In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.
[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.
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.
[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 where p is a multivariate polynomial.
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.
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.
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.
Often, it is left to the context whether the term hypersurface refers to all points or only to the real part.
For example, the imaginary n-sphere defined by the equation is a real hypersurface without any real point, which is defined over the rational numbers.
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
As usual, homogeneous polynomial means that all monomials of P have the same degree, or, equivalently that
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
with where d is the degree of P. These two processes projective completion and restriction to an affine subspace are inverse one to the other.
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 in the affine space of dimension three has a unique singular point, which is at infinity, in the direction x = 0, y = 0.