Polynomials Least Deviating from Zero on a Square of the Complex Plane

Abstract

The Chebyshev problem on the square Π = {z = x + iy ∈ ℂ: max{∣x∣, ∣y∣} ≤ 1} of the complex plane ℂ is studied. Let $$p_{n} \in \mathfrak{P}_{n}$$ be the set of algebraic polynomials of a given degree n with the unit leading coefficient. The problem is to find the smallest value τn(Π) of the uniform norm ∥pn∥C(π) of polynomials $$\mathfrak{P}_{n}$$ on the square Π and a polynomial with the smallest norm, which is called a Chebyshev polynomial (for the square). The Chebyshev constant $$\tau \left( Q \right) = {\lim _{n \to \infty }}\root n \of {{\tau _n}\left( Q \right)} $$ for the square is found. Thus, the logarithmic asymptotics of the least deviation τn(Π) with respect to the degree of a polynomial is found. The problem is solved exactly for polynomials of degrees from 1 to 7. The class of polynomials in the problem is restricted; more exactly, it is proved that, for n = 4m + s, 0 ≤ s ≤ 3, it is sufficient to solve the problem on the set of polynomials zsqm(z), $$q_{n} \in \mathfrak{P}_{m}$$. Effective two-sided estimates for the value of the least deviation τn (Π) with respect to n are obtained.