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Displaying similar documents to “A conjecture of Schoenberg.”

Generalization of a Conjecture in the Geometry of Polynomials

Sendov, Bl. (2002)

Serdica Mathematical Journal

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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...

Integrability of a linear center perturbed by a fourth degree homogeneous polynomial.

Javier Chavarriga, Jaume Giné (1996)

Publicacions Matemàtiques

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In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.

Integrability of a linear center perturbed by a fifth degree homogeneous polynomial.

Javier Chavarriga, Jaume Giné (1997)

Publicacions Matemàtiques

Similarity:

In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.

Quadratic Mean Radius of a Polynomial in C(Z)

Ivanov, K., Sharma, A. (1996)

Serdica Mathematical Journal

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* Dedicated to the memory of Prof. N. Obreshkoff A Schoenberg conjecture connecting quadratic mean radii of a polynomial and its derivative is verified for some kinds of polynomials, including fourth degree ones.

Integrable systems in the plane with center type linear part

Javier Chavarriga (1994)

Applicationes Mathematicae

Similarity:

We study integrability of two-dimensional autonomous systems in the plane with center type linear part. For quadratic and homogeneous cubic systems we give a simple characterization for integrable cases, and we find explicitly all first integrals for these cases. Finally, two large integrable system classes are determined in the most general nonhomogeneous cases.