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Real plane algebraic curves. (English) Zbl 1057.14071

Summary: We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi-definite nondefinite or definite. We present a discussion about isolated points. By means of the \({\mathfrak p}\) operator, these points can be easily identified for curves defined by minimal polynomials of order bigger than one. We also discuss the conditions that a curve must satisfy in order to have a minimal polynomial. Finally, we list the most relevant topological properties of affine and projective, complex and real plane algebraic curves.

MSC:

14P05 Real algebraic sets
14H50 Plane and space curves
14P25 Topology of real algebraic varieties
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