Towards the Casas- Alvero conjecture

TL;DR

This paper explores conditions under which polynomials are trivial, contributing new results related to the Casas-Alvero conjecture, including determinantal representations, root identities, and generalizations of classical theorems.

Contribution

It provides new necessary and sufficient conditions for polynomial triviality, introduces determinantal representations of interpolation polynomials, and generalizes classical polynomial identities and theorems.

Findings

Derived determinantal representations of Abel-Goncharov polynomials

Established new Sz.-Nagy type identities for roots

Generalized Schoenberg's analog of Rolle's theorem

Abstract

We investigate necessary and sufficient conditions for an arbitrary polynomial of degree $n$ to be trivial, i.e. to have the form $a(z-b)^n$. These results are related to an open problem, conjectured in 2001 by E. Casas- Alvero. It 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. In particular, we establish determinantal representation of the Abel-Goncharov interpolation polynomials, related to the problem and having its own interest. Among other results are new Sz.-Nagy type identities for complex roots and a generalization of the Schoenberg conjectured analog of Rolle's theorem for polynomials with real and complex coefficients.