A Necessary and Sufficient Condition for Local Maxima of Polynomial Modulus Over Unit Disc

Abstract

An important quantity associated with a complex polynomial $p(z)$ is $\Vert p{\Vert }_{\infty }$, the maximum of its modulus over the unit disc $D$. We prove, $z_{\ast }\in D$ is a local maximum of $|p(z)|$ if and only if $a_{\ast }$ satisfies, $z_{\ast }=p(z_{\ast })|p^{\prime }(z_{\ast })|/p^{\prime }(z_{\ast })|p(z_{\ast })|$, i.e. it is proportional to its corresponding Newton direction. This explicit formula gives rise to novel iterative algorithms for computing $\Vert p{\Vert }_{\infty }$. We describe two such algorithms, including a Newton-like method and present some visualization of their performance.

arXiv comment6 pages, 4 figures