Here's an interesting variation on the problem of approximating a continuous function on a closed bounded interval using polynomials.
When can you approximate a shape on $\mathbb{R}^2$ using algebraic curves $p(x,y)=0$?
Feel free to share any related topics or papers!
A precise problem, although not the only problem:
For the image of any continuous injective function $f$ from the circle $S^1$ to $\mathbb{R}^2$, does there exist a sequence of polynomials $p_i(x,y)$ such that the "distance" between $U_i=\{(x,y)\in\mathbb{R}^2\;|\;p_i(x,y)=0\}$ and $f(S^1)$ tends to $0$? The distance between bounded sets $B,C\subset\mathbb{R}^2$ is defined as $\sup_{b\in B}d(b,C)+\sup_{c\in C}d(B,c)$.