Polynomial problems of the Casas-Alvero type
Abstract
We establish necessary and sufficient conditions for an arbitrary polynomial of degree , especially with only real roots, to be trivial, i.e. to have the form a(x-b)^n. To do this, we derive new properties of polynomials and their roots. In particular, it concerns new bounds and genetic sum's representations of the Abel -Goncharov interpolation polynomials. Moreover, we prove the Sz.-Nagy type identities, the Laguerre and Obreshkov-Chebotarev type inequalities for roots of polynomials and their derivatives. As applications these results are associated with the known problem, conjectured by Casas- Alvero in 2001, which says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. We investigate particular cases of the problem, when the conjecture holds true or, possibly, is false.
Cite
@article{arxiv.1308.5320,
title = {Polynomial problems of the Casas-Alvero type},
author = {Semyon Yakubovich},
journal= {arXiv preprint arXiv:1308.5320},
year = {2019}
}