Question : Let $f(x,y)$ be a polynomial with real coefficients having two variables. If $f(x,y)\gt 0$ for any real number $x,y$, then does $f(x,y)$ have the minimum?
Motivation : I found the following question in a book without any proof:
Let $f(x)$ be a polynomial with real coefficients having one variable. If $f(x)\gt 0$ for any real number $x$, then prove that $f(x)$ has the minimum.
Proof : If $\deg (f)=0$, then it's obvious that $f(x)$ has the minimum. So, let $\deg (f)=n\gt 0$ and $f(x)=a_nx^n+f_1(x)$ where $a_n\not=0, \deg(f_1)\lt n$.
We get $$\lim_{x\to \pm\infty}f(x)=\lim_{x\to\pm\infty}|f(x)|=\lim_{x\to\pm\infty}|x|^n\left(\left|a_n+\frac{f_1(x)}{x^n}\right|\right)=\infty.$$
Hence, supposing $G\gt 0$ such that $f(x)\gt f(0)$ if $|x|\gt G$, let $m=f(x_0)$ be the minimum of the continuous function $f(x)$ in a closed-interval $[-G,G]$. Then, since $m\le f(0),$ we know that $f(x)$ has the minimum $m$ at $x=x_0$. Now the proof is completed.
This got me interested the question about $f(x,y)$. I can neither prove that $f(x,y)$ has the minimum nor find any counterexample. Can anyone help?
