrhart) quasi-polynomials
TLDR
It is proved that the second leading coefficient of an Ehrhart quasi- polynomial always has maximal expected period and a general theorem is presented that yields maximal periods for the coefficients of certain quasi-polynomials. Expand
The embedding capacity of 4-dimensional symplectic ellipsoids
This paper calculates the function $c(a)$ whose value at $a$ is the infimum of the size of a ball that contains a symplectic image of the ellipsoid $E(1,a)$. (Here $a \ge 1$ is the ratio of the areaExpand
Symplectic embedding problems, old and new
We describe old and new motivations to study symplectic embedding problems, and we discuss a few of the many old and the many new results on symplectic embeddings.
Sur un problème de géométrie diophantienne linéaire. II.
Un Systeme diophantien lineaire est forme d'equations et d'inequations du premier degre, les coefficients et le terme constant de chacune etant des entiers relatifs, ainsi que les inconnues. Si sesExpand
The embedding capacity of 4dimensional symplectic ellipsoids
  • Annals of Mathematics,
  • 2012