In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a hypersurface is a generalization of the concepts of
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
,
plane curve
In mathematics, a plane curve is a curve in a plane that may be 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 c ...
, 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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
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 solution set, set of solutions of a system of polynomial equations over the real number, ...
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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, 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 relat ...
or a
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
.
Hypersurfaces share, with surfaces in a
three-dimensional space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...
, 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
:
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 coo ...
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 may ...
, and is called a
hypersphere 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 may ...
is called a ''smooth hypersurface''.
In , a smooth hypersurface is
orientable.
[ Hans Samelson (1969]
''n''">"Orientability of hypersurfaces in R''n''
, ''Proceedings of the American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. The journal is devoted to shorter research articles. As a requirement, all articles ...
'' 22(1): 301,2 Every
connected compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
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 call ...
, 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 solution set, set of solutions of a system of polynomial equations over the real number, ...
that may be defined by a single implicit equation of the form
:
where is a
multivariate polynomial. 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. 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 , 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 relat ...
where is an
algebraically closed extension 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 ge ...
, 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 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 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 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 coo ...
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 relat ...
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 an example of vertical extent, "This basketball player is 7 foot 1 inches in height." For an e ...
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 coefficients. In this case the algebraically closed field over which the points are defined is generally the field
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. The ''real points'' of a real hypersurface are the points that belong to
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 (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 (for example,
The set of all ...
s, a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
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 ...
), one says that the hypersurface is ''defined over'' , and the points that belong to
are ''
rational
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
'' over (in the case of the field of rational numbers, "over " is generally omitted).
For example, the imaginary
-sphere defined by the equation
:
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
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of dimension over a field is defined by a
homogeneous polynomial 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 a power product or primitive monomial, is a product of powers of variables with n ...
s of have the same degree, or, equivalently that
for every constant , where is the degree of the polynomial. The of the hypersurface are the points of the projective space whose
projective coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
are zeros of .
If one chooses the
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
of equation
as
hyperplane at infinity, 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 relat ...
, and the points of the projective hypersurface that belong to this affine space form an affine hypersurface of equation
Conversely, given an affine hypersurface of equation
it defines a projective hypersurface, called its , whose equation is obtained by
homogenizing . That is, the equation of the projective completion is
with
:
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, while its projective completion has singular points. In this case, one says that the affine surface is . For example, the
circular cylinder of equation
:
in the affine space of dimension three has a unique singular point, which is at infinity, in the direction .
See also
*
Affine sphere
*
Coble hypersurface
*
Dwork family
*
Null hypersurface
*
Polar hypersurface
References
*
*
Shoshichi Kobayashi and
Katsumi Nomizu (1969),
Foundations of Differential Geometry Vol II,
Wiley Interscience
* 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
Dimension theory