A sufficient condition for a complex polynomial to have only simple zeros and an analog of Hutchinson's theorem for real polynomials

TL;DR

This paper establishes a sharp constant condition ensuring all zeros of a complex polynomial or entire function are simple, and extends Hutchinson's theorem to polynomials with real nonzero coefficients.

Contribution

It introduces the minimal constant $b_{ ext{infty}}$ for simple zeros and generalizes Hutchinson's theorem to real coefficient polynomials.

Findings

Identified the exact constant $b_{ ext{infty}} \, \approx \, 4.81058280$ for simplicity of zeros.

Proved the minimality of this constant.

Extended Hutchinson's theorem to real polynomials with nonzero coefficients.

Abstract

We find the constant $b_{\infty}$ ($b_{\infty} \approx 4.81058280$) such that if a complex polynomial or entire function $f(z) = \sum_{k=0}^ \omega a_k z^k, $ $\omega \in \{2, 3, 4, \ldots \} \cup \{\infty\},$ with nonzero coefficients satisfy the conditions $\left|\frac{a_k^2}{a_{k-1} a_{k+1}}\right| >b_{\infty} $ for all $k =1, 2, \ldots, \omega-1,$ then all the zeros of $f$ are simple. We show that the constant $b_{\infty}$ in the statement above is the smallest possible. We also obtain an analog of Hutchinson's theorem for polynomials or entire functions with real nonzero coefficients.