In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra.
These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes.
The typical situations where these notions are used are the following:
The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree.
The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.
The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial.
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space.
The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations. In addition, they provide useful invariants for families of algebraic varieties because a flat family has the same Hilbert polynomial over any closed point . This is used in the construction of the Hilbert scheme and Quot scheme.
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In commutative algebra, the Hilbert function, the Hilbertpolynomial, and the Hilbertseries of a graded commutative algebra finitely generated over a...
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional)...
Hilbert ring Hilbert–Poincaré seriesHilbertseriesandHilbertpolynomialHilbert space Hilbert spectrum Hilbert system Hilbert transform Hilbert's arithmetic...
arises in the least squares approximation of arbitrary functions by polynomials. The Hilbert matrices are canonical examples of ill-conditioned matrices, being...
term Hilbert dimension may refer to: Hilbert space dimension Hilbert dimension in ring theory, see Hilbert's basis theorem HilbertseriesandHilbert polynomial...
variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbertpolynomials. The basic theory of Hilbert schemes was...
analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous...
of Bézout's theorem (For a proof, see HilbertseriesandHilbertpolynomial § Degree of a projective variety and Bézout's theorem). The degree is not an...
abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns. A Diophantine...
Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations...
Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and Wiener chaos expansion, is a method for representing a random variable in terms...
challenge posed by David Hilbertand Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "yes" or "no"...
order polynomial solutions to the Sturm–Liouville equations in the setting of Hilbert space. These include the Legendre and Laguerre polynomials as well...
generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f ( x ) {\displaystyle f(x)} with integer coefficients. Recall that first...
or mixed states in a Hilbert space; the transition function is replaced by a collection of unitary matrices that map the Hilbert space to itself. That...
Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)} and U n ( x...
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation...
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