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What can be said about pairs of polynomials $P, Q\in\mathbb{C}[x, y, z]$, such that $\frac{\partial P}{\partial y}\frac{\partial Q}{\partial z} - \frac{\partial P}{\partial z}\frac{\partial Q}{\partial y}\in\mathbb{C}[x]$?

I am interested in examples of such $P, Q$.

The similar question can be asked if we replace $\mathbb{C}[x]$ with any field $F$ of characteristic zero and $\mathbb{C}[x, y, z]$ with $F[y, z]$.

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    $\begingroup$ The pair of such polynomials forms a set? I don't know what this question aims at. $\endgroup$
    – HenrikRüping
    Commented May 22, 2020 at 7:48
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    $\begingroup$ Is that set of pairs closed under addition? That’s a nonlinear differential operator... $\endgroup$
    – Zach Teitler
    Commented May 22, 2020 at 8:12
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    $\begingroup$ To start with you could ask the question in $L[y,z]$ for $L$ a field of characteristic zero, then specify to $L=\mathbf{C}(x)$, and then intersect with $A[y,z]$ with $A=\mathbf{C}[x]$. $\endgroup$
    – YCor
    Commented May 22, 2020 at 8:13
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    $\begingroup$ @ZachTeitler is right. Actually, for every $(P,Q)$ we can write $(P,Q)=(P,P)+(0,Q-P)$, and both pairs $(P,P)$ and $(0,Q-P)$ satisfy the equation, but not $(P,Q)$ in general. $\endgroup$
    – YCor
    Commented May 22, 2020 at 8:17
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    $\begingroup$ For any scalar ring, on the $L$-algebra $L[y,z]$, the usual product along with $\{P,Q\}=\partial_yP\partial_zQ-\partial_zP\partial_yQ$ form a commutative Poisson algebra. That is, $\{\cdot,\cdot\}$ is a Lie bracket, and is a derivation (for the multiplication) with respect to each variable. I don't know to which extent this can answer the question, but at least it's a very well-studied kind of structure. See Wikipedia: Poisson algebra. $\endgroup$
    – YCor
    Commented May 22, 2020 at 10:37

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