algebra / Commutative Algebra, Transcendence
Let $\mathcal{L}(x^{a}y^{b})=a!\,b!$ on $\mathbb{C}[x,y]$. The Factorial Conjecture asks whether $\mathcal{L}(f^{m})=0$ for every $m\geq 1$ forces $f=0$. 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.
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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
Research memory
Let $\mathcal{L}(x^{a}y^{b})=a!\,b!$ on $\mathbb{C}[x,y]$. The Factorial Conjecture asks whether $\mathcal{L}(f^{m})=0$ for every $m\geq 1$ forces $f=0$. 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.
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
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