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hypersurface wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 smooth hypersurface 2 affine algebraic hypersurface toggle affine algebraic hypersurface subsection 2 1 properties 2 2 real and rational points 3 projective algebraic hypersurface 4 see also 5 references toggle the table of contents hypersurface 23 languages العربية català deutsch ελληνικά español eesti français हिन्दी ido italiano 日本語 қазақша 한국어 македонски nederlands polski português română русский slovenščina українська tiếng việt 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia manifold or algebraic variety of dimension n in a space of dimension n 1 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 n 1 which is embedded in an ambient space of dimension n 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 x 1 2 x 2 2 x n 2 1 0 displaystyle x_ 1 2 x_ 2 2 cdots x_ n 2 1 0 defines an algebraic hypersurface of dimension n 1 in the euclidean space of dimension n this hypersurface is also a smooth manifold and is called a hypersphere or an n 1 sphere smooth hypersurface edit a hypersurface that is a smooth manifold is called a smooth hypersurface in r n a smooth hypersurface is orientable 2 every connected compact smooth hypersurface is a level set and separates r n into two connected components this is related to the jordan brouwer separation theorem 3 affine algebraic hypersurface edit an algebraic hypersurface is an algebraic variety that may be defined by a single implicit equation of the form p x 1 x n 0 displaystyle p x_ 1 ldots x_ n 0 where p 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 k and the points of the hypersurface are the zeros of p in the affine space k n displaystyle k n where k is an algebraically closed extension of k 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 edit 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 n 1 of an affine space of dimension of n 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 n 1 real and rational points edit 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 displaystyle mathbb c of complex numbers the real points of a real hypersurface are the points that belong to r n c n displaystyle mathbb r n subset mathbb c n 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 k 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 k and the points that belong to k n displaystyle k n are rational over k in the case of the field of rational numbers over k is generally omitted for example the imaginary n sphere defined by the equation x 0 2 x n 2 1 0 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 the gaussian rationals projective algebraic hypersurface edit a projective algebraic hypersurface of dimension n 1 in a projective space of dimension n over a field k is defined by a homogeneous polynomial p x 0 x 1 x n displaystyle p x_ 0 x_ 1 ldots x_ n in n 1 indeterminates as usual homogeneous polynomial means that all monomials of p have the same degree or equivalently that p c x 0 c x 1 c x n c d p x 0 x 1 x n displaystyle p cx_ 0 cx_ 1 ldots cx_ n c d p x_ 0 x_ 1 ldots x_ n for every constant c where d is the degree of the polynomial the points of the hypersurface are the points of the projective space whose projective coordinates are zeros of p if one chooses the hyperplane of equation x 0 0 displaystyle x_ 0 0 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 p 1 x 1 x n 0 displaystyle p 1 x_ 1 ldots x_ n 0 conversely given an affine hypersurface of equation p x 1 x n 0 displaystyle p x_ 1 ldots x_ n 0 it defines a projective hypersurface called its projective completion whose equation is obtained by homogenizing p that is the equation of the projective completion is p x 0 x 1 x n 0 displaystyle p x_ 0 x_ 1 ldots x_ n 0 with p x 0 x 1 x n x 0 d p x 1 x 0 x n x 0 displaystyle p x_ 0 x_ 1 ldots x_ n x_ 0 d p x_ 1 x_ 0 ldots x_ n x_ 0 where d is the degree of p 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 singular at infinity for example the circular cylinder of equation x 2 y 2 1 0 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 direction x 0 y 0 see also edit affine sphere coble hypersurface dwork family null hypersurface polar hypersurface references edit lee jeffrey 2009 curves and hypersurfaces in euclidean space manifolds and differential geometry providence american mathematical society pp 143 188 isbn 978 0 8218 4815 9 hans samelson 1969 orientability of hypersurfaces in r n proceedings of the american mathematical society 22 1 301 2 lima elon l 1988 the jordan brouwer separation theorem for smooth hypersurfaces the american mathematical monthly 95 1 39 42 doi 10 1080 00029890 1988 11971963 hypersurface encyclopedia of mathematics ems press 2001 1994 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 v t e dimension dimensional spaces vector space euclidean space affine space projective space free module manifold algebraic variety spacetime other dimensions krull lebesgue covering inductive hausdorff minkowski fractal degrees of freedom polytopes and shapes hyperplane hypersurface hypercube hyperrectangle demihypercube hypersphere cross polytope simplex hyperpyramid number systems hypercomplex numbers cayley dickson construction dimensions by number zero one two three four five six seven eight n dimensions see also hyperspace codimension category retrieved from https en wikipedia org w index php title hypersurface oldid 1275236801 categories algebraic geometry multi dimensional geometry surfaces dimension theory hidden categories articles with short description short description is different from wikidata this page was last edited on 11 february 2025 at 21 22 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety 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