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Two-Variable Factorial Conjecture

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factorial-conjecture-two-variablesAlgebraposed by Arno van den Essen, David Wright, Wenhua Zhao, 2011recorded: candidate

1 attempt · no person has looked

Statement

Let L(xayb)=a! b!\mathcal{L}(x^{a}y^{b})=a!\,b! on C[x,y]\mathbb{C}[x,y]. The Factorial Conjecture asks whether L(fm)=0\mathcal{L}(f^{m})=0 for every m≥1m\geq 1 forces f=0f=0. The homogeneous two-variable case was settled by Liu and Sun; the inhomogeneous problem does not reduce to it, because radial integration couples the homogeneous layers through Gamma factors. A claimed proof settles the full two-variable case affirmatively.

Context

Claimed in a self-published research draft; a standalone by-product is the transcendence of the integral of exp(q) between distinct algebraic endpoints for nonconstant algebraic q

A named conjecture from the Image Conjecture circle around the Jacobian conjecture, with a small dedicated literature; the two-variable case specifically.

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Attempts

1 attempt

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  • #1

    Attempt 1

    proof attemptGPT-5.6 Sol, Claude Opus 5 with Christopher D. Long ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.6 Sol, Claude Opus 5
    people
    Christopher D. Long

    Per the author's disclosure, the manuscript was developed through interactive work with ChatGPT 5.6 Sol, which assisted in proof discovery, organization, symbolic checking, reference verification, adversarial auditing and drafting; Claude Opus 5 contributed the semisimple-projector strategy, the reduction to phase-polynomial moments, and further adversarial audits. The AI systems are not authors and the human author takes full responsibility.

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