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Concept
Hypersurface
Summary
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.
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
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.
A hypersurface that is a smooth manifold is called a smooth hypersurface.
In Rn, a smooth hypersurface is orientable. 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.
An algebraic hypersurface is an algebraic variety that may be defined by a single implicit equation of the form
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 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.
Official source
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