The Generalized Vanishing Conjecture fails in five variables
Assessment
The evidence, examined directly, establishes the claim.
Zhao's Generalized Vanishing Conjecture says that if a constant-coefficient differential operator Λ and a polynomial P satisfy Λ^m(P^m) = 0 for every m ≥ 1, then for every polynomial Q, Λ^m(Q P^m) = 0 for all sufficiently large m. In July 2026, after Levent Alpöge's announced counterexample to the three-dimensional Jacobian conjecture and Long's SU(2) construction, Alexander Dvorsky proposed an explicit five-variable counterexample: P = (t+c)(ad+bt), Λ = ∂_t(∂_a∂_d − ∂_b∂_c), Q = −c.
The construction checks out. Because P has degree 3 and Λ has order 3, Λ^m(P^m) is a constant. A direct expansion shows it equals (m!)^3 Σ_j (−1)^{m−j} C(m,j), which is zero for every m ≥ 1. Λ^m(Q P^m) is linear, and exact computation for m = 2 to 10 gives (−1)^{m+1}(m+1)!(m!)^2 · t, which is never zero. David Speyer could not reproduce the result at first, then traced the discrepancy to his own transcription error and confirmed the construction. One gap remains: nonvanishing for every m is established here by computation and a clear closed-form pattern, not by a written proof. Because the conjecture only requires Λ^m(Q P^m) to vanish for all large m, nonvanishing for infinitely many m is enough to refute it, and the pattern leaves little room for doubt. The main shortfall is that no refereed write-up has yet been located.
Full reasoning: the evidence and decisions behind this verdict
Own check (sympy, exact): with P=(t+c)(ad+bt), Λ=∂t(∂a∂d−∂b∂c), Λ^m(P^m)=0 for m=1..10, and Λ^m(−c·P^m) = c+2t (m=1), −24t, 864t, −69120t, 10368000t, −2612736000t, 1024192512000t, −589934886912000t, 477847258398720000t, −525631984238592000000t (m=2..10), matching (−1)^{m+1}(m+1)!(m!)^2 t. Hand proof of the vanishing part for all m: Λ^m(P^m) is the apolar pairing of ∂t^m D^m (D=∂a∂d−∂b∂c) with P^m. Only the monomials of P^m with t-degree m contribute: C(m,j)^2 (ad)^j (bc)^{m−j}. D^m pairs with (ad)^j(bc)^{m−j} as C(m,j)(−1)^{m−j}(j!)^2((m−j)!)^2, and ∂t^m gives m!. The sum reduces to (m!)^3 Σ_j (−1)^{m−j}C(m,j) = 0. The nonvanishing of Λ^m(QP^m) for all m≥2 rests on the computed pattern; an analogous pairing computation should give the closed form, but it was not written out here. Source: Dvorsky's comment of 2026-07-23 on Secret Blogging Seminar (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), with Speyer's confirmation after correcting his typo "(ad+bc)" to "(ad+bt)". The statement of GVC used is Zhao's: Λ any constant-coefficient differential operator, any Q. What would change the verdict: a different reading of GVC(n) that restricts Λ (e.g. to Laplacians or homogeneous quadratic operators), which would make this a counterexample to a different statement. The Laplacian-only Vanishing Conjecture is a separate claim; a 48-variable witness against it was posted in the same thread.
Decomposition
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Provenance
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Alpöge’s Jacobian counterexample and Long’s SU(2) construction yield a 5-variable counterexample to the Generalized Vanishing Conjecture.
Comment by Alexander Dvorsky giving explicit P, Λ, Q with Λᵐ(Pᵐ)=0 but Λᵐ(QPᵐ)≠0; Speyer initially failed to verify due to his own transcription error, then confirmed the construction.
The source's own evidence bears what it asserts. The comment gives the full construction, which can be checked by direct computation. The blog host initially failed to reproduce it, then confirmed it after finding his own transcription error.
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