Classification of algebraic curves
How many types of curves are there name them?
Answer: The different types of curves are Simple curve, Closed curve, Simple closed curve, Algebraic and Transcendental Curve.
What is a curve in algebraic geometry?
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.
What is a simple algebraic curve?
An algebraic curve C is the graph of an equation f(x, y) = 0, with points at infinity added, where f(x, y) is a polynomial, in two complex variables, that cannot be factored.
What is algebraic geometry in mathematics?
Today, algebraic geometry is an area with geometry with connections to other areas such as commutative algebra, complex analysis, topology and number theory in mathematics, cryptography in informatics and string theory in physics.
What is branch of a curve?
A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments.
What is the genus of a curve?
The genus of a curve is a birational invariant which plays an important role in the parametrization of algebraic curves (and in the geometry of algebraic curves in general). In fact, only curves of genus 0 can be rationally parametrized.
Why is it called algebraic geometry?
As its name suggests, algebraic geometry deals with curves or surfaces (or more abstract generalisations of these) which can be viewed both as geometric objects and as solutions of algebraic (specifically, polynomial) equations.
Who is the father of algebraic geometry?
| Alexander Grothendieck | |
|---|---|
| Alma mater | University of Montpellier University of Nancy |
| Known for | Renewing algebraic geometry and synthesis between it and number theory and topology List of things named after Alexander Grothendieck |
| Awards | 1966 Fields Medal 1977 Émile Picard Medal 1988 Crafoord Prize (declined) |
| Scientific career | |
What is the difference between algebraic geometry and algebraic topology?
Geometry concerns the local properties of shape such as curvature, while topology involves large-scale properties such as genus. Algebraic methods become important in topology when working in many dimensions, and increasingly sophisticated parts of algebra are now being employed.
Which is harder geometry or algebra?
Geometry is easier than algebra. Algebra is more focused on equations while the things covered in Geometry really just have to do with finding the length of shapes and the measure of angles.
Is algebraic geometry difficult?
Algebraic geometry sets out to answer these questions by applying the techniques of abstract algebra to the set of polynomials that define the curves (which are then called “algebraic varieties”). The mathematics involved is inevitably quite hard, although it is covered in degree-level courses.
Can a plane be curved?
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.
What does it mean for a curve to be smooth?
A smooth curve is a curve which is a smooth function, where the word “curve” is interpreted in the analytic geometry context. In particular, a smooth curve is a continuous map from a one-dimensional space to an. -dimensional space which on its domain has continuous derivatives up to a desired order.
Who is the father of algebraic geometry?
| Alexander Grothendieck | |
|---|---|
| Alma mater | University of Montpellier University of Nancy |
| Known for | Renewing algebraic geometry and synthesis between it and number theory and topology List of things named after Alexander Grothendieck |
| Awards | 1966 Fields Medal 1977 Émile Picard Medal 1988 Crafoord Prize (declined) |
| Scientific career | |
Who invented algebraic geometry?
The French mathematician Alexandre Grothendieck revolutionized algebraic geometry in the 1950s by generalizing varieties to schemes and extending the Riemann-Roch theorem. Arithmetic geometry combines algebraic geometry and number theory to study integer solutions of polynomial equations.
What is the difference between algebraic geometry and algebraic topology?
Geometry concerns the local properties of shape such as curvature, while topology involves large-scale properties such as genus. Algebraic methods become important in topology when working in many dimensions, and increasingly sophisticated parts of algebra are now being employed.
What are the different types of algebra?
Algebra is divided into different sub-branches such as elementary algebra, advanced algebra, abstract algebra, linear algebra, and commutative algebra.
Why algebra is called algebra?
The word “algebra” is derived from the Arabic word الجبر al-jabr, and this comes from the treatise written in the year 830 by the medieval Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī, whose Arabic title, Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, can be translated as The Compendious Book on Calculation …