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.
Consider a finitely generated graded commutative algebra S over a field K, which is finitely generated by elements of positive degree. This means that
and that .
The Hilbert function
maps the integer n to the dimension of the K-vector space Sn. The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series
If S is generated by h homogeneous elements of positive degrees , then the sum of the Hilbert series is a rational fraction
where Q is a polynomial with integer coefficients.
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In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem that asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings.
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.
In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring R = K[X0, X1, X2, ..., XN] / I where I is the homogeneous ideal defining V, K is the algebraically closed field over which V is defined, and K[X0, X1, X2, ..., XN] is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space).
Explores polynomial optimization, emphasizing SOS and nonnegative polynomials, including the representation of polynomials as quadratic functions of monomials.