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Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$

where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last term $p_{2n}$ is always nonnegative which has a unique root at origin $(0,0)$ and has no any critical point except origin.

We consider the polynomial function $p(x,y)$ as a function from $\mathbb{R}^2 \to \mathbb{R}$.

Is it true to say that for all $z$ sufficiently large, all curves $p^{-1}(z)$ are simple closed curves whose centroid $c(z)\in \mathbb{R}^2$ has a limit $L\in \mathbb{R}^2$, as $z$ goes to $+\infty$ ?

The Motivation:

In lower dimension, assume that we have an even degree polynomial $p(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n_1}+\ldots+a_1x+a_0$. Then for $y$ sufficiently large $p^{-1}(y)$ consists two points $\{A(y), B(y)\}$. Then $$\lim_{y\to \infty} (A(y)+B(y)) =\frac{-a_{2n-1}}{na_{2n}}$$

Note: The centroid $c(z)$ is actually the centroid of closed region in the plane which is surrounded by p^{-1}(z)$.

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$

where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last term $p_{2n}$ is always nonnegative which has a unique root at origin $(0,0)$ and has no any critical point except origin.

We consider the polynomial function $p(x,y)$ as a function from $\mathbb{R}^2 \to \mathbb{R}$.

Is it true to say that for all $z$ sufficiently large, all curves $p^{-1}(z)$ are simple closed curves whose centroid $c(z)\in \mathbb{R}^2$ has a limit $L\in \mathbb{R}^2$, as $z$ goes to $+\infty$ ?

The Motivation:

In lower dimension, assume that we have an even degree polynomial $p(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n_1}+\ldots+a_1x+a_0$. Then for $y$ sufficiently large $p^{-1}(y)$ consists two points $\{A(y), B(y)\}$. Then $$\lim_{y\to \infty} (A(y)+B(y)) =\frac{-a_{2n-1}}{na_{2n}}$$

Note: The centroid $c(z)$ is actually the centroid of closed region in the plane which is surrounded by $p^{-1}(z)$.

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form of $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$

where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last term $p_{2n}$ is always nonnegative which has a unique root at origin $(0,0)$ and has no any critical point except origin.

We define the polynomial function $f(x,y)=p(x,y)$, as a function from $\mathbb{R}^2 \to \mathbb{R}$.

Is it true to say that for all $z$ sufficiently large, all curves $f^{-1}(z)$ are simple closed curves whose centroid $c(z)\in \mathbb{R}^2$ has a limit $L\in \mathbb{R}^2$ as $z$ goes to $+\infty$ ?

The Motivation:

In lower dimension, assume that we have an even degree polynomial $p(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n_1}+\ldots+a_1x+a_0$. Then for $y$ sufficiently large $p^{-1}(y)$ consists two points $\{A(y), B(y)\}$. Then $$\lim_{y\to \infty} (A(y)+B(y)) =\frac{-a_{2n-1}}{na_{2n}}$$

Note: The centroid $c(z)$ is actually the centroid of clised region in the plane which is surrounded by p^{-1}(z)$.

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$

where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last term $p_{2n}$ is always nonnegative which has a unique root at origin $(0,0)$ and has no any critical point except origin.

We consider the polynomial function $p(x,y)$ as a function from $\mathbb{R}^2 \to \mathbb{R}$.

Is it true to say that for all $z$ sufficiently large, all curves $p^{-1}(z)$ are simple closed curves whose centroid $c(z)\in \mathbb{R}^2$ has a limit $L\in \mathbb{R}^2$, as $z$ goes to $+\infty$ ?

The Motivation:

In lower dimension, assume that we have an even degree polynomial $p(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n_1}+\ldots+a_1x+a_0$. Then for $y$ sufficiently large $p^{-1}(y)$ consists two points $\{A(y), B(y)\}$. Then $$\lim_{y\to \infty} (A(y)+B(y)) =\frac{-a_{2n-1}}{na_{2n}}$$

Note: The centroid $c(z)$ is actually the centroid of closed region in the plane which is surrounded by p^{-1}(z)$.

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form of $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$

where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last term $p_{2n}$ has a unique root at origin $(0,0)$ and has no any critical point except origin.

We define the polynomial function $f(x,y)=p(x,y)$, as a function from $\mathbb{R}^2 \to \mathbb{R}$.

Is it true to say that for all $z$ sufficiently large, all curves $f^{-1}(z)$ are simple closed curves whose centroid $c(z)\in \mathbb{R}^2 \times \{z\}\simeq \mathbb{R}^2$ has a limit $L\in \mathbb{R}^2$ as $z$ goes to $+\infty$ ?

The Motivation:

In lower dimension, assume that we have an even degree polynomial $p(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n_1}+\ldots+a_1x+a_0$. Then for $y$ sufficiently large $f^{-1}(y)$ consists two points $\{A(y), B(y)\}$. Then $$\lim_{y\to \infty} (A(y)+B(y)) =\frac{-a{2n-1}}{na_{2n}}.

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form of $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$

where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last term $p_{2n}$ is always nonnegative which has a unique root at origin $(0,0)$ and has no any critical point except origin.

We define the polynomial function $f(x,y)=p(x,y)$, as a function from $\mathbb{R}^2 \to \mathbb{R}$.

Is it true to say that for all $z$ sufficiently large, all curves $f^{-1}(z)$ are simple closed curves whose centroid $c(z)\in \mathbb{R}^2$ has a limit $L\in \mathbb{R}^2$ as $z$ goes to $+\infty$ ?

The Motivation:

In lower dimension, assume that we have an even degree polynomial $p(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n_1}+\ldots+a_1x+a_0$. Then for $y$ sufficiently large $p^{-1}(y)$ consists two points $\{A(y), B(y)\}$. Then $$\lim_{y\to \infty} (A(y)+B(y)) =\frac{-a_{2n-1}}{na_{2n}}$$

Note: The centroid $c(z)$ is actually the centroid of clised region in the plane which is surrounded by p^{-1}(z)$.