Note on the real rootedness of polynomials

TL;DR

This paper investigates conditions under which polynomials are real rooted using generalized Bell umbra and Rolle's theorem, with applications to partition and sigma polynomials of graphs.

Contribution

It introduces new criteria for real rootedness of polynomials and applies these results to specific classes like partition and graph sigma polynomials.

Findings

Established new conditions for real rootedness of polynomials.

Applied results to partition polynomials.

Applied results to sigma polynomials of graphs.

Abstract

In this paper, by the generalized Bell umbra and Rolle's theorem, we give some results on the real rootedness of polynomials. Some applications on partition polynomials and the sigma polynomials of graphs are given.