Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathbb{R}^n$?

share|improve this question
    
Something for your perusal: journals.cambridge.org/action/displayAbstract?aid=4903004 –  J. M. Sep 1 '10 at 23:38
    
...and also jstor.org/stable/2324459 –  J. M. Sep 1 '10 at 23:50

2 Answers 2

up vote 27 down vote accepted

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.

share|improve this answer

@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.

share|improve this answer
    
I just realised your question was meant for Chen only. Sorry. –  Weltschmerz Sep 2 '10 at 2:32
    
Don't worry: it was just a silly question. Maybe I should delete it. –  a.r. Sep 2 '10 at 11:15

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.