On the magic positivity of Ehrhart polynomials of dilated polytopes

TL;DR

This work investigates when Ehrhart polynomials become magic positive after dilation and whether magic positivity persists under further dilation. It proves that any polynomial with positive real coefficients can be dilated to become magic positive and that magic positivity is preserved under larger dilations, but shows that in dimension $d\ge3$ no fixed dilation suffices for all polytopes. The authors introduce the m-index to quantify the minimal dilation needed and compute or bound it for several families of polytopes, including simplices, matroid-base polytopes, hypersimplices, edge polytopes, and CL-polytopes. The results connect magic positivity to real-rootedness of $h^*$-polynomials and provide structural insights into Ehrhart positivity via dilations and factorization properties.

Abstract

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.

Theorems & Definitions (26)

• Theorem 1.1: beck2019hbranden2004linearferroni2024examples
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['main1']}
• Lemma 3.1: Matouek2006under
• proof : Proof of Theorem \ref{['main2']}
• proof : Proof of Theorem \ref{['main3']}