ojasiewicz Inequality
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Since the real number field is not algebraically closed, the geometry of even a plane curve C in the real projective plane is not a very easy topic. Assuming no singular points, the real points of C form a number of ovals, in other words submanifolds that are topologically a circle. The real projective plane has a fundamental group that is a cyclic group with two elements. Such an oval may represent either group element; in other words we may or may not be able to contract it down in the plane. Taking out the line at infinity L, any oval that stays in the finite part of the affine plane will b...
Since the real number field is not algebraically closed, the geometry of even a plane curve C in the real projective plane is not a very easy topic. Assuming no singular points, the real points of C form a number of ovals, in other words submanifolds that are topologically a circle. The real projective plane has a fundamental group that is a cyclic group with two elements. Such an oval may represent either group element; in other words we may or may not be able to contract it down in the plane. Taking out the line at infinity L, any oval that stays in the finite part of the affine plane will be contractible, and so represent the identity element of the fundamental group; the other type of oval must therefore intersect L.There is still the question of how the various ovals are nested. This was the topic of a Hilbert problem, Hilbert's sixteenth problem. See Harnack's curve theorem for a classical result.
