Hilbert polynomial
Encyclopedia
In commutative algebra
, the Hilbert
polynomial of a graded commutative algebra
or graded module is a polynomial
in one variable that measures the rate of growth of the dimensions of its homogeneous components. The degree and the leading coefficient of the Hilbert polynomial of a graded commutative algebra S are related with the dimension and the degree of the projective algebraic variety Proj
S.
over a field
K that is generated by the finite dimensional space S1 is the unique polynomial
HS(t) with rational coefficients such that
for all but finitely many positive integers n. In other words, the term 'Hilbert polynomial' refers to the 'Hilbert function', in those cases where the function's values are given by a polynomial for all but finitely many natural n: it is a form of polynomial interpolation
, though not usually referred to as such.
The Hilbert polynomial is a numerical polynomial
, since the dimensions are integers, but the polynomial almost never has integer coefficients .
Similarly, one can define the Hilbert polynomial HM of a finitely generated graded module M, at least, when the grading is positive.
The Hilbert polynomial of a projective variety V in Pn is defined as the Hilbert polynomial of the homogeneous coordinate ring
of V.
of M is defined as the generating function
of the dimensions of the graded components of M. In good cases, the Hilbert–Poincaré series is a rational function
.
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, the Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
polynomial of a graded commutative algebra
Graded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
or graded module is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
in one variable that measures the rate of growth of the dimensions of its homogeneous components. The degree and the leading coefficient of the Hilbert polynomial of a graded commutative algebra S are related with the dimension and the degree of the projective algebraic variety Proj
Proj construction
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties...
S.
Definition
The Hilbert polynomial of a graded commutative algebraGraded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K that is generated by the finite dimensional space S1 is the unique polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
HS(t) with rational coefficients such that
- HS(n) = dimk Sn
for all but finitely many positive integers n. In other words, the term 'Hilbert polynomial' refers to the 'Hilbert function', in those cases where the function's values are given by a polynomial for all but finitely many natural n: it is a form of polynomial interpolation
Polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial: given some points, find a polynomial which goes exactly through these points.- Applications :...
, though not usually referred to as such.
The Hilbert polynomial is a numerical polynomial
Numerical polynomial
In mathematics, a numerical polynomial is a polynomial with rational coefficients that takes integer values on integers. They are also called integer-valued polynomials....
, since the dimensions are integers, but the polynomial almost never has integer coefficients .
Similarly, one can define the Hilbert polynomial HM of a finitely generated graded module M, at least, when the grading is positive.
The Hilbert polynomial of a projective variety V in Pn is defined as the Hilbert polynomial of the homogeneous coordinate ring
Homogeneous coordinate ring
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...
of V.
Examples
- The Hilbert polynomial of the polynomial ring in k+1 variables, S = K[x0, x1,…, xk], where each xi is homogeneous of degree 1, is the binomial coefficientBinomial coefficientIn mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
- If M is a finite-dimensional graded module then all its homogeneous components of sufficiently high degree are zero, therefore, the Hilbert polynomial of M is identically zero.
Generalizations
When the ring S is not generated by its degree one part, the Hilbert function of a finitely generated graded module over S is still well-defined, but may no longer be a polynomial. The Hilbert–Poincaré seriesHilbert–Poincaré series
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series , named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures...
of M is defined as the generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
of the dimensions of the graded components of M. In good cases, the Hilbert–Poincaré series is a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
.