Does a non-negative polynomial of three variables have minimum?

Question

I was wondering, does a non-negative polynomial of three variables (in $\mathbb{R}^3$) have a minimum point? I understand that for example $(0,0,0)$ is a minimum point for some of them, but what could be the answer in the general case?

Answer 1

Such a polynomial can, but does not need to have a minimum point.

$f(x,y,z)=x^2+y^2+z^2$ is an example of the former, which takes the minum value of $0$ at $(0,0,0)$.

But the polynomial $g(x,y,z)=x^2+y^2+(xyz-1)^2$ does not have a minimum point. We have $g(x,y,z) \ge 0$ obviously as sum of $3$ squares, but the equality can't be reached, as that would require $x=0, y=0$ and $xyz=1$, which is impossible.

So we have that $g(x,y,z) > 0$ but also $g\left(\frac1n,\frac1n, n^2\right)=\frac2{n^2}$ for each $n > 0, n \in \mathbb Z$. That means $g(x,y,z)$ can take arbitrary small positive values but can never reach $0$ exactly, so it does not have a minimum point.

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Note that this construction works for polynomials of $2$ or more variables, while for just one variable a lower bounded polynomial will always have a minimum point.

That's because for one variable, a polynomial always tends to either $\pm\infty$ when the argument tends to $+\infty$ and $-\infty$. If it has a lower bound, it means for the minimum only a finite interval is interesting, then the usual theorem for a continuous function attaining a minimum value on a finite, closed interval proves the conclusion.