Given a polynomial $p: \mathbb{R}^2 \rightarrow \mathbb{R}$ in two real variables where $p(x,y) \geq 0$ for all $x, y \in \mathbb{R}$, must $p$ be coercive?
Note that the definition of coercive that I am using is that for any function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, $$\lim_{||x|| \rightarrow \infty} f(x) = \infty.$$
I think any such nonnegative polynomial must be coercive. My attempt at the proof is below:
Let $p(x_1,x_2)$ be a polynomial of the form $$p(x_1,x_2) = a_0 + a_1 x_1 + a_2 x_1^2 + \cdots + a_n x_1^n + b_1 x_2 + \cdots + b_m x_2^m.$$ By the assumption of nonnegativity for all $x_1$ and $x_2$, $a_i, b_j \geq 0$ for all $i, j$. Taking the limit, $$\lim_{||x|| \rightarrow \infty} p(x_1, x_2) = a_0 + \lim_{||x|| \rightarrow \infty} a_1 x_1 + \cdots a_n x_1^n + \cdots + b_1 x_2 + \cdots + b_m y^m = \infty.$$
Is that all that is necessary, or am I missing something important?
