Hypersurface

Template:Short description 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 Template:Math, which is embedded in an ambient space of dimension Template:Math, generally a Euclidean space, an affine space or a projective space.<ref>Template:Cite book</ref> 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

<math>x_1^2+x_2^2+\cdots+x_n^2-1=0</math>

defines an algebraic hypersurface of dimension Template:Math in the Euclidean space of dimension Template:Math. This hypersurface is also a smooth manifold, and is called a hypersphere or an [[n-sphere|Template:Math-sphere]].

Smooth hypersurface

A hypersurface that is a smooth manifold is called a smooth hypersurface.

In Template:Math, a smooth hypersurface is orientable.<ref>Hans Samelson (1969) "Orientability of hypersurfaces in Rⁿ", Proceedings of the American Mathematical Society 22(1): 301,2 </ref> 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.<ref name="Lima">Template:Cite journal</ref>

Affine algebraic hypersurface Template:Anchor

An algebraic hypersurface is an algebraic variety that may be defined by a single implicit equation of the form

<math>p(x_1, \ldots, x_n)=0,</math>

where Template:Mvar 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 Template:Mvar, and the points of the hypersurface are the zeros of Template:Mvar in the affine space <math>K^n,</math> where Template:Mvar is an algebraically closed extension of Template:Mvar.

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 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 Template:Math of an affine space of dimension of Template:Mvar. 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 Template:Math.

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 <math>\mathbb C</math> of complex numbers. The real points of a real hypersurface are the points that belong to <math>\mathbb R^n \subset \mathbb C^n.</math> 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 Template:Mvar 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 Template:Mvar, and the points that belong to <math>k^n</math> are rational over Template:Mvar (in the case of the field of rational numbers, "over Template:Mvar" is generally omitted).

For example, the imaginary [[n-sphere|Template:Mvar-sphere]] defined by the equation

<math>x_0^2 +\cdots+x_n^2 +1=0</math>

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 hypersurfaceTemplate:Anchor

A Template:Em of dimension Template:Math in a projective space of dimension Template:Mvar over a field Template:Mvar is defined by a homogeneous polynomial <math>P(x_0, x_1, \ldots, x_n)</math> in Template:Math indeterminates. As usual, Template:Em means that all monomials of Template:Mvar have the same degree, or, equivalently that <math>P(cx_0, cx_1, \ldots, cx_n)=c^dP(x_0, x_1, \ldots, x_n)</math> for every constant Template:Mvar, where Template:Math is the degree of the polynomial. The Template:Em of the hypersurface are the points of the projective space whose projective coordinates are zeros of Template:Mvar.

If one chooses the hyperplane of equation <math>x_0=0</math> as hyperplane at infinity, the complement of this hyperplane is an affine space, and the points of the projective hypersurface that belong to this affine space form an affine hypersurface of equation <math>P(1, x_1, \ldots, x_n) = 0.</math> Conversely, given an affine hypersurface of equation <math>p(x_1, \ldots, x_n)=0,</math> it defines a projective hypersurface, called its Template:Em, whose equation is obtained by homogenizing Template:Mvar. That is, the equation of the projective completion is <math>P(x_0, x_1, \ldots, x_n) = 0,</math> with

<math>P(x_0, x_1, \ldots, x_n) = x_0^dp(x_1/x_0, \ldots, x_n/x_0),</math>

where Template:Mvar is the degree of Template:Mvar.

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 Template:Em. For example, the circular cylinder of equation

<math>x^2+y^2-1=0</math>

in the affine space of dimension three has a unique singular point, which is at infinity, in the direction Template:Math.

See also

• Affine sphere
• Coble hypersurface
• Dwork family
• Null hypersurface
• Polar hypersurface

References

• Template:Springer
• 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.