Ingeometry, ahypersurface is a generalization of the concepts ofhyperplane,plane curve, andsurface. A hypersurface is amanifold or analgebraic variety of dimensionn − 1, which isembedded in anambient space of dimensionn, generally aEuclidean space, anaffine space or aprojective space.^[1]Hypersurfaces share, with surfaces in athree-dimensional space, the property of being defined by a singleimplicit 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

${\displaystyle x_1^2+x_2^2+\cdots +x_n^2-1=0}$

defines an algebraic hypersurface ofdimensionn − 1 in the Euclidean space of dimensionn. This hypersurface is also asmooth manifold, and is called ahypersphere or an(n – 1)-sphere.

A hypersurface that is asmooth manifold is called asmooth hypersurface.

InRⁿ, a smooth hypersurface isorientable.^[2] Everyconnectedcompact smooth hypersurface is alevel set, and separatesRⁿ into two connected components; this is related to theJordan–Brouwer separation theorem.^[3]

Analgebraic hypersurface is analgebraic variety that may be defined by a single implicit equation of the form

${\displaystyle p(x_1,\dots ,x_n)=0,}$

wherep is amultivariate polynomial. Generally the polynomial is supposed to beirreducible. When this is not the case, the hypersurface is not an algebraic variety, but only analgebraic set. It may depend on the authors or the context whether a reducible polynomial defines a hypersurface. For avoiding ambiguity, the termirreducible hypersurface is often used.

As for algebraic varieties, the coefficients of the defining polynomial may belong to any fixedfieldk, and the points of the hypersurface are thezeros ofp in theaffine space${\displaystyle K^n,}$ whereK is analgebraically closed extension ofk.

A hypersurface may havesingularities, 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 isHilbert's Nullstellensatz, which asserts that a hypersurface contains a givenalgebraic set if and only if the defining polynomial of the hypersurface has a power that belongs to theideal generated by the defining polynomials of the algebraic set.

A corollary of this theorem is that, if twoirreducible polynomials (or more generally twosquare-free polynomials) define the same hypersurface, then one is the product of the other by a nonzero constant.

Hypersurfaces are exactly the subvarieties ofdimensionn – 1 of anaffine space of dimension ofn. This is the geometric interpretation of the fact that, in a polynomial ring over a field, theheight of an ideal is 1 if and only if the ideal is aprincipal 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 dimensionn – 1.

Areal hypersurface is a hypersurface that is defined by a polynomial withreal coefficients. In this case the algebraically closed field over which the points are defined is generally the field${\displaystyle \mathbb{C}}$ ofcomplex numbers. Thereal points of a real hypersurface are the points that belong to${\displaystyle {\mathbb{R}}^n\subset {\mathbb{C}}^n.}$ The set of the real points of a real hypersurface is thereal part of the hypersurface. Often, it is left to the context whether the termhypersurface refers to all points or only to the real part.

If the coefficients of the defining polynomial belong to a fieldk that is notalgebraically closed (typically the field ofrational numbers, afinite field or anumber field), one says that the hypersurface isdefined overk, and the points that belong to${\displaystyle k^n}$ arerational overk (in the case of the field of rational numbers, "overk" is generally omitted).

For example, the imaginaryn-sphere defined by the equation

${\displaystyle x_0^2+\cdots +x_n^2+1=0}$

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 theGaussian rationals.

Aprojective (algebraic) hypersurface of dimensionn – 1 in aprojective space of dimensionn over a fieldk is defined by ahomogeneous polynomial${\displaystyle P(x_0,x_1,\dots ,x_n)}$ inn + 1 indeterminates. As usual,homogeneous polynomial means that allmonomials ofP have the same degree, or, equivalently that${\displaystyle P(c{x}_0,c{x}_1,\dots ,c{x}_n)=c^{d}P(x_0,x_1,\dots ,x_n)}$ for every constantc, whered is the degree of the polynomial. Thepoints of the hypersurface are the points of the projective space whoseprojective coordinates are zeros ofP.

If one chooses thehyperplane of equation${\displaystyle x_0=0}$ ashyperplane at infinity, the complement of this hyperplane is anaffine space, and the points of the projective hypersurface that belong to this affine space form an affine hypersurface of equation${\displaystyle P(1,x_1,\dots ,x_n)=0.}$ Conversely, given an affine hypersurface of equation${\displaystyle p(x_1,\dots ,x_n)=0,}$ it defines a projective hypersurface, called itsprojective completion, whose equation is obtained byhomogenizingp. That is, the equation of the projective completion is${\displaystyle P(x_0,x_1,\dots ,x_n)=0,}$ with

${\displaystyle P(x_0,x_1,\dots ,x_n)=x_0^{d}p(x_1/x_0,\dots ,x_n/x_0),}$

whered is the degree ofP.

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 isnonsingular, while its projective completion has singular points. In this case, one says that the affine surface issingular at infinity. For example, thecircular cylinder of equation

${\displaystyle x^2+y^2-1=0}$

in the affine space of dimension three has a unique singular point, which is at infinity, in the directionx = 0,y = 0.

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

• "Hypersurface",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Shoshichi Kobayashi andKatsumi 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.

• Algebraic geometry
• Multi-dimensional geometry
• Surfaces
• Dimension theory

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