# Polynomial $P(x,y)$ with $\inf_{\mathbb{R}^2} P=0$, but without any point where $P=0$

Recently I've came across such problem: give a polynomial $P(x,y)$, with $\inf_{\mathbb{R}^2} P=0$, but there is no point on the plane where $P=0$. I couldn't solve it after a day, and seriously doubt whether such a function exists, however its source claims that there is. Is that really possible?

• Is this in the real plane $\mathbb{R}^2$, or could it perhaps be over the rationals? – hardmath Jan 15 '13 at 19:51
• The real plane. – aplavin Jan 15 '13 at 19:51
• I edited "$in f(P)$ to $\inf f(P)$, as that seems to be the more common notation. Nice question, by the way. – Fredrik Meyer Jan 15 '13 at 19:51

## 1 Answer

$P(x,y)=(1-xy)^2+x^2$ has this property. Clearly $P>0$ and also the sequence $(x_n,y_n)=(1/n,n)$ shows that $\inf P=0$.

• Very nice example. – Brian M. Scott Jan 15 '13 at 19:54
• Thanks! Did you hear about the answer before or created it now? – aplavin Jan 15 '13 at 20:02
• I read about it somewhere before --- I'm not sure where exactly. If I recall correctly it is a counter-example to the claim that $\text{im } P(x,y)$ is closed. – JP McCarthy Jan 15 '13 at 20:05
• @JpMcCarthy: sorry, but what is $im$? – aplavin Jan 15 '13 at 20:12
• Sorry. The image of the real plane under a polynomial in two variables. I suppose one would think it would be closed. – JP McCarthy Jan 15 '13 at 20:16