论文标题
在一个可变复合物多项式的标准的最陡下降的路径上
On the paths of steepest descent for the norm of a one variable complex polynomial
论文作者
论文摘要
我们考虑了一个非恒定的一个变量多项式$ f $的标准，在复杂平面中，最陡后下降的路径。我们表明，从$ f $的对数导数的零开始，并以$ f $的根结束，在复杂的平面上画一棵树，我们对其长度进行了上限的估计。在某些情况下，我们获得的估算值仅取决于$ f $的根集，而不取决于它们的多重性，我们想知道这是否可以做到这一点。我们还将这个问题扩展到用于单元磁盘的有限blaschke产品。
We consider paths of steepest descent, in the complex plane, for the norm of a non-constant one variable polynomial $f$. We show that such paths, starting from a zero of the logarithmic derivative of $f$ and ending in a root of $f$, draw a tree in the complex plane, and we give an upper bound estimate on their lengths. In some cases, we obtain a finer estimate that depends only on the set of roots of $f$, not on their multiplicity, and we wonder if this can be done in general. We also extend this question to finite Blaschke products for the unit disk.
