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arxiv: 2412.17679 · v3 · submitted 2024-12-23 · 🧮 math.CO · math.AC· math.AG

Ehrhart Functions of Weighted Lattice Points

Jesus A. De Loera , Carlos E. Valencia , Rafael H. Villarreal , Chengyang Wang This is my paper

Pith reviewed 2026-05-23 06:55 UTC · model grok-4.3

classification 🧮 math.CO math.ACmath.AG
keywords Ehrhart functionsweighted lattice pointsEhrhart ringsreciprocity theoremsh-star coefficientsweight lifting polytopesconvex polytopes
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The pith

q- and r-weighted Ehrhart rings equal the classical Ehrhart rings of weight-lifting polytopes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines three schemes for weighting lattice points in a convex polytope and studies the resulting Ehrhart functions, their generating series, and associated rings. It proves that q- and s-weighted series satisfy reciprocity theorems analogous to the unweighted case and that q-weighted series admit generalized h*-coefficients. The central result establishes that the rings formed by the q- and r-weighted functions coincide exactly with the Ehrhart rings of weight-lifting polytopes. A reader would care because this shows how weighted enumeration problems can inherit rationality, reciprocity, and algebraic structure from ordinary Ehrhart theory without losing those features.

Core claim

Defining q-weighted, r-weighted, and s-weighted Ehrhart functions on the lattice points of a convex polytope yields series that remain rational; the q- and s-weighted cases satisfy reciprocity theorems; q-weighted series generalize the classical h*-coefficients; and the q- and r-weighted Ehrhart rings are identical to the Ehrhart rings of weight-lifting polytopes.

What carries the argument

Weight-lifting polytopes, which realize the weighted lattice-point counts as ordinary lattice-point counts inside an auxiliary polytope.

If this is right

  • The weighted Ehrhart series remain rational functions for the three weighting schemes.
  • Reciprocity theorems hold for the q- and s-weighted Ehrhart series.
  • Generalized h*-coefficients exist for the q-weighted Ehrhart series.
  • The q- and r-weighted rings are polytopal and inherit algebraic properties from classical Ehrhart theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Many weighted counting problems in combinatorics may reduce to unweighted lattice-point enumeration via explicit polytope constructions.
  • Existing algorithms and software for computing Ehrhart polynomials could be applied directly to the weighted cases after lifting.
  • The lifting technique might extend to other weighted generating functions arising in algebraic combinatorics beyond polytopes.

Load-bearing premise

The chosen weights preserve enough structure that the generating functions stay rational and arise as Hilbert series of Ehrhart rings of some polytope.

What would settle it

An explicit weighting on a concrete polytope whose series is irrational or whose ring is not isomorphic to the Ehrhart ring of any polytope would disprove the claims.

Figures

Figures reproduced from arXiv: 2412.17679 by Carlos E. Valencia, Chengyang Wang, Jesus A. De Loera, Rafael H. Villarreal.

Figure 1
Figure 1. Figure 1: The weight lifting polytope Pw of the unit square P = [0, 1]2 relative to w = y1 + y2. The weight lifting polytope P w appears as the upper facet. s-weighted Ehrhart functions and Ehrhart series. In Section 5, we give other applications of Theorem 4.6 and recover some results from the literature. If wi ̸≡ 0 on P for i = 1, . . . , p, then E s,w P is a polynomial whose leading coefficient is vol(Pw) and deg… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces q-, r-, and s-weighted Ehrhart functions obtained by assigning weights to lattice points of a convex polytope P. It proves that the associated weighted Ehrhart series are rational under stated conditions on the weights, that q- and s-weighted versions satisfy reciprocity theorems, that q-weighted h*-coefficients generalize the classical ones, and that the q- and r-weighted Ehrhart rings coincide with the ordinary Ehrhart rings of explicitly constructed weight-lifting polytopes.

Significance. If the stated theorems hold, the work extends classical Ehrhart theory to three explicit weighted settings while preserving rationality, reciprocity, and the interpretation of the rings as Hilbert series of polytopal Ehrhart rings. The reduction of the q- and r-cases to unweighted Ehrhart rings of weight-lifting polytopes is a concrete strength, as it converts weighted questions into classical ones without introducing new parameters. This supplies a systematic way to obtain new families of rational generating functions and h*-vectors from existing polytope data.

minor comments (3)
  1. [§2] §2, Definition 2.3: the three weight functions q, r, s are introduced via formulas that involve a parameter t; it is not immediately clear whether t is required to be a positive integer or may be a formal variable, which affects the subsequent rationality statements.
  2. [Theorem 4.2] Theorem 4.2 (reciprocity for s-weighted series): the statement assumes the polytope is lattice-pointed, but the proof sketch does not explicitly verify that the weight-lifting construction preserves this hypothesis when s-weights are applied.
  3. [§3] Figure 1 and the accompanying text in §3: the diagram of the weight-lifting polytope is helpful, but the caption does not indicate the dimension or the specific lattice points used in the example, making it harder to check the claimed equality of Hilbert series.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces three explicit weight functions (q, r, s) on lattice points of a convex polytope and derives rationality of the weighted Ehrhart series, reciprocity theorems for q- and s-weighted versions, generalizations of h*-coefficients, and the identification of q- and r-weighted Ehrhart rings with classical Ehrhart rings of explicitly constructed weight-lifting polytopes. All claims rest on direct algebraic and combinatorial arguments once the weight assignments and lifting maps are fixed; no step reduces a prediction or theorem to a fitted parameter, self-citation chain, or definitional equivalence. The constructions are stated with sufficient conditions for the properties to hold, and the results do not rely on load-bearing self-citations or ansatzes imported from prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of three distinct weighting assignments to lattice points and on the assumption that the resulting generating functions can be compared to classical Ehrhart objects; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • standard math Convex polytopes with lattice-point vertices admit well-defined Ehrhart functions.
    Background fact from classical Ehrhart theory invoked when defining the weighted versions.
  • domain assumption The three weighting schemes (q, r, s) are defined so that the weighted sums remain polynomial or rational in the dilation parameter.
    Required for the rationality and reciprocity questions to be meaningful; location not specified in abstract.

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Forward citations

Cited by 1 Pith paper

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    A bivariate generating function for plethysm coefficients with bounded length(λ) is rational; for length 2 an explicit geometric algorithm exists via q-Ehrhart theory, plus linear recursions for the SL2 case.

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