The search for the maximum of a polynomial (Q1264466)

For a real polynomial \(f(X)\) of \(K\) variables the problem of finding \(\max_{X\in \mathbb{R}^K}f(X)\) is investigated by reducing it to that of searching for the real roots of the univariate polynomial \({\mathcal F}(z):= \prod_j(z- f(\Lambda_j))\), where the product is carried out over all the critical points \(\Lambda_j\) of \(f(X)\). Employment of the Hermite method of separation of real solutions of a system of algebraic equations permits one to construct along with \({\mathcal F}(z)\) its Sturm series, and to restore the coordinates of the corresponding critical point. The problem of finding the \(\max f\) in the set defined by the real polynomial inequality \(G(X)\geq 0\) is also discussed.