I encountered a mathematical problem while studying mathematical analysis.
The problem is as follows:
Let $n\geq 1$ and consider a polynomial $R$ of degree $2n$ in two variables that is only equal to zero at the point $(0,0)$. Additionally, let $P$ and $Q$ be polynomials of degree at most $2n-2$ in two variables. Suppose that outside of the point $(0,0)$, the following condition holds: $$ \frac{\partial}{\partial x}\frac{Q}{R}=\frac{\partial}{\partial y}\frac{P}{R}. $$ Using Green's formula, this condition guarantees that for any simple closed curve $C$ that encloses the point $(0,0)$ and is sufficiently smooth, the line integral $$\int_C \frac{Pdx+Qdy}{R}$$ is independent of the choice of $C$. The question is, does this line integral always equal zero?
I have been attempting to prove it, but I have not made much progress. It is possible that techniques from complex analysis may be useful. Any insights or counterexamples would be greatly appreciated. Thank you in advance for your help!