Remarks on Rational Vector Fields on $\mathbb {C}\mathbb {P}^{1}$
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Abstract
Abstract In this paper, we introduce geometric tools to study the families of rational vector fields of a given degree over $\mathbb {C}\mathbb {P}^{1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℂ</mml:mi> <mml:msup> <mml:mrow> <mml:mi>ℙ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . To a generic vector field of such a parametric family, we associate several geometric objects: a periodgon, a star domain, and a translation surface. These objects generalize objects with the same name introduced in previous works on polynomial vector fields. They are used to describe the bifurcations inside the families. We specialize to the case of rational vector fields of degree 4.
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