Inspired by the seminal paper of  Andre Weil on the number of  solutions of equations on  finite fields we would like to present the  following question:

Let $P(x,y), Q(x,y)$ be two polynomials of degree $n$ with coefficient in a  finite field $k$ of  $q$ elements.Then we  have  a linear operator on $k[x,y]$ via $$D(f)=Pf_x+Qf_y$$.

Inspired by this  question: [Codimension of the range of certain linear operators](https://mathoverflow.net/questions/164059/codimension-of-the-range-of-certain-linear-operators) we are interested in the  codimension of the range of the  above differential operator.

Denote by $I(n,q)$ the uniform upper bounded for the codimension of the range of  the  above differential operator where $P,Q$ are arbitrary $n$ degree polynomials in two variables and $k$ is a field of order $q$.

>What is the asymptotic behavior of  $I(q,n)$?


**Note:** The  codimension of the  range of this operator in an appropriate algebra could be  an upper bound for the number of  isolated closed orbits of the  differential equation $P\partial_x +Q\partial_y$ on the real plane. But we realized that in some case this  codimension is infinite(When we apply this on the real polynomial algebra or the algebra of  smooth functions provided we have at least one isolated closed orbits). So this leads us to consider a finite  field analogy. On the other hand as it is written in some  comment  conversation of the above linke of MO conversation, we possibly have more luck with consideration of Holomorphic objects. On the other hand it  would be interesting to study the  kernel of  the  above  operator $D$ then study the algebraic  curve $u(x,y)=0$  where $u$  belongs to Ker of $D$.(its genus, etc). So these  situations leads me to be immersed in both wonderful books of Griffiths and  Harris, Principles of Algebraic  Geometry and  Hartshorne  Algebraic Geometry.  So  your  help, hint and answer or comments are very appreciated to find an appropriate way in either finite or Holomorphic case. Thank you!