Must a polynomial function $f \in \mathbb{R}[x_1, \ldots, x_n]$ that's lower bounded by some $\lambda \in \mathbb{R}$ have a global minimum over $\mathbb{R}^n$?
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Something for your perusal: http://journals.cambridge.org/action/displayAbstract?aid=4903004 – J. M. ain't a mathematician Sep 01 '10 at 23:38
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...and also http://www.jstor.org/stable/2324459 – J. M. ain't a mathematician Sep 01 '10 at 23:50
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No. The polynomial $f(x, y) = (1 - xy)^2 + x^2$ is bounded below by $0$, cannot actually take the value $0$, but can take the value $\epsilon$ for any $\epsilon > 0$. I learned this example from Richard Dore on MO.
Qiaochu Yuan
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@Agusti Roig. $f$ has no critical points in the interior of the disc, so you have to look for the minimum at the unit circle. One way would be to parametrize $x=\cos\theta, y=\sin\theta$, and minimize.
Weltschmerz
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