Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map
$D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the differential operator corresponding to the vector field $P\partial_{x}+Q\partial_{y}$.
Are there polynomials $P$ and $Q$ such that the codimension of the range of $D$ is finite but different from $0$ and $1$?
Note This codimension is $0$, $1$ and $\infty$ for $\partial_{x}$, $x\partial_{x}+y\partial_{y}$ and $x\partial_{x}-y\partial_{y}$, respectively.
Motivation It can be easily shown that this codimension is an upper bound for the number of closed orbits of the polynomial vector field $P\partial_{x}+Q\partial_{y}$.
However there is a Painful fact: If an smooth vector field $X$ on $\mathbb{R}^{2}$ has a limit cycle which surrounds a nondegenerate singularity (or a singularity which has a local smooth first integral at a deleted neighborhood of the singularity and is discontinuous at the singularity), then the codimension of the range of the differential operator $D_{X}$ on $C^{\infty} (\mathbb{R}^{2})$ is infinite ! The reason is as follows.
Assume that a limit cycle $\gamma$ is attractor and surounds a source singularity at the origin.
Non degeneracy of the singularity help us to find an open set D around origin whose boundary $S$ is a smooth closed curve and the vector field is transverse to $S$ toward the exterior of $S$. So the flow of $X$ defines a smooth retraction $r:\bar{D}\setminus \{0\} \to S$. In fact $r(x)$ is the intersection point of the orbit of $x$ with $S$ This shows that there is an smooth real valued function $\phi$ on $\mathbb{R}^2 \setminus \{0\}$ such that locally around the origin we have $X. \phi =0$ and $P\circ \phi$ is discontinuous at origin for every polynomial $P(x)$. Such $\phi$ can be constructed locally by $\psi \circ
r$ where $\psi:S \to \mathbb{R}$ is an arbitrary non constant smooth function.Now we choose an arbitrary smooth extension of this locally defined $\phi$ to whole punctured plane. This extension is denoted again
by $\phi$. Obviously every nontrivial linear combination $\sum \lambda_i \phi^i$ is discontinuos at origin. Because its restriction to every small neighborhood of the origin has the same range(image) as the range of $\sum \lambda_i \psi^i: S \to \mathbb{R}$. However $X.\phi^i$ is a smooth function on whole plane, for every $i$ since it vanish locally around the origin.
Now for every $n$,the following set represents an independent subset of $C^{\infty}(\mathbb{R}^2)/Image(D_X)$
$$\{X.\phi, X.\phi^2, \ldots, X.\phi^n \}$$ To prove this, we strongly use the fact that we have at least one limit cycle.
More precisely: assume that $\sum_{i=1}^n \lambda_i (X.\phi^i)= X.F$ for some smooth function $F\in C^{\infty} (\mathbb{R}^2)$. Then $F-\sum \lambda_i \phi^i$ is a first integral(constant of motion) on the puntured plane. So it is constant on the basian of attraction of the limit cycle. A punctured neighborhood of the origin is contained in the basian of attraction of limit cycle. Thus locally around the origin,$F$ differs $\sum_{i=1}^n \lambda_i \phi^i$ by a constant(In a deleted neighborhood). This is a contradiction because $F$ is smooth at origin but $\sum \lambda_i \phi^i$ is discontinuous at origin.
So unfortunately we encounter with the following two inconsistent and opposite but true results:
1)The codimension of the range of $D_X$ is an upper bound for the number of limit cycles.
2.If the codimension is finite then there is no any limit cycle.
So we should choose an appropriate function algebra, different from $C^{\infty} (\mathbb{R}^{2})$, which is invariant under the differential operator corresponding to an algebraic vector field.
#Added:
According to the answer of Loic Teyssier we put $X=x^k\partial_x+y\partial_y$.
Then $X$ can act on various spaces $C^{\infty}(\mathbb{R}^2),\; C^{\omega}(\mathbb{R}^2)$ or the space of holomorphic functions from $\mathbb{C}^2$ to $\mathbb{C}$. What can be said about the dimension of the cokernels in each of these actions?
Best Answer
The question admits a positive answer when one looks at it for the ring of formal power series $\mathbb R[[x,y]]$. For instance the vector field $x^k\partial_x+y\partial_y$ has a formal cokernel of dimension $k$. Yet when restricted to polynomials the property disappears: the polynomial cokernel is infinite-dimensional (you can never reach monomials of the form $x^{k-1}y^m$ for $m\in\mathbb N$ nor $x^n$ with $n<k$, and those are the only problems).
Somehow I believe the answer should also be positive in the polynomial case. But I have no example offhand.