Polynomial problems of the Casas-Alvero type

TL;DR

This paper characterizes when polynomials, especially with real roots, are trivial of the form a(x-b)^n, and explores properties and inequalities related to roots and derivatives to address the Casas-Alvero conjecture.

Contribution

It establishes new conditions, bounds, and identities for polynomials and their roots, advancing understanding of the Casas-Alvero conjecture and polynomial root behavior.

Findings

Derived new bounds and sum representations for Abel-Goncharov polynomials.

Proved Sz.-Nagy type identities and Laguerre inequalities for polynomial roots.

Investigated cases where the Casas-Alvero conjecture holds or fails.

Abstract

We establish necessary and sufficient conditions for an arbitrary polynomial of degree $n$, 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.