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arXiv · 2609.06465

When is a polynomial in three variables cylindrical?

Xiaojin Lin
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Abstract

We prove that a nonconstant polynomial $f\in\mathbb{C}[x_1,x_2,x_3]$ is cylindrical (that is, $f\in\mathbb{C}[x_1,x_2]$ after an invertible linear change of coordinates) if and only if its bordered Hessian determinant vanishes identically. The same conclusion holds over $\mathbb{R}$. We also give explicit counterexamples to this criterion in every dimension $n\ge4$.

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Xiaojin Lin. 2026-09-06. When is a polynomial in three variables cylindrical?. https://arxiv.org/abs/2609.06465

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