On pairs of commuting derivations of the polynomial ring in two variables

Details On pairs of commuting derivations of the polynomial ring in two variables On pairs of commuting derivations of the polynomial ring in two variables

2009-11-11

arxiv.org/abs/0911.2073v1

Mathematics

Commutative Algebra

5 pages

Scientific paper

Let $k$ be an arbitrary field of characteristic zero, $k[x, y]$ be the polynomial ring and $D$ a $k$-derivation of the ring $k[x, y]$. Recall that a nonconstant polynomial $F\in k[x, y]$ is said to be a Darboux polynomial of the derivation $D$ if $D(F)=\lambda F$ for some polynomial $\lambda \in k[x, y]$. We prove that any two linearly independent over the field $k$ commuting $k$-derivations $D_{1}$ and $D_{2}$ of the ring $k[x, y]$ either have a common Darboux polynomial, or $D_{1}=D_{u_{1}}, D_{2}=D_{u_{2}}$ are Jacobian derivations i.e., $D_{i}(f)=\det J(u_{i}, f)$ for every $f\in k[x, y], i=1, 2,$ where the polynomials $u_{1}, u_{2}$ satisfy the condition $\det J(u_{1}, u_{2})=c\in k^{\star}.$ This statement about derivations is an analogue of the known fact from Linear Algebra about common eigenvectors of pairs of commuting linear operators.

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