Recognition: 1 theorem link
· Lean TheoremEhrhart quasi-polynomials of rational polytopes by real dilations
Pith reviewed 2026-05-12 04:19 UTC · model grok-4.3
The pith
The Ehrhart function of a rational polytope extends to real dilations t as a quasi-polynomial whose coefficients are periodic piecewise polynomials when the affine hull contains the origin.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The Ehrhart function L(P,t) for a rational n-polytope P and real t greater than or equal to zero equals the sum from k equals zero to n of c_k(P,t) times t to the power k, where the c_k(P,t) are periodic piecewise polynomials of degree n minus k if the affine hull of P contains the origin and are periodic functions vanishing almost everywhere otherwise. When P is a rational simplex the coefficient functions c_k are given explicitly in terms of vertex information, and the reciprocity law still holds.
What carries the argument
The quasi-polynomial decomposition L(P,t) equals sum c_k(P,t) t^k carried by the periodic coefficient functions c_k(P,t) that depend on whether the affine hull contains the origin.
Load-bearing premise
The polytope must be rational so that its dilations produce periodic coefficients in the lattice-point count.
What would settle it
Pick any rational simplex with known vertices, evaluate the explicit formulas for the c_k at a non-integer real t, and compare the resulting value against the actual number of lattice points inside that dilated simplex.
read the original abstract
This paper is to study the Ehrhart function $L(P,t)$ of a rational $n$-polytope $P$, defined as the number of lattice points of dilated polytopes $tP$ with real numbers $t\geq 0$. It turns out that $L(P,t)$ is a quasi-polynomial of real variable $t$ in the sense that \[ L(P,t)=\sum_{k=0}^{n} c_k(P,t)t^k, \quad t\geq 0, \] where $c_k(P,t)$ are periodic piecewise polynomials of degree $n-k$ if ${\rm aff}\,P$ contains the origin, and are periodic functions vanishing almost everywhere otherwise. When $P$ is a rational simplex $\sigma$, the coefficient functions $c_k(\sigma,t)$ are given explicitly in terms of vertex information of the simplex $\sigma$. Moreover, the reciprocity law still holds.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Ehrhart function L(P,t) counting lattice points in real dilations tP (t ≥ 0) of a rational n-polytope P. It claims that L(P,t) equals the sum from k=0 to n of c_k(P,t) t^k, where the coefficient functions c_k(P,t) are periodic piecewise polynomials of degree n-k if aff P contains the origin and are periodic functions vanishing almost everywhere otherwise. For a rational simplex σ the c_k(σ,t) are given explicitly in terms of vertex information, and the reciprocity law continues to hold.
Significance. If the counting arguments in the full text are correct, the result extends classical Ehrhart theory from integer to real dilations by exhibiting an explicit quasi-polynomial structure whose coefficients are periodic and piecewise polynomial (or vanish a.e.). The explicit vertex formulas for simplices and the verification that reciprocity persists supply concrete, usable statements that could support further work in combinatorial geometry and integer programming.
minor comments (2)
- Abstract: the phrase 'periodic piecewise polynomials' is used without a local definition; a one-sentence clarification of what 'piecewise polynomial' means for a periodic function of real t would improve readability.
- The manuscript would benefit from a short paragraph in the introduction comparing the new real-dilation quasi-polynomials with the classical integer quasi-polynomials of Ehrhart.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript, the positive assessment of its significance, and the recommendation for minor revision. No specific major comments were provided in the report.
Circularity Check
No significant circularity detected
full rationale
The paper defines L(P,t) directly as the lattice-point count in the real dilation tP and derives its quasi-polynomial representation with periodic piecewise-polynomial coefficients from the rationality of P and the geometry of aff P. The explicit vertex formulas for simplices and the reciprocity statement follow from the same counting argument. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the derivation is self-contained against the standard lattice-point definition.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard definition of the Ehrhart function as the number of lattice points in tP
- domain assumption P is a rational polytope
Reference graph
Works this paper leans on
-
[1]
Velleda Baldoni, Nicole Berline, Matthias K¨ oppe, and Mich` ele Vergne,Intermediate sums on polyhedra: computation and real ehrhart theory, Mathematika59(2013), no. 1, 1–22
work page 2013
-
[2]
Alexander Barvinok,Computing the ehrhart quasi-polynomial of a rational simplex, Mathematics of Computation75(2006), no. 255, 1449–1466
work page 2006
-
[3]
Alexander Barvinok and James E Pommersheim,An algorithmic theory of lattice points in polyhedra, New perspectives in algebraic combinatorics38(1999), 91–147
work page 1999
- [4]
-
[5]
Matthias Beck and Sinai Robins,Computing the continuous discretely: Integer-point enumeration in polyhedra, Vol. 2, Springer, 2007
work page 2007
-
[6]
Michel Brion and Michele Vergne,Lattice points in simple polytopes, Journal of the American Mathe- matical Society (1997), 371–392
work page 1997
- [7]
-
[8]
B. Chen,Weight functions, double reciprocity laws, and volume formulas for lattice polyhedra, Proceed- ings of the National Academy of Sciences95(1998), no. 16, 9093–9098
work page 1998
-
[9]
,The gauss-bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature, Proceedings of the American Mathematical Society137(2009), no. 5, 1601– 1611
work page 2009
-
[10]
Beifang Chen,Lattice points, dedekind sums, and ehrhart polynomials of lattice polyhedra, Discrete & Computational Geometry28(2002), 175–199
work page 2002
-
[11]
,Ehrhart polynomials of lattice polyhedral functions, Contemporary Mathematics374(2005), 37–64
work page 2005
- [12]
-
[13]
Ricardo Diaz and Sinai Robins,The ehrhart polynomial of a lattice polytope, Annals of Mathematics 145(1997), no. 3, 503–518
work page 1997
-
[14]
Eugene Ehrhart,Sur les poly` edres rationnels homoth´ etiques ` a n dimensions, CR Acad. Sci. Paris254 (1962), 616
work page 1962
-
[15]
,Sur un probleme de g´ eom´ etrie diophantienne lin´ eaire i, J. reine angew. Math226(1967), 1–29
work page 1967
-
[16]
,Sur un probleme de g´ eom´ etrie diophantienne lin´ eaire ii, J. reine angew. Math227(1967), no. 25, C49
work page 1967
-
[17]
Eva Linke,Rational ehrhart quasi-polynomials, Journal of Combinatorial Theory, Series A118(2011), no. 7, 1966–1978
work page 2011
-
[18]
Ian G Macdonald,The volume of a lattice polyhedron, Mathematical proceedings of the cambridge philosophical society, 1963, pp. 719–726
work page 1963
-
[19]
,Polynomials associated with finite cell-complexes, Journal of the London Mathematical Society 2(1971), no. 1, 181–192
work page 1971
-
[20]
Peter McMullen,Lattice invariant valuations on rational polytopes, Archiv der Mathematik31(1978), no. 1, 509–516
work page 1978
-
[21]
R. Stanley,Enumerative combinatorics volume 1 second edition, Cambridge studies in advanced math- ematics (2011)
work page 2011
-
[22]
Alan Stapledon,Counting lattice points in free sums of polytopes, Journal of Combinatorial Theory, Series A151(2017), 51–60. 20
work page 2017
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.