Characterize classes with vanishing real Taylor-nonnegativity index

Characterize natural classes of lattice polytopes for which the real Taylor-nonnegativity index satisfies \(\rho(P)=0\), especially among matroid base polytopes and smooth polytopes.

Background

The paper defines ρ(P)\rho(P) as the smallest nonnegative real center at which every Taylor coefficient of the Ehrhart polynomial is nonnegative. The authors establish general bounds and identify several classes and examples with ρ(P)=0\rho(P)=0 or ρ(P)>0\rho(P)>0, but they do not classify the natural families for which nonnegative Taylor coefficients already occur at the origin.

The concluding remarks specifically identify matroid base polytopes and smooth polytopes as particularly relevant families in which a structural characterization of the condition ρ(P)=0\rho(P)=0 remains unresolved.

References

Several classification problems remain open. It would be useful to characterize natural classes for which \rho(P)=0, especially within the families of matroid base polytopes and smooth polytopes.

— Taylor Positivity of Ehrhart Polynomials  (2609.03327 - Liu et al., 3 Sep 2026) in Section 10, Concluding remarks

Another motivation for introducing the relevant concepts in \cref{def:taylor-positive} is the following open problem listed on the website of the American Institute of Mathematics. Let $P$ be a lattice polytope. Fix $k\in \mathbb{Z}{>0}$ and write $L_P(t)$ in the basis ${ (t-k)i}_i$. For example, if $\deg(h_P*(z))=s$, then $L_P(t)$ is nonnegative in the basis ${ (t-s+1)i}{i=0}d$. Investigate what happens when changing $k$ from 0 to 1 for a non-Ehrhart positive polytope (or other values of $k$).

— Taylor Positivity of Ehrhart Polynomials  (2609.03327 - Liu et al., 3 Sep 2026) in Section 1, Subsection Motivation; Problem 1.6 (cited from the American Institute of Mathematics website)