Algebraic curve

Algebraic curve

Algebraic Geometry

Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022

The number of handles is an important topological invariant called the topological genus of the surface. Hence, the genus of an algebraic curve is the number of handles on the surface. Let us consider next some examples of some very recognizable families of curves. We will get into more details about some of these families in the sections that follow.

Transcritical bifurcation at infinity in planar piecewise polynomial differential systems with two zones

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Published in Dynamical Systems, 2022

Denis de Carvalho Braga, Jaume Llibre, Luis Fernando Mello

The Poincaré compactification can be extended to piecewise polynomial differential systems in the plane with two (or more) zones whose separation boundary is an algebraic or even a piecewise algebraic curve. Since the Poincaré compactification allows the study of the dynamics away from compact sets, this is one of the ways used to study the mechanism of generating periodic orbits from infinity in the phase portrait of such systems. In general, this mechanism is better described through a bifurcation or a perturbation problem.

Simultaneity of centres in double-reversible planar differential systems

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Published in Dynamical Systems, 2021

Jaume Giné, Claudia Valls

An invariant algebraic curve of system (4) is given by a polynomial satisfying where K is called the cofactor of the invariant algebraic curve and is a polynomial of degree at most n−1.

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Algebraic Geometry

Transcritical bifurcation at infinity in planar piecewise polynomial differential systems with two zones

Simultaneity of centres in double-reversible planar differential systems