On the paths of steepest descent for the norm of a one variable complex polynomial

TL;DR

This paper studies the geometric paths of steepest descent for polynomial norms in the complex plane, revealing they form trees from zeros of the derivative to roots, with bounds on their lengths and extensions to Blaschke products.

Contribution

It demonstrates that these steepest descent paths form trees connecting derivative zeros to roots and provides length bounds, including cases independent of root multiplicities.

Findings

Paths form trees in the complex plane

Upper bounds on path lengths are established

Finer estimates depend only on roots, not multiplicities

Abstract

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.