Displaying similar documents to “Darbouxian integrability for polynomial vector fields on the 2-dimensional sphere.”

Homogeneous polynomial vector fields of degree 2 on the 2-dimensional sphere.

Jaume Llibre, Claudio Pessoa (2006)

Extracta Mathematicae

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Let X be a homogeneous polynomial vector field of degree 2 on S having finitely many invariant circles. Then, we prove that each invariant circle is a great circle of S, and at most there are two invariant circles. We characterize the global phase portrait of these vector fields. Moreover, we show that if X has at least an invariant circle then it does not have limit cycles.

Irreducibility of the iterates of a quadratic polynomial over a field

Mohamed Ayad, Donald L. McQuillan (2000)

Acta Arithmetica

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1. Introduction. Let K be a field of characteristic p ≥ 0 and let f(X) be a polynomial of degree at least two with coefficients in K. We set f₁(X) = f(X) and define f r + 1 ( X ) = f ( f r ( X ) ) for all r ≥ 1. Following R. W. K. Odoni [7], we say that f is stable over K if f r ( X ) is irreducible over K for every r ≥ 1. In [6] the same author proved that the polynomial f(X) = X² - X + 1 is stable over ℚ. He wrote in [7] that the proof given there is quite difficult and it would be of interest to have an elementary proof....

A class of integrable polynomial vector fields

Javier Chavarriga (1995)

Applicationes Mathematicae

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We study the integrability of two-dimensional autonomous systems in the plane of the form = - y + X s ( x , y ) , = x + Y s ( x , y ) , where Xs(x,y) and Ys(x,y) are homogeneous polynomials of degree s with s≥2. First, we give a method for finding polynomial particular solutions and next we characterize a class of integrable systems which have a null divergence factor given by a quadratic polynomial in the variable ( x 2 + y 2 ) s / 2 - 1 with coefficients being functions of tan−1(y/x).