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An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
Fermat curve; Bézout's theorem; Brill–Noether theory; Genus (mathematics) Riemann surface; Riemann–Hurwitz formula; Riemann–Roch theorem; Abelian integral; Differential of the first kind; Jacobian variety. Generalized Jacobian; Moduli of algebraic curves; Hurwitz's theorem on automorphisms of a curve; Clifford's theorem on special divisors
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
Arithmetic surfaces arise naturally in diophantine geometry, when an algebraic curve defined over K is thought of as having reductions over the fields R/P, where P is a prime ideal of R, for almost all P; and are helpful in specifying what should happen about the process of reducing to R/P when the most naive way fails to make sense.
In topology, a curve is defined by a function from an interval of the real numbers to another space. [49] In differential geometry, the same definition is used, but the defining function is required to be differentiable. [53] Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one. [54]
In the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being ...
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