On the magic positivity of Ehrhart polynomials of dilated polytopes
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912354649767936 |
|---|---|
| author | Konoike, Masato |
| author_facet | Konoike, Masato |
| contents | A polynomial $f(x)$ of degree $d$ is said to be magic positive if all the coefficients are non-negative when $f(x)$ is expanded with respect to the basis $\{x^i(x+1)^{d-i}\}_{i=0}^d$. It is known that if $f(x)$ is magic positive, then the polynomial appearing in the numerator of its generating function is real-rooted. In this paper, we show that for a polynomial $f(x)$ with positive real coefficients, there exists a positive real number $k$ such that $f(k'x)$ is magic positive for any $k' \geq k$. Furthermore, for any integer $d\geq3$, we show the existence of a $d$-dimensional polytope $P$ such that the Ehrhart polynomial of $kP$ is not magic positive for a given integer $k$. Finally, we investigate how much certain polytopes need to be dilated to make their Ehrhart polynomials magic positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21395 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the magic positivity of Ehrhart polynomials of dilated polytopes Konoike, Masato Combinatorics Primary: 05A15, Secondary: 05A10, 52B20 A polynomial $f(x)$ of degree $d$ is said to be magic positive if all the coefficients are non-negative when $f(x)$ is expanded with respect to the basis $\{x^i(x+1)^{d-i}\}_{i=0}^d$. It is known that if $f(x)$ is magic positive, then the polynomial appearing in the numerator of its generating function is real-rooted. In this paper, we show that for a polynomial $f(x)$ with positive real coefficients, there exists a positive real number $k$ such that $f(k'x)$ is magic positive for any $k' \geq k$. Furthermore, for any integer $d\geq3$, we show the existence of a $d$-dimensional polytope $P$ such that the Ehrhart polynomial of $kP$ is not magic positive for a given integer $k$. Finally, we investigate how much certain polytopes need to be dilated to make their Ehrhart polynomials magic positive. |
| title | On the magic positivity of Ehrhart polynomials of dilated polytopes |
| topic | Combinatorics Primary: 05A15, Secondary: 05A10, 52B20 |
| url | https://arxiv.org/abs/2504.21395 |