Smooth polytopes with negative Ehrhart coefficients (Q1671785)

This paper is motivated by a question of Bruns whether the Ehrhart coefficients of smooth lattice polytopes are all positive. The authors prove that there exists a smooth \(n\)-dimensional lattice polytope with all negative \(t^i\)-coefficients of its Ehrhart polynomial where \( 1 \leq i \leq n-2 \). For its proof they utilize the technique called chiseling from [\textit{W. Bruns}, in: Commutative algebra and algebraic geometry. Proceedings of the international conference (CAAG-2010), Bangalore, India, December 6--10, 2010, in honour of Balwant Singh, Uwe Storch and Rajendra V. Gurjar. Mysore: Ramanujan Mathematical Society. 45--61 (2013; Zbl 1317.14112)]. The case of a dilated \(n\)-cube is studied more closely. It has been showed that one needs at least dimension 7 in order to obtain negative linear coefficients. This result aims to illustrate the method of Berline-Vergne valuations as a useful tool for the study of positivity of Ehrhart coefficients. Moreover, this result can be interpretated as a result for the Hilbert polynomial of the projective manifold obtained by blowing up of \((\mathbb{P}^1)^n\) in all its torus-invariant fixpoints. They also weaken the question and investigate the coefficients in the case of smooth reflexive polytopes. Although the Ehrhart coefficients are all positive up to dimension 8, they present an example in dimension 9 with negative Ehrhart coefficients. Followed by this example, the authors pose the question whether there exist smooth \(n\)-dimensional reflexive polytopes with the maximal possible number of \(n - 2\) negative Ehrhart coefficients.