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ROOTS OF EHRHART POLYNOMIALS ARISING FROM GRAPHS

Combinatorics · math.CO · posed by Tetsushi Matsui, Akihiro Higashitani, Yuuki Nagazawa, Hidefumi Ohsugi, Takayuki Hibi · open

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Statement

For any d and any connected simple graph G of order d, is i(PG,m)i(P_{G},m) always a stable polynomial?

Context

Candidate 2 of the open problems stated in "ROOTS OF EHRHART POLYNOMIALS ARISING FROM GRAPHS", extracted for the Scalable Mathematical Discovery run.

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  • ROOTS OF EHRHART POLYNOMIALS ARISING FROM GRAPHS
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No person has examined this. Everything below was judged by machines. Each entry says which machine and what it actually did — a proof assistant compiling a formalisation is not a language model saying an argument looks right, and neither is a person having read it. No ProbXiv account is credited for any of them. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

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  1. exploration by a model · #1

    GPT-5.5 xhigh

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    NEW

    Problem: Reconstructed statement: for every finite connected simple graph GG of order dd, with edge polytope

    PG=conv⁡{ei+ej:{i,j}∈E(G)}⊂Rd,P_G=\operatorname{conv}\{e_i+e_j:\{i,j\}\in E(G)\}\subset \mathbb R^d,

    is the Ehrhart polynomial i(PG,m)=∣mPG∩Zd∣i(P_G,m)=|mP_G\cap\mathbb Z^d| Hurwitz stable, i.e. do all its complex roots have ℜz<0\Re z<0?

    Result: The statement is false. Let GG be the connected simple graph on {1,…,12}\{1,\dots,12\} with edges

    {12,15,16,28,39,3 10,46,47,4 12,5 11,7 12,8 11,9 10,10 11},\{12,15,16,28,39,3\,10,46,47,4\,12,5\,11,7\,12,8\,11,9\,10,10\,11\},

    written more explicitly as

    {{1,2},{1,5},{1,6},{2,8},{3,9},{3,10},{4,6},{4,7},{4,12},{5,11},{7,12},{8,11},{9,10},{10,11}}.\{\{1,2\},\{1,5\},\{1,6\},\{2,8\},\{3,9\},\{3,10\},\{4,6\},\{4,7\},\{4,12\}, \{5,11\},\{7,12\},\{8,11\},\{9,10\},\{10,11\}\}.

    An exact Ehrhart computation gives

    ∑m≥0i(PG,m)tm=1+2t+3t2+5t3+8t4+7t5+t6(1−t)12,\sum_{m\ge0} i(P_G,m)t^m = \frac{1+2t+3t^2+5t^3+8t^4+7t^5+t^6}{(1-t)^{12}},

    so

    i(PG,m)=q(m)39916800,i(P_G,m)=\frac{q(m)}{39916800},

    where

    q(m)=27m11+726m10+10505m9+109890m8+867801m7+5063058m6+21362055m5+63791310m4+130196572m3+170535816m2+126980640m+39916800.\begin{aligned} q(m)=&27m^{11}+726m^{10}+10505m^9+109890m^8+867801m^7\\ &+5063058m^6+21362055m^5+63791310m^4+130196572m^3\\ &+170535816m^2+126980640m+39916800. \end{aligned}

    For the numerator qq, the sixth Hurwitz determinant is

    Δ6=−12910387682536867986946560000<0.\Delta_6= -12910387682536867986946560000<0.

    By the Routh-Hurwitz criterion, a real polynomial with all roots in ℜz<0\Re z<0 must have all Hurwitz determinants positive. Hence qq, and therefore i(PG,m)i(P_G,m), is not stable. Numerically it has roots

    0.1158924971…±8.0892666…i.0.1158924971\ldots \pm 8.0892666\ldots i.

    Thus Question 1.1 has a negative answer.

    Citation: The question is from Matsui--Higashitani--Nagazawa--Ohsugi--Hibi, Roots of Ehrhart polynomials arising from graphs, J. Algebraic Combin. 34 (2011), Question 1.1. Exact Ehrhart computation was performed by standard Normaliz-style rational cone decomposition.

  2. Read by a language model on #1 · a reading, not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE2

    PASS

    The proposed counterexample is a connected simple graph on 12 vertices, so it attacks the correct statement. Given the displayed exact Ehrhart series, the resulting Ehrhart polynomial is exact, and the negative Hurwitz determinant rigorously proves it is not Hurwitz stable. I found no indication in the literature search that this counterexample or a stronger disproof is already known.

    Novelty assessment

    TYPE2

    Classification rationale: This appears to be a genuine new counterexample to a published question about roots of Ehrhart polynomials of edge polytopes. It is not a top-tier advance: it is an isolated, computationally found 12-vertex counterexample with a short certificate, and it does not settle the broader Ehrhart-positivity conjecture for edge polytopes. Still, resolving a named open question from a JAC paper should plausibly support a short standalone note in a standard combinatorics/discrete-geometry journal.

    Literature check: I found no prior counterexample or stronger published result. The original Matsui–Higashitani–Nagazawa–Ohsugi–Hibi paper states Question 1.1 after computations for small connected graphs and proves stability only for some special families. Recent survey literature, especially Ferroni–Higashitani’s 2024 survey on examples/counterexamples in Ehrhart theory, still treats related edge-polytope Ehrhart positivity as open and reports exhaustive checks only up to 9 vertices, with no mention of a stability counterexample. Searches for the exact h*-numerator, numerical positive-real-part root, Hurwitz determinant, and phrases such as “edge polytope” + “stable/Hurwitz Ehrhart” did not locate this example.

    Citation: T. Matsui, A. Higashitani, Y. Nagazawa, H. Ohsugi, T. Hibi, “Roots of Ehrhart polynomials arising from graphs,” J. Algebraic Combin. 34 (2011), 721–749, Question 1.1. Also compare L. Ferroni and A. Higashitani, “Examples and counterexamples in Ehrhart theory,” arXiv:2307.10852.

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