  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.

Classical Analysis and ODEs