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Orlik-Solomon algebras and Tutte polynomials*

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orlik-solomon-algebras-and-tutte-polynomialsAlgebraic Topologymath.ATmath.CVposed by Carrie J. Eschenbrenner, Michael J. Falkrecorded: open · 1 machine check, unexamined

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Statement

For central arrangements whose underlying matroid is connected, the homotopy type of the complement determines the underlying matroid.

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Candidate 1 of the open problems stated in "Orlik-Solomon algebras and Tutte polynomials*", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: Reconstructed conjecture: In ZFC, for finite central complex hyperplane arrangements A,B\mathcal A,\mathcal B, let

    C(A)=V∖⋃H∈AHC(\mathcal A)=V\setminus\bigcup_{H\in\mathcal A}H

    and let M(A)M(\mathcal A) be the matroid of the defining linear forms. If M(A)M(\mathcal A) and M(B)M(\mathcal B) are connected matroids and C(A)≃C(B)C(\mathcal A)\simeq C(\mathcal B), then M(A)≅M(B)M(\mathcal A)\cong M(\mathcal B). This matches the quoted Conjecture 5.4 and the paper’s central-arrangement/matroid setting.

    Result: The conjecture is false, even for simple essential central arrangements.

    Let A0\mathcal A_0 in C3\mathbb C^3, with coordinates x,y,zx,y,z, have defining forms

    a=x,b=y,c=x+y,d=z,e=x+2y+3z.a=x,\quad b=y,\quad c=x+y,\quad d=z,\quad e=x+2y+3z.

    For p=a,dp=a,d, define central arrangements Ap\mathcal A_p in C4\mathbb C^4, coordinates x,y,z,tx,y,z,t, by adding

    f=t,gp=t+p.f=t,\qquad g_p=t+p.

    Thus

    Aa: x,y,x+y,z,x+2y+3z,t,t+x,\mathcal A_a:\ x,y,x+y,z,x+2y+3z,t,t+x, Ad: x,y,x+y,z,x+2y+3z,t,t+z.\mathcal A_d:\ x,y,x+y,z,x+2y+3z,t,t+z.

    Their complements are biholomorphic. Let

    C0={(x,y,z):xy(x+y)z(x+2y+3z)≠0}.C_0=\{(x,y,z):xy(x+y)z(x+2y+3z)\neq 0\}.

    For p=a,dp=a,d,

    C(Ap)={(v,t):v∈C0, t≠0, t+p(v)≠0}.C(\mathcal A_p)=\{(v,t):v\in C_0,\ t\neq0,\ t+p(v)\neq0\}.

    Since p(v)≠0p(v)\neq0 on C0C_0, the map

    (v,t)↦(v,−tp(v))(v,t)\mapsto \left(v,-\frac{t}{p(v)}\right)

    is a biholomorphism

    C(Ap)≅C0×(C∖{0,1}).C(\mathcal A_p)\cong C_0\times(\mathbb C\setminus\{0,1\}).

    Hence C(Aa)≅C(Ad)C(\mathcal A_a)\cong C(\mathcal A_d).

    The matroids are not isomorphic. In the five-element matroid of A0\mathcal A_0, the only 3-circuit is {a,b,c}\{a,b,c\}. In M(Aa)M(\mathcal A_a), the 3-circuits are exactly

    {a,b,c},{a,f,ga}.\{a,b,c\},\quad \{a,f,g_a\}.

    In M(Ad)M(\mathcal A_d), the 3-circuits are exactly

    {a,b,c},{d,f,gd}.\{a,b,c\},\quad \{d,f,g_d\}.

    Thus the multisets of numbers of 3-circuits through an element are

    M(Aa): {2,1,1,1,1,0,0},M(\mathcal A_a):\ \{2,1,1,1,1,0,0\},

    but

    M(Ad): {1,1,1,1,1,1,0}.M(\mathcal A_d):\ \{1,1,1,1,1,1,0\}.

    This invariant is preserved by matroid isomorphism, so the matroids are non-isomorphic.

    Both matroids are connected: {a,b,c}\{a,b,c\} is a circuit, {a,b,d,e}\{a,b,d,e\} is a circuit connecting d,ed,e to a,b,ca,b,c, and the added circuit {p,f,gp}\{p,f,g_p\} connects f,gpf,g_p. Therefore both examples satisfy the conjecture’s connectedness hypothesis.

    Verification audit: the arrangements are finite, central, essential, simple, complex hyperplane arrangements; their complements are biholomorphic, hence homotopy equivalent; their underlying matroids are connected but non-isomorphic. This directly refutes the reconstructed conjecture.

    Citation: No published disproof is invoked here. The conjecture is from Eschenbrenner–Falk, “Orlik-Solomon algebras and Tutte polynomials,” arXiv:math/9805128.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE2

      PASS

      The proposed counterexample attacks the stated conjecture directly. The biholomorphism C(Ap)≅C0×(C∖{0,1})C(\mathcal A_p)\cong C_0\times(\mathbb C\setminus\{0,1\}) is valid since pp is nonzero on C0C_0, so the two complements are homotopy equivalent. The matroid computations are also correct: the two arrangements have different multisets of incidences with 3-circuits, hence non-isomorphic matroids, while the displayed circuits connect all elements, so the connectedness hypothesis is satisfied. Thus this rigorously disproves the conjecture as stated.

      Novelty assessment

      TYPE2

      Classification rationale: The accepted construction gives connected, simple, essential central arrangements with biholomorphic complements but non-isomorphic matroids, directly refuting Eschenbrenner–Falk Conjecture 5.4. The argument is elementary and closely related to standard deconing/product ideas, so this is not a top-journal advance. But a clean counterexample to an explicit arrangement-theoretic conjecture is plausibly publishable as a short note in a standard combinatorics/arrangements venue. Borderline with TYPE1, but I grade TYPE2 for resolving the stated conjecture.

      Literature check: I found no published connected-central counterexample. The closest known results are already in Eschenbrenner–Falk: their Theorem 5.3 gives diffeomorphic complements for A0⊕A1\mathcal A_0\oplus\mathcal A_1 and S⊕P(A0,A1)\mathcal S\oplus P(\mathcal A_0,\mathcal A_1), but the relevant matroids are disconnected; the paper then states Conjecture 5.4 precisely to exclude such examples. Earlier Falk/Rose–Terao examples and affine/decone constructions similarly do not settle the connected central case. Searches for the exact conjecture phrase, the paper title, “homotopy equivalent complements” with “non-isomorphic matroids,” and “connected central arrangements” did not reveal a later proof or disproof.

      Citation: C. J. Eschenbrenner and M. J. Falk, “Orlik-Solomon algebras and Tutte polynomials,” arXiv:math/9805128, especially Theorem 5.3 and Conjecture 5.4. Also related: M. Falk, “Homotopy types of line arrangements,” Invent. Math. 111 (1993), 139–150.

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