Displaying similar documents to “Smale's Conjecture on Mean Values of Polynomials and Electrostatics”

Generalization of a Conjecture in the Geometry of Polynomials

Sendov, Bl. (2002)

Serdica Mathematical Journal

Similarity:

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by...

Extention of Apolarity and Grace Theorem

Sendov, Blagovest, Sendov, Hristo (2013)

Mathematica Balkanica New Series

Similarity:

MSC 2010: 30C10 The classical notion of apolarity is defined for two algebraic polynomials of equal degree. The main property of two apolar polynomials p and q is the classical Grace theorem: Every circular domain containing all zeros of p contains at least one zero of q and vice versa. In this paper, the definition of apolarity is extended to polynomials of different degree and an extension of the Grace theorem is proved. This leads to simplification of the conditions of...

Differential equations associated with generalized Bell polynomials and their zeros

Seoung Cheon Ryoo (2016)

Open Mathematics

Similarity:

In this paper, we study differential equations arising from the generating functions of the generalized Bell polynomials.We give explicit identities for the generalized Bell polynomials. Finally, we investigate the zeros of the generalized Bell polynomials by using numerical simulations.

Remarks on some recent results about polynomials with restricted zeros

M. A. Qazi (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

We point out certain flaws in two papers published in Ann. Univ. Mariae Curie-Skłodowska Sect. A, one in 2009 and the other in 2011. We discuss in detail the validity of the results in the two papers in question.

Growth of polynomials whose zeros are outside a circle

K. Dewan, Sunil Hans (2008)

Annales UMCS, Mathematica

Similarity:

If p(z) be a polynomial of degree n, which does not vanish in |z| < k, k < 1, then it was conjectured by Aziz [Bull. Austral. Math. Soc. 35 (1987), 245-256] that [...] In this paper, we consider the case k < r < 1 and present a generalization as well as improvement of the above inequality.