**Updated:**
>A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogenous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The answer to this posts provide a proof for the following theorem;

>**Theorem:** If $p$ is an elliptic polynomial whose last homogenius part is positive definitive, then for $c$ sufficiently large
, $p^{-1}(c)$ is a convex simple closed curve. Moreover if the centroid of  interior of $p^{-1}(c)$ is denited by $z_c$ then $z_c$ is convergent as $c$ goes to $+\infty$. The limit can be determined in terms of coefficient of $p$. If we drop the ellipticity condition then this result is not necessarily true.



**The previous version of the post:**



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 $\gamma_c$, homeomorphic to $S^1$, but as $c$ goes to $+\infty$. the centroid $e_c$ of the interior of $\gamma_c$
 does not converge to any point of $\mathbb{R}^2$.

**Motivation:** The answer is negative if we consider this question for polynomials $p:\mathbb{R} \to \mathbb{R}$ whose eventual level sets are $2$-pointed set, i.e. $S^0$.(Namely a polynomial of even degree). The motivation comes from line -3, item III,  page 4 of [Taghavi - On periodic solutions of Liénard equations](https://arxiv.org/abs/math/0409594v1), which can be generalized to every even degree polynomial with one variable.