Plane real algebraic curves (Q2702170)

Elementary texts on algebraic curves renounce from the very beginning to the study of curves defined over \({\mathbb R}\) because they are considered as pathological objects. However, it seems more reasonable to understand first the real curves and then to go into the curves defined over \({\mathbb C}\). In this work the author sketches how this approach can be achieved. NEWLINENEWLINENEWLINEAlgebraic curves over \({\mathbb C}\) have a minimal polynomial, that is, any polynomial vanishing on the curve is in the ideal generated by this minimal polynomial. This is no longer true over \({\mathbb R}\). The author characterizes the real plane algebraic curves with a minimal polyomial and shows their properties which turn out to be very similar to that of complex curves. For example, curves with a minimal polynomial have always an infinite number of points and a real Study lemma can be stated. Projective curves are also considered and characterized. NEWLINENEWLINENEWLINEThe paper ends comparing the algebraic and topological properties of complex and real curves.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00066].