Towards the Casas- Alvero conjecture

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.