Polynomials least deviating from zero with a constraint on the location of roots

A. E. Pestovskaya

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 3, pp. 166-175

Résumé

We consider Chebyshev's problem on polynomials least deviating from zero on a compact set $K$ with a constraint on the location of their roots. More exactly, the problem is considered on the set $\mathcal{P}_n(G)$ of polynomials of degree $n$ that have unit leading coefficient and do not vanish on an open set $G$. An exact solution is obtained for $K=[-1, 1]$ and $G=\{z\in\mathbb{C}\,:\, |z|$, $R\ge \varrho_n$, where $\varrho_n$ is a number such that $\varrho_n^2\le (\sqrt{5}-1)/2$. In the case ${\rm Conv}\,K \subset \overline{G}$, the problem is reduced to similar problems for the set of algebraic polynomials all of whose roots lie on the boundary $\partial G$ of the set $G$. The notion of Chebyshev constant $\tau(K, G)$ of a compact set $K$ with respect to a compact set $G$ is introduced, and two-sided estimates are found for $\tau(K, G)$.

Détail

Keywords: Chebyshev polynomial of a compact set, Chebyshev constant of a compact set; constraints on the roots of a polynomial.