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.[1] 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
2-1=0 | |
x | |
n |
A hypersurface that is a smooth manifold is called a smooth hypersurface.
In, a smooth hypersurface is orientable.[2] Every connected compact smooth hypersurface is a level set, and separates Rn into two connected components; this is related to the Jordan–Brouwer separation theorem.[3]
An algebraic hypersurface is an algebraic variety that may be defined by a single implicit equation of the form
p(x1,\ldots,xn)=0,
Kn,
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.
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 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 of an affine space of dimension of . This is the geometric interpretation of the fact that, in a polynomial ring over a field, the 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 .
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
C
Rn\subsetCn.
If the coefficients of the defining polynomial belong to a field that is not 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
kn
For example, the imaginary -sphere defined by the equation
2 | |
x | |
0 |
2 | |
+ … +x | |
n |
+1=0
P(x0,x1,\ldots,xn)
P(cx0,cx1,\ldots,
dP(x | |
cx | |
0, |
x1,\ldots,xn)
If one chooses the hyperplane of equation
x0=0
P(1,x1,\ldots,xn)=0.
p(x1,\ldots,xn)=0,
P(x0,x1,\ldots,xn)=0,
P(x0,x1,\ldots,xn)=
dp(x | |
x | |
1/x |
0,\ldots,xn/x0),
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
x2+y2-1=0