Is there a polynomial function $P:\mathbb{R}^2 \to \mathbb{R}$ with the following property?: For sufficiently large $c>0$, $P^{-1}(c)$ is a simple closed curve $\gamma_c$ but the centroid $e_c$ of the interior of $\gamma_c$ does not converge to any point of $\mathbb{R}^2$ as $c$ goes to $+\infty$

Motivation: The motivation comes from line $-3$, item III page $4$ of this paper which can be generalize to every even degree polynomial with one variable:

https://arxiv.org/pdf/math/0409594.pdf