A refined realization theorem in the context of the Schur-Szeg\H{o} composition

TL;DR

This paper refines a realization theorem for polynomials and entire functions, characterizing root distributions and composition representations via Schur-Szeg ext{"o} composition, with implications for polynomial and exponential function structures.

Contribution

It provides necessary and sufficient conditions linking root counts of polynomials and their Schur-Szeg ext{"o} components, extending the theorem to entire functions of the form e^x R.

Findings

Characterization of root distributions for polynomials in Schur-Szeg ext{"o} form

Extension of the realization theorem to entire functions e^x R

Unique representation of polynomials as compositions with specific root properties

Abstract

Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable as Schur-Szeg\H{o} composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are unique up to permutation. We give necessary and sufficient conditions upon the possible values of the $8$-vector whose components are the number of positive, zero, negative and complex roots of a real polynomial $P$ and the number of positive, zero, negative and complex among the quantities $a_i$ corresponding to $P$. A similar result is proved about entire functions of the form $e^xR$, where $R$ is a polynomial.