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 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, see this answer 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 link 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!
