Polynomials least deviating from zero on a square of the complex plane

The Chebyshev problem is studied on the square $\Pi=\left\{z=x+iy\in\mathbb{C}\colon\max\{|x|,|y|\}\le 1\right\}$ of the complex plane $\mathbb{C}$. Let $\mathfrak{P}_n$ be the set of algebraic polynomials of a given degree $n$ with the unit leading coefficient. The problem is to find the smallest value $\tau_n(\Pi)$ of the uniform norm $\|p_n\|_{C(\Pi)}$ of polynomials $p_n\in \mathfrak{P}_n$ on the square $\Pi$ and a polynomial with the smallest norm, which is called the Chebyshev polynomial (for the squire). The Chebyshev constant $\tau(Q)=\lim_{n\rightarrow\infty} \sqrt[n]{\tau_n(Q)}$ for the squire is found. Thus, the logarithmic asymptotics of the least deviation $\tau_n(\Pi)$ 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 $n=4m+s$, $0\le s\le 3$, it is sufficient to solve the problem on the set of polynomials $z^sq_m(z)$, $q_m\in \mathfrak{P}_m$. Effective two-sided estimates for the value of the least deviation $\tau_n(\Pi)$ with respect to $n$ are obtained.