In
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 c ...
, a hypersurface is a generalization of the concepts of
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 hyper ...
,
plane curve, and
surface. 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 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 Euclidea ...
, 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
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
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, 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 coord ...
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
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. As a requirement, all articles must be at most 15 printed pages.
According to the ' ...
'' 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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
smooth hypersurface is a
level set, 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
:
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 related ...
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, 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
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 considered ...
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 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 coord ...
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 coefficients. In this case the algebraically closed field over which the points are defined is generally the field
of complex numbers. 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 (e.g. ). The set of all ra ...
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), one says that the hypersurface is ''defined over'' , and the points that belong to
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
:
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, 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 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; ...
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 expon ...
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 coordinate system, Cartesian coordinates are u ...
are zeros of .
If one chooses the
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 hyper ...
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 related ...
, 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
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 multiplic ...
, while its projective completion has singular points. In this case, one says that the affine surface is . For example, the
circular cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
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