The Tschirnhausen cubic is an algebraic curve of degree three.
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In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
If the defining polynomial of a plane algebraic curve is irreducible, then one has an irreducible plane algebraic curve. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its components, that are defined by the irreducible factors.
More generally, an algebraic curve is an algebraic variety of dimension one. (In some contexts, an algebraic set of dimension one is also called an algebraic curve, but this will not be the case in this article.) Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an irreducible algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a birational equivalence.
These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula).
A non-plane curve is often called a space curve or a skew curve.
an algebraic set of dimension one is also called an algebraiccurve, but this will not be the case in this article.) Equivalently, an algebraiccurve is...
space and skew curves apply also to real algebraiccurves, although the above definition of a curve does not apply (a real algebraiccurve may be disconnected)...
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...
mathematics, an elliptic curve is a smooth, projective, algebraiccurve of genus one, on which there is a specified point O. An elliptic curve is defined over...
smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose...
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism...
In algebraic geometry, a quartic plane curve is a plane algebraiccurve of the fourth degree. It can be defined by a bivariate quartic equation: A x 4...
gonality of an algebraiccurve C is defined as the lowest degree of a nonconstant rational map from C to the projective line. In more algebraic terms, if C...
precise definition of a singular point depends on the type of curve being studied. Algebraiccurves in the plane may be defined as the set of points (x, y)...
mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with...
singularities were first noticed in the study of algebraiccurves. The double point at (0, 0) of the curve y 2 = x 2 + x 3 {\displaystyle y^{2}=x^{2}+x^{3}}...
containing no such points). This is an example of an algebraiccurve. Every elliptic curve is an algebraiccurve, given by (the compactification of) the locus...
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraiccurve, constructed as a quotient of...
the curve, curvilinear asymptotes have also been used although the term asymptotic curve seems to be preferred. The asymptotes of an algebraiccurve in...
In algebraic geometry, a lemniscate (/lɛmˈnɪskɪt/ or /ˈlɛmnɪsˌkeɪt, -kɪt/) is any of several figure-eight or ∞-shaped curves. The word comes from the...
In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric...
In geometry, a circular algebraiccurve is a type of plane algebraiccurve determined by an equation F(x, y) = 0, where F is a polynomial with real coefficients...
references to algebraic geometry codes throughout 1980s and 1990s coding theory literature. In this section the construction of algebraic geometry codes...
of algebraic varieties are: plane algebraiccurves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and...
(mathematics)#Behavior of a polynomial function near a multiple root Algebraiccurve#Tangent at a point In "Nova Methodus pro Maximis et Minimis" (Acta...
{\displaystyle k(Y)} ). We see that the degree of an algebraic function field is not a well-defined notion. The algebraic function fields over k form a category; the...
In algebraic geometry, a hyperelliptic curve is an algebraiccurve of genus g > 1, given by an equation of the form y 2 + h ( x ) y = f ( x ) {\displaystyle...
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