This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing this.
Let $P(x,y)$ be a real polynomial in 2 variables. Show that:
(a) the set $\{(x,y)\in\mathbb{R}^{2}|P(x,y)=0\}$ can be written as finite union of sets of the form $\{(f(t),g(t))|t\in\mathbb{R}\}$ for some infinitely differentiable $f,g$
(b) if any 2 such curves intersect non-tangentially (ie. the tangent vectors of the 2 curves at the intersection does not lie on an 1-dimensional subspace), then the gradient of $P(x,y)$ is $0$ (I think there might be some extra conditions here for (b) to be true)
Thank you.