On some properties of the Abel–Goncharov polynomials and the Casas-Alvero problem. Issue 8 (2nd August 2016)

• Record Type: Journal Article
• Authors: Yakubovich, S.
• Is Part Of: Integral transforms and special functions. Volume 27:Issue 8(2016)
• Page Start: 599
• Page End: 610
• Publication Date: 2016-08-02
• Subjects: Casas-Alvero problem -- Abel–Goncharov interpolation polynomials; Primary: 26C05 -- 12D10 -- 41A05 -- Secondary: 13F20; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2016.1167689
• Languages: English
• ISSNs: 1065-2469

Abstract

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.