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On the magic positivity of Ehrhart polynomials of dilated polytopes

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Pith's one-line read Sufficiently dilating a polynomial with positive coefficients makes it magic positive, and the property is permanent under further dilation.

desk verdict Main theorems 1.3–1.4 are correct and the m-index is a nice invariant; the paper needs a revision to fill in an unproved identity and fix a wrong coefficient in Prop. 5.4. read the letter →

arxiv 2504.21395 v1 pith:YJFDSQTP submitted 2025-04-30 math.CO

Masato Konoike This is my paper · ORCID
classification math.CO MSC 05A1505A1052B20
keywords Ehrhartpolynomialdilatedpolytopemagicpositivelatticeh*-polynomialreal-rootednessm-indexreflexive
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that any polynomial with strictly positive coefficients becomes magic positive after a sufficiently large dilation, and that this property is monotone: once a dilation is magic positive, every larger dilation is too. For lattice polytopes whose Ehrhart polynomials have positive coefficients, this makes the m-index — the smallest dilation at which magic positivity begins — a well-defined invariant. The paper also shows that no fixed dilation size works for all polytopes in any dimension at least 3: for every integer k there is a polytope whose k-th dilation is not magic positive. A closing section computes m-index for simplices, minimal matroid polytopes, hypersimplices, cross polytopes, and other families, and leaves several conjectures.

What carries the argument

The central object is magic positivity: nonnegativity of the coefficients when a degree-$d$ polynomial is expanded in the basis $x^i(x+1)^{d-i}$. The device carrying the argument is Lemma 2.1, which expresses the coefficients of $f(kx)$ in this basis as $g_i(k)=\sum_{j=0}^{i}(-1)^{i-j}\binom{d-j}{i-j} b_j k^j$; since the leading term $b_i k^i$ dominates for large $k$, eventual positivity follows. Monotonicity is proved through a linear-programming alternative (Farkas' lemma): if a larger dilation had a negative coefficient, the alternative would contradict the positivity of the smaller dilation. For the no-universal-bound claim, a family of simplices with $h^*$-polynomial $1+qt+\cdots+qt^d$ is used, chosen so that certain low-degree coefficients of the dilated Ehrhart polynomial grow with $q$ and force the required dilation to grow without bound.

What would settle it

A direct lattice-point count for the polytope $P_{2,3}$ and its dilations determines the numerator of its Ehrhart generating function; if it is not $1+2t+2t^2+2t^3$, the counterexample family behind the unboundedness theorem is invalid.

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Extended reading notes

Core claim

The central claim is that magic positivity is a stable, eventual property of dilation. Writing a degree-d polynomial $f(x)$ with all coefficients positive in the basis $x^i(x+1)^{d-i}$, there is a threshold $k$ such that $f(kx)$ has all nonnegative basis coefficients, and once that threshold is passed, no further dilation can spoil it. In Ehrhart theory, for an Ehrhart-positive lattice polytope $P$ this means $E_{kP}(n)$ is magic positive for all $k$ at least some $m$-index($P$). The paper further claims that for each $d\ge 3$ and each integer $k$, there is a $d$-dimensional Ehrhart-positive lattice polytope whose $k$-th dilation is not magic positive, so no dimension-dependent universal bound exists; the two-dimensional case is settled positively, with $E_{2P}$ always magic positive.

Load-bearing premise

The unboundedness claim in dimensions 3 and above relies on a published formula for the numerator polynomial of a specific simplex-like polytope family and on a binomial-coefficient identity having only positive coefficients; if either of these fails, the conclusion that no fixed dilation works for all such polytopes does not follow.

Editorial extensions

If this is right

  • Every Ehrhart-positive lattice polytope has a finite m-index: some finite dilation makes its Ehrhart polynomial magic positive, and all larger dilations keep it magic positive.
  • Because magic positivity forces the numerator of the Ehrhart generating function to be real-rooted, every sufficiently dilated Ehrhart-positive polytope has a real-rooted numerator polynomial.
  • In dimension 2, the second dilation always works: $E_{2P}$ is magic positive for every lattice polytope $P$, so the m-index is at most 2 there.
  • For each dimension at least 3, the m-index is unbounded: given any integer $k$, there is a $d$-dimensional Ehrhart-positive polytope whose $k$-th dilation fails magic positivity.
  • For CL-polytopes, reflexive polytopes whose Ehrhart roots all lie on the vertical line through $-1/2$ in the complex plane, the m-index is bounded by a function of the dimension alone, in contrast to the general unboundedness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The threshold behavior invites comparison with other dilation-flavoured properties, such as the integer decomposition property and unimodular triangulations of large dilations; magic positivity is unusual in being monotone once attained, so its threshold is a genuine invariant rather than a one-time event.
  • The monotonicity theorem may extend to polynomials with some zero coefficients if the positive leading terms still dominate in each basis coefficient; identifying the exact class of nonnegative polynomials that are eventually magic positive would be a natural next step.
  • The counterexample family $P_{q,d}$ suggests that in fixed dimension the worst-case m-index grows at least linearly with $q$; quantifying the maximum m-index among, say, polytopes with bounded volume or bounded $h^*$-coefficients could give a sharp asymptotic picture.
  • The conjectures on minimal matroids, complete multipartite edge polytopes, and hypersimplices point to a possible closed formula for the m-index in these families; if true, those formulas would give a rich supply of test cases for the general threshold phenomenon.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies 'magic positivity' of a real polynomial of degree d, meaning nonnegative coefficients in the basis {x^i(x+1)^{d-i}}. The main results are: (1) Theorem 1.3: every polynomial with strictly positive coefficients becomes magic positive after a sufficiently large scaling f(kx); (2) Theorem 1.4: if f(kx) is magic positive, then f(k'x) is magic positive for all k' ≥ k; (3) Theorem 1.5: for each d≥3 and each prescribed integer k, there exists an Ehrhart-positive d-dimensional lattice polytope P whose Ehrhart polynomial E_{kP}(n) is not magic positive, while for d=2 the second dilation is always magic positive. The paper then defines the m-index of an Ehrhart-positive polytope as the minimal dilation achieving magic positivity, computes it for several families (standard simplices, minimal matroid base polytopes, complete multipartite edge polytopes, hypersimplices, cross polytopes, standard reflexive simplices, CL-polytopes), and poses several conjectures.

Significance. The positive results are clean and their proofs are mostly elementary and checkable: Theorem 1.3 is a leading-term argument, and Theorem 1.4 uses a neat Farkas-lemma construction. Theorem 1.5 gives a convincing negative answer to the question of a dimension-uniform dilation bound, which is a natural structural question in Ehrhart theory. The m-index is a natural new invariant, and the connection to real-rootedness through Theorem 1.1 gives the notion independent interest. The main caveat is that a key positivity identity, Eq. (4.1) used in the proof of Theorem 1.5(2), is asserted without proof; the manuscript should supply it before publication.

major comments (3)
  1. [Section 4, Eq. (4.1)] The identity Σ_{i=1}^d binom(n+d-i,d) = binom(n+d,d+1) - binom(n,d+1) = Σ_{j=0}^{⌊(d-1)/2⌋} a_j n^{d-2j} with all a_j>0 is asserted without proof or reference. This is load-bearing: the proof of Theorem 1.5(2) uses it to conclude that the coefficient b_1 (for odd d) or b_2 (for even d) is a linear function of q with positive slope, which in turn produces the negative coefficient c_2 or c_3 for large q. Please add a proof, for example via an explicit coefficient formula or a generating-function argument, or give a precise citation. Also reconcile the notation 'a_0,...,a_{⌊d/2⌋}' with the summation upper limit ⌊(d-1)/2⌋ for even d.
  2. [Section 5.2, Proposition 5.4(2)] The displayed coefficient of x^{n-2}(x+1) in D_{2,n}((n-3)x) is incorrect. Direct expansion gives -(n^2-6n+7)/((n-1)(n-2)) rather than -(n^2-6n+7)(n-4)!; for n=5 the value is -1/6, not -2. The coefficient is still negative, so the non-magic-positivity conclusion survives, but the numerical value must be corrected and the expansion of binom((n-3)x+n-2,n-2) should be written out carefully.
  3. [Section 3, proof of Theorem 1.4] The Farkas argument silently normalizes b_0=1: the vector b contains binom(d,i) rather than binom(d,i)b_0, and the inequalities Ax≤b follow from the coefficient inequalities only after this normalization. Since magic positivity is invariant under positive scaling of the polynomial, the normalization is harmless, but it should be stated explicitly.
minor comments (4)
  1. [Section 5.1, Proposition 5.2] The proof that m-index(Δ_d)=d is too terse; the sentence 'Since 1≤d-j≤d' by itself does not show that k=d-1 fails. Please argue explicitly that for k=d every factor has nonnegative coefficients in the magic basis and that for k=d-1 the factor with j=0 introduces a negative coefficient.
  2. [Section 5.4, Proposition 5.8] The proof invokes 'by a similar argument' for many subsets I in the hypersimplex computation. Please provide a uniform argument covering all subsets, or spell out the remaining cases, so that the reader can verify the positivity of C_I.
  3. [Sections 5.5 and 5.6] The tables for m-index(♢_d) and m-index(Δ'_d) are presented without describing the computational method or providing code. If these are experimental values, they should be labeled as such and the method of computation should be indicated.
  4. [Section 5.3, Question 5.7] The notation 'max{q1,...,q_d}' should presumably be 'max{q1,...,q_n}', and the phrase 'equal to one of' could be clarified to indicate that the invariant is conjectured to take one of the three listed values.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorems 1.3-1.4 are derived from coefficient positivity and Farkas' lemma, with no fitted parameters or self-referential premises; Theorem 1.5(2) rests on external cited h*-polynomials and an unproved but non-circular identity.

full rationale

The paper's central claims do not reduce to their inputs. Theorem 1.3 proves existence of a dilation by the leading-term argument in Lemma 2.1: each coefficient g_i(k) has positive leading term b_i k^i, so no magic-positivity assumption is smuggled in. Theorem 1.4 is a Farkas-lemma separation argument: assuming f(kx) is magic positive and f(k'x) is not, the constructed y gives y^T A = 0 and y^T b < 0, contradicting Lemma 3.1; the proof is self-contained apart from citing the standard Farkas lemma. Theorem 1.5(2) depends on the h*-polynomial of P_{q,d} cited from [17] and on identity (4.1) asserted with positive a_j; both are external mathematical premises, not conclusions restated as inputs, and neither involves the present author's prior work. The m-index is defined after Theorems 1.3-1.4 and does not feed back into their proofs. The unproved positivity of a_j in (4.1) and the apparent numerical slip in Proposition 5.4 are correctness risks, not circularity. There are no fitted parameters called predictions and no renaming of known results as new ones.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The paper uses standard results from Ehrhart theory and convex geometry as inputs. The only new object introduced is the m-index, a definitional invariant. No free parameters are fitted to data; the parameter q in the counterexample family P_{q,d} is a construction variable, not a fitted constant.

assumptions (6)
  • standard math Ehrhart's theorem: |nP∩Z^d| is a polynomial for lattice polytope P
    Used throughout to define Ehrhart polynomials.
  • standard math Farkas' lemma (variant with x >= 0)
    Used in the proof of Theorem 1.4 in Section 3.
  • domain assumption h*-polynomial of P_{q,d} is 1 + qt + ... + q t^d
    Cited from [17]; load-bearing for the counterexample in Theorem 1.5(2).
  • domain assumption 2D Ehrhart polynomial formula E_{kP}(n) = binom(kn+2,2)+a binom(kn+1,2)+b binom(kn,2) with a,b non-negative and a >= b
    Used in Theorem 1.5(1), cited from [15, Proposition 36.2].
  • domain assumption Ehrhart positivity of hypersimplices, minimal matroids, edge polytopes, and CL-polytopes
    Cited from [11,12,13,14]; needed for m-index to be well-defined on these families.
  • domain assumption Root bound |alpha + 1/2| <= d(d - 1/2) for Ehrhart polynomials
    Used in Corollary 5.12 to bound m-index of CL-polytopes, cited from [6].
invented entities (1)
  • m-index(P)
    purpose: Minimal positive integer k such that the Ehrhart polynomial of kP is magic positive
    Introduced in Section 5 as a measure of how much a polytope must be dilated. It is a definition, not a falsifiable physical entity, so it has no independent evidence outside the paper.

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Pith. "Pith review of On the magic positivity of Ehrhart polynomials of dilated polytopes." pith.science (2026). https://pith.science/paper/YJFDSQTP

@misc{pith2026250421395,
  author       = {Pith},
  title        = {Pith review of: On the magic positivity of Ehrhart polynomials of dilated polytopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJFDSQTP}},
  note         = {Machine review of arXiv:2504.21395}
}
abstract

A polynomial $f(x)$ of degree $d$ is said to be magic positive if all the coefficients are non-negative when $f(x)$ is expanded with respect to the basis $\{x^i(x+1)^{d-i}\}_{i=0}^d$. It is known that if $f(x)$ is magic positive, then the polynomial appearing in the numerator of its generating function is real-rooted. In this paper, we show that for a polynomial $f(x)$ with positive real coefficients, there exists a positive real number $k$ such that $f(k'x)$ is magic positive for any $k' \geq k$. Furthermore, for any integer $d\geq3$, we show the existence of a $d$-dimensional polytope $P$ such that the Ehrhart polynomial of $kP$ is not magic positive for a given integer $k$. Finally, we investigate how much certain polytopes need to be dilated to make their Ehrhart polynomials magic positive.

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Works this paper leans on

29 extracted references · 28 canonical work pages

  1. [1]

    Athanasiadis

    Christos A. Athanasiadis. On the real-rootedness of the Eulerian transformation. J. Lond. Math. Soc. (2), 111(2):Paper No. e70083, 23, 2025

    work page 2025

  2. [2]

    h∗-polynomials of zonotopes

    Matthias Beck, Katharina Jochemko, and Emily McCullough. h∗-polynomials of zonotopes. Trans. Amer. Math. Soc., 371(3):2021–2042, 2019

    work page 2021

  3. [3]

    On linear transformations preserving the P´ olya frequency property.Trans

    Petter Br¨ and´ en. On linear transformations preserving the P´ olya frequency property.Trans. Amer. Math. Soc., 358(8):3697–3716, 2006

    work page 2006

  4. [4]

    Unimodality, log-concavity, real-rootedness and beyond

    Petter Br¨ and´ en. Unimodality, log-concavity, real-rootedness and beyond. InHandbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), pages 437–483. CRC Press, Boca Raton, FL, 2015

    work page 2015

  5. [5]

    The Eulerian transformation.Trans

    Petter Br¨ and´ en and Katharina Jochemko. The Eulerian transformation.Trans. Amer. Math. Soc. , 375(3):1917–1931, 2022

    work page 1917

  6. [6]

    Norm bounds for Ehrhart polynomial roots

    Benjamin Braun. Norm bounds for Ehrhart polynomial roots. Discrete Comput. Geom., 39(1-3):191– 193, 2008

    work page 2008

  7. [7]

    Normal polytopes, triangulations, and Koszul algebras

    Winfried Bruns, Joseph Gubeladze, and Ngˆ o Viˆ et Trung. Normal polytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. , 485:123–160, 1997

    work page 1997

  8. [8]

    An extremal problem for non-separable matroids

    CP Bruter and George W Dinolt. An extremal problem for non-separable matroids. In Th´ eorie des Matro¨ ıdes: Rencontre Franco-Britannique Actes 14–15 Mai 1970, pages 31–49. Springer, 1971

    work page 1970

Show all 29 references
  1. [9]

    Cox, Christian Haase, Takayuki Hibi, and Akihiro Higashitani

    David A. Cox, Christian Haase, Takayuki Hibi, and Akihiro Higashitani. Integer decomposition prop- erty of dilated polytopes. Electron. J. Combin., 21(4):Paper 4.28, 17, 2014

    2014

  2. [10]

    Sur les poly` edres rationnels homoth´ etiques ` an dimensions

    Eug` ene Ehrhart. Sur les poly` edres rationnels homoth´ etiques ` an dimensions. C. R. Acad. Sci. Paris , 254:616–618, 1962

    1962

  3. [11]

    Hypersimplices are Ehrhart positive

    Luis Ferroni. Hypersimplices are Ehrhart positive. J. Combin. Theory Ser. A , 178:Paper No. 105365, 13, 2021

    2021

  4. [12]

    On the Ehrhart polynomial of minimal matroids

    Luis Ferroni. On the Ehrhart polynomial of minimal matroids. Discrete Comput. Geom. , 68(1):255– 273, 2022. 13

    2022

  5. [13]

    Examples and counterexamples in Ehrhart theory

    Luis Ferroni and Akihiro Higashitani. Examples and counterexamples in Ehrhart theory. EMS Surveys in Mathematical Sciences, 2024

    2024

  6. [14]

    Ehrhart polynomials of rank two matroids

    Luis Ferroni, Katharina Jochemko, and Benjamin Schr¨ oter. Ehrhart polynomials of rank two matroids. Adv. in Appl. Math. , 141:Paper No. 102410, 26, 2022

    2022

  7. [15]

    Algebraic combinatorics on convex polytopes

    Takayuki Hibi. Algebraic combinatorics on convex polytopes . Carslaw Publications, Glebe, 1992

    1992

  8. [16]

    Dual polytopes of rational convex polytopes

    Takayuki Hibi. Dual polytopes of rational convex polytopes. Combinatorica, 12(2):237–240, 1992

    1992

  9. [17]

    Shifted symmetric δ-vectors of convex polytopes

    Akihiro Higashitani. Shifted symmetric δ-vectors of convex polytopes. Discrete Math., 310(21):2925– 2934, 2010

    2010

  10. [18]

    Arithmetic aspects of symmetric edge polytopes

    Akihiro Higashitani, Katharina Jochemko, and Mateusz Micha l ek. Arithmetic aspects of symmetric edge polytopes. Mathematika, 65(3):763–784, 2019

    2019

  11. [19]

    Symmetric decompositions and the Veronese construction

    Katharina Jochemko. Symmetric decompositions and the Veronese construction. Int. Math. Res. Not. IMRN, (15):11427–11447, 2022

    2022

  12. [20]

    The Hilbert series of algebras of the Veronese type

    Mordechai Katzman. The Hilbert series of algebras of the Veronese type. Comm. Algebra, 33(4):1141– 1146, 2005

    2005

  13. [21]

    On positivity of Ehrhart polynomials

    Fu Liu. On positivity of Ehrhart polynomials. In Recent trends in algebraic combinatorics, volume 16 of Assoc. Women Math. Ser. , pages 189–237. Springer, Cham, 2019

    2019

  14. [22]

    Unimodular triangulations of sufficiently large dilations

    Gaku Liu. Unimodular triangulations of sufficiently large dilations. arXiv preprint arXiv:2112.04654 , 2021

    2021 arXiv

  15. [23]

    Understanding and Using Linear Programming (Universitext)

    Jir´ ı Matouek and Bernd G¨ artner. Understanding and Using Linear Programming (Universitext) . Springer-Verlag, Berlin, Heidelberg, 2006

    2006

  16. [24]

    U. S. R. Murty. On the number of bases of a matroid. In Proceedings of the Second Louisiana Con- ference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1971), pages 387–410. Louisiana State University, Baton Rouge, LA, 1971

    1971

  17. [25]

    Compressed polytopes, initial ideals and complete multipartite graphs

    Hidefumi Ohsugi and Takayuki Hibi. Compressed polytopes, initial ideals and complete multipartite graphs. Illinois J. Math. , 44(2):391–406, 2000

    2000

  18. [26]

    Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics

    James Oxley. Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, second edition, 2011

    2011

  19. [27]

    Permutohedra, associahedra, and beyond

    Alexander Postnikov. Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN, (6):1026– 1106, 2009

    2009

  20. [28]

    Richard P. Stanley. Decompositions of rational convex polytopes. Ann. Discrete Math. , 6:333–342, 1980

    1980

  21. [29]

    Richard P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In Graph theory and its applications: East and West (Jinan, 1986) , volume 576 of Ann. New York Acad. Sci., pages 500–535. New York Acad. Sci., New York, 1989. (M. Konoike) Departme...

    1986

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