A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its highest homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.

Inspired by the first part of the Hilbert 16th problem we ask that:

Is there an elliptic polynomial $P(x,y)\in \mathbb{R}[x,y]$ of degree $n$ which has a level set $P^{-1}(c)$ with more than $n$ connected components? Can elliptic polynomials produce $M$-curves?