Polya-Chebotarev问题与逆多项式象
The Polya-Chebotarev problem and inverse polynomial images
摘要：
考虑这个问题，通常被称为波利亚-切博塔列夫问题，在复平面上找到一个连续体，包括一些给定的点，使得这个连续体的对数容量最小。我们证明了多项式$\T_n$的每个连通逆像$\T_n^{-1}([-1,1])$总是某个Pólya-Chebotarev问题的解。通过求解$\T_n^2-1$的零点的非线性方程组，我们能够构造具有连接的逆像的多项式$\T_n$。
英文摘要：
Consider the problem, usually called the Pólya-Chebotarev problem, of finding a continuum in the complex plane including some given points such that the logarithmic capacity of this continuum is minimal. We prove that each connected inverse image $\T_n^{-1}([-1,1])$ of a polynomial $\T_n$ is always the solution of a certain Pólya-Chebotarev problem. By solving a nonlinear system of equations for the zeros of $\T_n^2-1$, we are able to construct polynomials $\T_n$ with a connected inverse image.
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