On some properties of the Abel-Goncharov polynomials and the Casas-Alvero problem

Abstract

We derive new properties of the Abel-Goncharov interpolation polynomials, relating them to 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 associated with 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, 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.