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Polynomial $P(x,y)$ takes only positive values for all x,y . Can it take all the positive values?

My thoughts : This is quite a strange one. I tried proving this by contradiction but I got nowhere fast.

Bart Michels
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foxietamine
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2 Answers2

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$$f(x,y)=x^2+(xy-1)^2$$ is the standard answer.

Gerry Myerson
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Take $P(x,y)=x^2+y^2+3$ for example and try to find $x$ and $y$ such that $P(x,y)=1$.

user37238
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    Actually you just found an example. In order to prove this you must work with an abstract polynomial not with a particular one – prometheus21 Mar 24 '15 at 10:23
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    Alternatively, you must find a particular one that takes on all, and only, positive values. – Gerry Myerson Mar 24 '15 at 11:25
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    Maybe I misread the question. But for me the question is : $P(x,y)>0$ for all $x,y$ implies $P(\mathbb{R}^2) = (0,+\infty)$. Consequently, my example answers the question. – user37238 Mar 24 '15 at 11:37
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    I think the question is whether there exists a polynomial with range --- it's rather too easy to find a polynomial that doesn't have that range, isn't it? – Gerry Myerson Mar 24 '15 at 12:00