Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a kth dilation is a sum of k lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the \(h^*\)-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that \(h^*\)-vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.

In this article we investigate the property of complete monotonicity within a special family Fs of functions in s variables involving logarithms. The main result of this work provides a linear isomorphism between Fs and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in Fs is semi-algebraic. This gives a finite time algorithm to decide whether a function in Fs is completely monotone.

We introduce a new multiplication for the polytope algebra, defined via the intersection of polytopes. After establishing the foundational properties of this intersection product, we investigate finite-dimensional subalgebras that arise naturally from this construction. These subalgebras can be regarded as volumetric analogues of the graded Möbius algebra, which appears in the context of the Dowling-Wilson conjecture. We conjecture that they also satisfy the injective hard Lefschetz property and the Hodge-Riemann relations, and we prove these in degree one.

This paper deals with sequences of random variables \(X_n\) only taking values in \(\{0,\ldots,n\}\). The probability generating functions of such random variables are polynomials of degree \(n\). Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for \(X_n\) is established in a unified way. In the real rooted case the result is classical and only involves the variances of \(X_n\), while in the cyclotomic case the fourth cumulants or moments of \(X_n\) appear in addition. The proofs are elementary and based on the Stein--Tikhomirov method.

An arrangement of hypersurfaces in projective space is SNC if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomogeneous prime ideal of this variety.

Recently, Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph G, so called connected subgraph arrangements AG. In this note we strengthen some of the result from their work and prove new ones for members of this class. For instance, we show that aspherical members withing this class stem from a rather restricted set of graphs. Specifically, if AG is an aspherical connected subgraph arrangement, then AG is free with the unique possible exception when the underlying graph G is the complete graph on 4 nodes.

For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for distributions whose support is a natural distributive lattice: that is, any distribution which has natural lattice support and satisfies the pairwise Markov statements of a graph must factor according to the graph. We also show a connection to the Hibi ideals of lattices. dui fames leo sodales risus, posuere justo curae nostra, luctus vitae montes potenti, lectus sem.