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Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is orienta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties. For instance, a hyperplane of an -dimensional affine space is a flat subset with dimension and it separates the space into two half spaces. While a hyperplane of an -dimensional projective space does not have this property. The difference in dimension between a subspace and its ambient space is known as the codimension of with respect to . Therefore, a necessary and sufficient condition for to be a hyperplane in is for to have codimension one in . Technical description In geometry, a hyperplane of an ''n''-dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (mathematics)
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a ''coordinate patch'' on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles (ideally) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Point Of An Algebraic Variety
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth. Definition A plane curve defined by an implicit equation :F(x,y)=0, where is a smooth function is said to be ''singular'' at a point if the Taylor series of has order at least at this point. The reason for this is that, in differential calculus, the tangent at the point of such a curve is defined by the equation :(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0)=0, whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem). Formulation Let ''k'' be a field (such as the rational numbers) and ''K'' be an algebraically closed field extension (such as the complex numbers). Consider the polynomial ring k _1, \ldots, X_n/math> and let ''I'' be an ideal in this ring. The algebraic set V(''I'') defined by this ideal consists of all ''n''-tuples x = (''x''1,...,''x''''n'') in ''Kn'' such that ''f''(x) = 0 for all ''f'' in ''I''. Hilbert's Nullstellensatz st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Extension
In mathematics, a field (mathematics), field is algebraically closed if every Degree of a polynomial, non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a Zero of a function, root in . Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation ''x''2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Set
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a datatype in computer programming each of whose values is data from other datatypes wrapped in one of the constructors of the datatype * Algebraic numbers, a complex number that is a root of a non-zero polynomial in one variable with integer coefficients * Algebraic functions, functions satisfying certain polynomials * Algebraic element, an element of a field extension which is a root of some polynomial over the base field * Algebraic extension, a field extension such that every element is an algebraic element over the base field * Algebraic definition, a definition in mathematical logic which is given using only equalities between terms * Algebraic structure, a set with one or more finitary operations defined on it * Algebraic, the order of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as \left(x - \sqrt\right)\left(x + \sqrt\right) if it is considered as a polynomial with real coefficients. One says that the polynomial is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common definitions. Most often, a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joins tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
