Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2412.17679

Ehrhart Functions of Weighted Lattice Points

Jesus A. De Loera·Carlos E. Valencia·Rafael H. Villarreal·Chengyang Wang
View original source ↗Read source document ↗Export metadata ↓

Also available from

Abstract

This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions and their generating functions and associated rings. These will be called $q$-weighted, $r$-weighted, and $s$-weighted Ehrhart functions, respectively. The key questions we investigate are \emph{When are the weighted Ehrhart series rational functions and which classical Ehrhart theory properties are preserved? And, when are the abstract formal power series the Hilbert series of Ehrhart rings of some polytope?} We prove generalizations about weighted Ehrhart $h^*$-coefficients of $q$-weighted Ehrhart series, and show $q$- and $s$-weighted Ehrhart reciprocity theorems. Then, we show the $q$- and $r$-weighted Ehrhart rings are the (classical) Ehrhart rings of weight lifting polytopes.

Explore related subjects

math.COmath.ACmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jesus A. De Loera, Carlos E. Valencia, Rafael H. Villarreal, Chengyang Wang. 2026-04-08. Ehrhart Functions of Weighted Lattice Points. https://doi.org/10.1007/s40590-025-00844-3

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗