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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

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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

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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.