Harnack's curve theorem

A nonsingular real plane algebraic curve of degree d has at most (d−1)(d−2)/2 + 1 connected components, and curves attaining this bound exist.

Connect

Real algebraic geometry: The study of sets defined by polynomial equations and inequalities over the real numbers. Harnack's bound is a foundational constraint on the topology of real algebraic sets.
Plane curve: A curve in a projective or affine plane defined by a polynomial equation in two coordinates. The theorem applies specifically to nonsingular plane curves.
Harnack curve: A real plane algebraic curve attaining the maximum number of connected components allowed by Harnack's theorem. Such curves demonstrate that the theorem's upper bound is attainable in every degree.
Axel Harnack: A German mathematician known for foundational work on real algebraic curves and their topology. His 1876 work established the bound and constructed curves attaining it.
Genus (mathematics): A topological or algebraic invariant that measures the number of handles of a surface or curve. The component bound equals the curve's genus plus one.
Nonsingular algebraic variety: An algebraic variety whose local points have no singularities. Nonsingularity lets the complex curve have a well-defined genus and smooth real components.
Oval (algebraic geometry): A compact connected component of a real plane algebraic curve. The separate ovals make up the components counted by Harnack's theorem.
Nineteenth-century mathematics: The mathematical research and developments produced during the nineteenth century. The theorem emerged as algebraic geometry began studying real curves through topology.
Riemann surface: A one-dimensional complex manifold on which complex analysis can be defined. A real plane curve sits inside a complex curve whose genus governs Harnack's bound.
Projective plane: The geometric plane obtained by adding a line at infinity to the affine plane. Projective closure provides the setting in which degree and genus formulas apply uniformly.