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 $\mathbb{R}^n$?
Does a polynomial that's bounded below have a global minimum?
16
$\begingroup$
real-analysis
multivariable-calculus
polynomials
maxima-minima
-
0Something for your perusal: http://journals.cambridge.org/action/displayAbstract?aid=4903004 – 2010-09-01
-
0...and also http://www.jstor.org/stable/2324459 – 2010-09-01
2 Answers
31
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.
1
@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.
-
0I just realised your question was meant for Chen only. Sorry. – 2010-09-02
-
0Don't worry: it was just a silly question. Maybe I should delete it. – 2010-09-02