PreprintCC BY 4.0arXiv
Novel properties of polynomials
Abstract
We establish necessary and sufficient conditions for a complex polynomial to have a prescribed multiset of roots, expressed in terms of its coefficients and complete homogeneous symmetric polynomials. These conditions extend the classical Vieta formulas and yield explicit formulas for the coefficients of the corresponding quotient. We apply our results to obtain characterizations of Kronecker polynomials through identities involving symmetric polynomials and construct Borwein multiples of products of cyclotomic polynomials.
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BibTeX
@misc{carli2026novel,
title = {{Novel properties of polynomials}},
author = {Laura De Carli and Maurizio Laporta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37743},
eprint = {2610.37743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37743},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Novel properties of polynomials
AU - Laura De Carli
AU - Maurizio Laporta
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37743
UR - https://arxiv.org/abs/2610.37743
AB - We establish necessary and sufficient conditions for a complex polynomial to have a prescribed multiset of roots, expressed in terms of its coefficients and complete homogeneous symmetric polynomials. These conditions extend the classical Vieta formulas and yield explicit formulas for the coefficients of the corresponding quotient. We apply our results to obtain characterizations of Kronecker polynomials through identities involving symmetric polynomials and construct Borwein multiples of products of cyclotomic polynomials.
ER -