An Explicit Five-Variable Counterexample to the Generalized Vanishing Conjecture

Abstract

We give an explicit counterexample in five variables to the Generalized Vanishing Conjecture. The construction is motivated by the recent counterexample to the Mathieu conjecture for SU(2). In the polynomial ring C[a,b,c,d,t], set P = (t+c)(ad+bt), Q = c, and let Λ = d/dt (d/da d/dd - d/db d/dc). We prove that Λ^m(P^m) = 0 for every m >= 1, whereas for every m >= 2, Λ^m(QP^m) = (-1)^m (m!)^2 (m+1)! t, which is nonzero. Thus the Generalized Vanishing Conjecture fails in dimension 5.

Disclosure

“form of the construction. The verification below is entirely algebraic and does not require a general implication from the Mathieu conjecture to the Generalized Vanishing Conjecture. During the preparation of this note, the author used OpenAI ChatGPT (model GPT-5.6 Sol, July 2026) as an interactive research assistant. ChatGPT assisted in testing the homogenized five- variable construction, generating computational verification code and locating relevant literature. All computations and”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T04:59:56.233612Z

Structural counts