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<p>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, their generating functions, and their associated rings. These will be called <i>q</i>-weighted, <i>r</i>-weighted, and <i>s</i>-weighted Ehrhart functions, respectively. The key questions we investigate are: <i>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?</i> We prove generalizations about weighted Ehrhart <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(h^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-coefficients of <i>q</i>-weighted Ehrhart series and show <i>q</i>- and <i>s</i>-weighted Ehrhart reciprocity theorems. Then we show the <i>q</i>- and <i>r</i>-weighted Ehrhart rings are the (classical) Ehrhart rings of weight lifting polytopes.</p>

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Ehrhart Functions of Weighted Lattice Points

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月5日

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  • Jesús A. De Loera,
  • Carlos E. Valencia,
  • Rafael H. Villarreal,
  • Chengyang Wang

摘要

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, their generating functions, and their associated rings. These will be called q-weighted, r-weighted, and s-weighted Ehrhart functions, respectively. The key questions we investigate are: 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 \(h^*\) 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.