Return to Question

Updated:

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

 

Theorem: If $p$ is an elliptic polynomial whose last homogeneous part is positive definitive, then for $c$ sufficiently large , $p^{-1}(c)$ is a simple closed curve. Moreover if the centroid of interior of $p^{-1}(c)$ is denoted by $e_c$ then $e_c$ is convergent as $c$ goes to $+\infty$. The limit $\lim_{c\to \infty} e_c$ can be written in terms of coefficients of $p$. If we drop the ellipticity condition then this convergence 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, which can be generalized to every even degree polynomial with one variable.

Updated:

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

Theorem: If $p$ is an elliptic polynomial whose last homogeneous part is positive definitive, then for $c$ sufficiently large , $p^{-1}(c)$ is a simple closed curve. Moreover if the centroid of interior of $p^{-1}(c)$ is denoted by $e_c$ then $e_c$ is convergent as $c$ goes to $+\infty$. The limit $\lim_{c\to \infty} e_c$ can be written in terms of coefficients of $p$. If we drop the ellipticity condition then this convergence 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, which can be generalized to every even degree polynomial with one variable.

Updated:

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

Theorem: If $p$ is an elliptic polynomial whose last homogeneous 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 denoted by $e_c$ then $e_c$ is convergent as $c$ goes to $+\infty$. The limit $\lim_{c\to \infty} e_c$ can be written in terms of coefficients of $p$. If we drop the ellipticity condition then this convergence 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, which can be generalized to every even degree polynomial with one variable.

Updated:

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

Theorem: If $p$ is an elliptic polynomial whose last homogeneous part is positive definitive, then for $c$ sufficiently large , $p^{-1}(c)$ is a simple closed curve. Moreover if the centroid of interior of $p^{-1}(c)$ is denoted by $e_c$ then $e_c$ is convergent as $c$ goes to $+\infty$. The limit $\lim_{c\to \infty} e_c$ can be written in terms of coefficients of $p$. If we drop the ellipticity condition then this convergence 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, which can be generalized to every even degree polynomial with one variable.

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 two answers to this post 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 denoted by $e_c$ then $e_c$ is convergent as $c$ goes to $+\infty$. The limit $\lim_{c\to \infty} e_c$ can be written in terms of coefficients of $p$. If we drop the ellipticity condition then this convergence 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, which can be generalized to every even degree polynomial with one variable.

Updated:

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

Theorem: If $p$ is an elliptic polynomial whose last homogeneous 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 denoted by $e_c$ then $e_c$ is convergent as $c$ goes to $+\infty$. The limit $\lim_{c\to \infty} e_c$ can be written in terms of coefficients of $p$. If we drop the ellipticity condition then this convergence 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, which can be generalized to every even degree polynomial with one variable.