โ Scribed by A.Yu Uteshev; T.M Cherkasov
No coin nor oath required. For personal study only.
For a real polynomial f (X) of K variables the problem of finding max XโR K f (X) is investigated by reducing it to that of searching for the real roots of the univariate polynomial F (z) := j (zf (ฮ j )), where the product is extended over all the critical points ฮ j of f (X). Employment of the Hermite method of separation of real solutions of an algebraic equation system permits one to construct along with 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) โฅ 0 is also discussed.
๐ SIMILAR VOLUMES
A well-known result of Rivlin states that if \(p(z)\) is a polynomial of degree \(n\), such that \(p(z) \neq 0\) in \(|z|<1\), then \(\max _{1:} \quad, \quad|p(z)| \geqslant((r+1) / 2)^{n} \max _{1: 1},|p(z)|\). In this paper, we prove a generalization and refinements of this result. 1994 Academic P