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 τ_n(Π) of the uniform norm ∥p_n∥_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 τ_n(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 z^sq_m(z), \(q_{n} \in \mathfrak{P}_{m}\). Effective two-sided estimates for the value of the least deviation τ_n (Π) with respect to n are obtained.

中文翻译：

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在复平面的平方上最小偏离零的多项式

研究了复平面ℂ的平方= { z = x + iy∈ℂ上的Chebyshev问题：max {∣ x ∣，∣ y ∣}≤1}。令\（p_ {n} \ in \ mathfrak {P} _ {n} \）为给定阶数为n的单位前导系数的代数多项式的集合。的问题是要找到最小值τ _Ñ均匀范数（Π）∥ p _Ñ ∥ _Ç（π）多项式的\（\ mathfrak {P} _ {N} \）的平方Π和具有最小的多项式范数，称为Chebyshev多项式（对于平方）。切比雪夫常数找到平方的\（\ tau \ left（Q \ right）= {\ lim _ {n \ to \ infty}} \ root n \ {{\ tau _n} \ left（Q \ right）} \）。因此，最小偏差的对数渐近τ _Ñ（Π）相对于一个多项式的阶数被发现。对于度数为1到7的多项式，可以精确地解决该问题。更确切地说，它证明了，对于Ñ = 4米+小号，0≤小号≤3，它足以解决该组多项式的问题Ž^小号q_米（Ž），\（Q_ {N} \在\ mathfrak {P} _ {m} \）。有效双面估计最小偏差的值τ _Ñ（Π）相对于Ñ获得。