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

The Chebyshev problem on the square. = {z = x + iy. C: max{|x|, |y|} = 1} of the complex plane C is studied. Let Pn be the set of algebraic polynomials of a given degree n with the unit leading coefficient. The problem is to find the smallest value tn(.) of the uniform norm pn C(.) of polynomials pn. Pn on the square. and a polynomial with the smallest norm, which is called a Chebyshev polynomial (for the square). The Chebyshev constant t(Q) = limn.8 n tn(Q) for the square is found. Thus, the logarithmic asymptotics of the least deviation tn 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), qm. Pm. Effective two-sided estimates for the value of the least deviation tn(.) with respect to n are obtained.