global minimum of the square of a multivariate polynomial

Question

I consider the square of a multivariate polynomial $f\in\mathbb{R}[x_1,\cdots,x_n]$ with real coefficients in $n$ variables of maximum order $m$.

(e.g. $f=x^3+2yx^2-3y^2-xy+2 \rightarrow m=3,n=2$)

My goal is to minimize $g=f^2$, so I want to find the (global) minimum of $g=f^2$. My question:

• Since $g$ is lower bounded ($g\geq0$), is it guaranteed that there is always a global minimum?
• Can i somehow estimate (given $n$ and $m$) the number of local extremas?

Thanks in advance :)

Answer 1

Every polynomial of even degree and positive coefficient has a global minimum and since squaring doubles the degree of $f$ in every dimension you can definitively say that it has a global minimum. Proving it is easily possible since $g$ is finite for all finite coordinates, but $\forall n \lim_{x_n \rightarrow \infty}{g} \rightarrow \infty$.

Your second question: There is no trivial way to estimate the actual number of local minima. You can calculate the maximum amount of extrema/minima possible, but that's it. -> $\nabla = 0$, in 1D see Sturm's theorem, Descartes Rule of Signs. You cannot predict how the maximum amount of extrema have merged -> think about merging minima, maxima and sattlepoints in 1D of degree 6 and up -> Abel-Ruffini.