Hilbert polynomial

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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.

[edit] Definition

The Hilbert polynomial of a graded commutative algebra

S = ⊕Sn

over a field K that is generated by the finite dimensional space S1 is the unique polynomial 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.

Note that the Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial does not necessarily have integer coefficients (Schenck 2003, pp. 41).

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.

[edit] 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 coefficient
H_S(t) = {{t+k}\choose{k}} = \frac{(t+1)\ldots(t+k)}{k!}.
  • 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.

[edit] References