Polynomial $P(x,y)$ with $\inf_{\mathbb{R}^2} P=0$, but without any point where $P=0$
Solution 1:
$P(x,y)=(1-xy)^2+x^2$ has this property. Clearly $P>0$ and also the sequence $(x_n,y_n)=(1/n,n)$ shows that $\inf P=0$.
Polynomial $P(x,y)$ with $\inf_{\mathbb{R}^2} P=0$, but without any point where $P=0$
Solution 1:
$P(x,y)=(1-xy)^2+x^2$ has this property. Clearly $P>0$ and also the sequence $(x_n,y_n)=(1/n,n)$ shows that $\inf P=0$.