Hilbert polynomial

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".

Definition

The Hilbert polynomial of a graded commutative algebra

: "S" = ⊕"S"_"n"

over a field "K" that is generated by the finite dimensional space "S"_"1" is the unique polynomial "H"_"S"("t") with rational coefficients such that

:"H"_"S"("n") = dim_"k" "S"_"n"

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".

The Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial does not necessarily have integer coefficients harv|Schenck|2003|pp=41.

Similarly, one can define the Hilbert polynomial "H"_"M" of a finitely generated graded module "M", at least, when the grading is positive.

The Hilbert polynomial of a projective variety "V" in "P^{n" is defined as the Hilbert polynomial of the homogeneous coordinate ring of "V".}

Examples

* The Hilbert polynomial of the polynomial ring in "k"+1 variables, "S" = "K" ["x"₀, "x"₁,…"x"_k] , where each "x"_i 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.

References

* | isbn=0-387-94268-8.
* | year=2003
* .