hypersurface

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In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, a hypersurface is a generalization of the concepts of
hyperplane In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
,
plane curve In mathematics, a plane curve is a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought o ...
, and
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
. A hypersurface is a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
of dimension , which is embedded in an ambient space of dimension , generally a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensi ...
, an
affine space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
or a
projective space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Hypersurfaces share, with surfaces in a
three-dimensional space Three-dimensional space (also: 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 informal meaning of the ...
, the property of being defined by a single
implicit equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...
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 linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an a ...
, and is called a
hypersphere In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an a ...
is called a ''smooth hypersurface''. In , a smooth hypersurface is Orientability, orientable.Hans Samelson (1969
''n''">"Orientability of hypersurfaces in R''n''
, ''Proceedings of the American Mathematical Society'' 22(1): 301,2
Every connected space, connected compact space, compact smooth hypersurface is a level set, and separates R''n'' into two connected components; this is related to the Jordan curve theorem#Proof and generalizations, 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 Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
that may be defined by a single implicit equation of the form :$p\left(x_1, \ldots, x_n\right)=0,$ where is a multivariate polynomial. Generally the polynomial is supposed to be irreducible polynomial, irreducible. When this is not the case, the hypersurface is not an algebraic variety, but only an algebraic set. 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 (mathematics), field , and the points of the hypersurface are the zero of a function, zeros of in the
affine space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
$K^n,$ where is an algebraically closed extension of . A hypersurface may have singular point of an algebraic variety, 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, which asserts that a hypersurface contains a given algebraic set if and only if the defining polynomial of the hypersurface has a power that belongs to the ideal (ring theory), ideal generated by the defining polynomials of the algebraic set. A corollary of this theorem is that, if two irreducible polynomials (or more generally two square-free polynomials) define the same hypersurface, then one is the product of the other by a nonzero constant. Hypersurfaces are exactly the subvarieties of
dimension In and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...
of an
affine space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
of dimension of . This is the geometric interpretation of the fact that, in a polynomial ring over a field, the height (ring theory), height of an ideal is 1 if and only if the ideal is a principal ideal. 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 number, real 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 field, algebraically closed (typically the field of rational numbers, a finite field or a number field), 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 rationals.

# Projective algebraic hypersurface

A of dimension in a
projective space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of dimension over a field is defined by a homogeneous polynomial $P\left(x_0, x_1, \ldots, x_n\right)$ in indeterminates. As usual, means that all monomials of have the same degree, or, equivalently that $P\left(cx_0, cx_1, \ldots, cx_n\right)=c^dP\left(x_0, x_1, \ldots, x_n\right)$ 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
of equation $x_0=0$ as hyperplane at infinity, the complement of this hyperplane is an
affine space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, and the points of the projective hypersuface that belong to this affine space form an affine hypersurface of equation $P\left(1, x_1, \ldots, x_n\right) = 0.$ Conversely, given an affine hypersurface of equation $p\left(x_1, \ldots, x_n\right)=0,$ it defines a projective hypersurface, called its , whose equation is obtained by Homogeneous polynomial#Homogenization, homogenizing . That is, the equation of the projective completion is $P\left(x_0, x_1, \ldots, x_n\right) = 0,$ with :$P\left(x_0, x_1, \ldots, x_n\right) = x_0^dp\left(x_1/x_0, \ldots, x_n/x_0\right),$ 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 singular point of an algebraic variety, nonsingular, 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 .