So, I am exploring functions on the circle understood as the elements of $$\frac{\mathbb{R}[x,y]}{\langle x^2+y^2-1\rangle}.$$ I am particularly interested in the image and kernel of the angular vector field $$x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}.$$
In polar coordinates it is easy to see that the functions in the kernel are constant on the circle. Thus they are isomorphic to $\mathbb{R}$. On the other hand, given some function $f(r,\theta)$ in polar coordinates we have $$f(r,\theta)=\frac{\partial}{\partial\theta}\int_0^\theta\text{d}\varphi\,f(r,\varphi).$$ Thus, one would expect that the image of this vector field is then all of the functions on the circle. However, this is only true if we work on the continuous functions on the circle. For example, the function whose image under the vector field would be $3$ is $3\theta$, which is however not a polynomial.
Is there a way to arrive at this results without going to polar coordinates so that the polynomial nature of the functions is explicit? More generally, in the little algebraic geometry that I know (which is extremely little) I have seen people in general describing the circle through these polynomials. How do they introduce polar coordinates?