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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 . 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.