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In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties , analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables .
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry . Analytic geometry is used in physics and engineering , and also in aviation , rocketry , space science , and spaceflight .
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.
In the complex case, algebraic geometry begins by defining the complex affine space to be . For each , we define , the ring of analytic functions on to be the ring of holomorphic functions, i.e. functions on that can be written as a convergent power series in a neighborhood of each point.
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions . Functions of each type are infinitely differentiable , but complex analytic functions exhibit properties that do not generally hold for real analytic functions.
In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing p -adic elliptic curves with bad reduction using the multiplicative group .
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. [1] [2] These theories are usually studied in the context of real and complex numbers and functions.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement.
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