# Question about polynomials in $\mathbb{C}[x, y, z]$

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 $$F[y, z]$$.

• The pair of such polynomials forms a set? I don't know what this question aims at. – HenrikRüping May 22 at 7:48
• Is that set of pairs closed under addition? That’s a nonlinear differential operator... – Zach Teitler May 22 at 8:12
• 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 $A=\mathbf{C}[x]$. – YCor May 22 at 8:13
• @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. – YCor May 22 at 8:17
• 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. – YCor May 22 at 10:37