affine algebraic hypersurface
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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 ...
, a hypersurface is a generalization of the concepts of hyperplane,
plane curve 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 ...
, and
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. A hypersurface is a manifold or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
of dimension , which is embedded in an ambient space of dimension , generally a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
or a projective space. Hypersurfaces share, with surfaces in a
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
, the property of being defined by a single
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
, 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 physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
in the Euclidean space of dimension . This hypersurface is also a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, and is called a
hypersphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...
or an -sphere.


Smooth hypersurface

A hypersurface that is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is called a ''smooth hypersurface''. In , a smooth hypersurface is
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
. Hans Samelson (1969
''n''">"Orientability of hypersurfaces in R''n''
, '' Proceedings of the American Mathematical Society'' 22(1): 301,2
Every
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
smooth hypersurface is a
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
, and separates R''n'' into two connected components; this is related to the Jordan–Brouwer separation theorem.


Affine algebraic hypersurface

An algebraic hypersurface is an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
that may be defined by a single implicit equation of the form :p(x_1, \ldots, x_n)=0, where is a
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 exampl ...
. Generally the polynomial is supposed to be irreducible. When this is not the case, the hypersurface is not an algebraic variety, but only an
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 data ...
. It may depend on the authors or the context whether a reducible polynomial defines a hypersurface. For avoiding ambiguity, the term ''irreducible hypersurface'' is often used. As for algebraic varieties, the coefficients of the defining polynomial may belong to any fixed
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, and the points of the hypersurface are the zeros of in the
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
K^n, where is an
algebraically closed extension In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
of . A hypersurface may have singularities, which are the common zeros, if any, of the defining polynomial and its partial derivatives. In particular, a real algebraic hypersurface is not necessarily a manifold.


Properties

Hypersurfaces have some specific properties that are not shared with other algebraic varieties. One of the main such properties is
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 ...
, which asserts that a hypersurface contains a given
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 data ...
if and only if the defining polynomial of the hypersurface has a power that belongs to the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
generated by the defining polynomials of the algebraic set. A corollary of this theorem is that, if two
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 f ...
s (or more generally two
square-free polynomial In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if i ...
s) define the same hypersurface, then one is the product of the other by a nonzero constant. Hypersurfaces are exactly the subvarieties of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
of dimension of . This is the geometric interpretation of the fact that, in a polynomial ring over a field, the
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
of an ideal is 1 if and only if the ideal is a
principal ideal In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
. In the case of possibly reducible hypersurfaces, this result may be restated as follows: hypersurfaces are exactly the algebraic sets whose all irreducible components have dimension .


Real and rational points

A ''real hypersurface'' is a hypersurface that is defined by a polynomial with
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
coefficients. In this case the algebraically closed field over which the points are defined is generally the field \mathbb C of complex numbers. The ''real points'' of a real hypersurface are the points that belong to \mathbb R^n \subset \mathbb C^n. The set of the real points of a real hypersurface is the ''real part'' of the hypersurface. Often, it is left to the context whether the term ''hypersurface'' refers to all points or only to the real part. If the coefficients of the defining polynomial belong to a field that is not
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
(typically the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s, a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
or a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
), one says that the hypersurface is ''defined over'' , and the points that belong to k^n are ''rational'' over (in the case of the field of rational numbers, "over " is generally omitted). For example, the imaginary -sphere defined by the equation :x_0^2 +\cdots+x_n^2 +1=0 is a real hypersurface without any real point, which is defined over the rational numbers. It has no rational point, but has many points that are rational over the
Gaussian rational In mathematics, a Gaussian rational number is a complex number of the form ''p'' + ''qi'', where ''p'' and ''q'' are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(''i''), obtained b ...
s.


Projective algebraic hypersurface

A of dimension in a projective space of dimension over a field is defined by a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
P(x_0, x_1, \ldots, x_n) in indeterminates. As usual, means that all
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
s of have the same degree, or, equivalently that P(cx_0, cx_1, \ldots, cx_n)=c^dP(x_0, x_1, \ldots, x_n) for every constant , where is the degree of the polynomial. The of the hypersurface are the points of the projective space whose projective coordinates are zeros of . If one chooses the hyperplane of equation x_0=0 as
hyperplane at infinity In geometry, any hyperplane ''H'' of a projective space ''P'' may be taken as a hyperplane at infinity. Then the set complement is called an affine space. For instance, if are homogeneous coordinates for ''n''-dimensional projective space, then ...
, the complement of this hyperplane is an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
, and the points of the projective hypersurface that belong to this affine space form an affine hypersurface of equation P(1, x_1, \ldots, x_n) = 0. Conversely, given an affine hypersurface of equation p(x_1, \ldots, x_n)=0, it defines a projective hypersurface, called its , whose equation is obtained by homogenizing . That is, the equation of the projective completion is P(x_0, x_1, \ldots, x_n) = 0, with :P(x_0, x_1, \ldots, x_n) = x_0^dp(x_1/x_0, \ldots, x_n/x_0), where is the degree of . These two processes projective completion and restriction to an affine subspace are inverse one to the other. Therefore, an affine hypersurface and its projective completion have essentially the same properties, and are often considered as two points-of-view for the same hypersurface. However, it may occur that an affine hypersurface is
nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplica ...
, while its projective completion has singular points. In this case, one says that the affine surface is . For example, the circular cylinder of equation :x^2+y^2-1=0 in the affine space of dimension three has a unique singular point, which is at infinity, in the direction .


See also

*
Affine sphere In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point. The term affine sphere is used because they play an analogous role in affine differential geometr ...
* Coble hypersurface *
Dwork family In algebraic geometry, a Dwork family is a one-parameter family of hypersurfaces depending on an integer ''n'', studied by Bernard Dwork. Originally considered by Dwork in the context of local zeta-function In number theory, the local zeta function ...
* Null hypersurface *
Polar hypersurface In algebraic geometry, given a projective algebraic 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 ...


References

* *
Shoshichi Kobayashi was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie alg ...
and
Katsumi Nomizu was a Japanese-American mathematician known for his work in differential geometry. Life and career Nomizu was born in Osaka, Japan on the first day of December, 1924. He studied mathematics at Osaka University, graduating in 1947 with a Maste ...
(1969),
Foundations of Differential Geometry ''Foundations of Differential Geometry'' is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publis ...
Vol II,
Wiley Interscience John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in p ...
* P.A. Simionescu & D. Beal (2004
Visualization of hypersurfaces and multivariable (objective) functions by partial globalization
''The Visual Computer'' 20(10):665–81. {{Dimension topics Algebraic geometry Multi-dimensional geometry Surfaces