# The convergence of a sequence of polynomials and the distribution of their zeros

## TL;DR

This paper investigates the conditions under which sequences of polynomials with zeros constrained to the upper half-plane or with bounded non-real zeros converge uniformly, extending classical theorems and conjectures in complex analysis.

## Contribution

It establishes new convergence criteria for polynomial sequences with zeros in specific regions, confirming a theorem of Benz and a conjecture of Pólya.

## Key findings

- Sequences with zeros in the upper half-plane converge uniformly if derivatives at zero stabilize.
- Real polynomial sequences with bounded non-real zeros also exhibit uniform convergence.
- Results extend classical theorems and confirm longstanding conjectures in polynomial zero distribution.

## Abstract

Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$ lie in the closed upper half plane $\rm{Im}\ z\geq 0$, or if $f_1, f_2, \dots$ are real polynomials and the numbers of their non-real zeros are uniformly bounded, then the sequence converges uniformly on compact sets in the complex plane. The results imply a theorem of Benz and a conjecture of P\'olya.