Hilbert polynomial - Citizendia

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. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations 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. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of Affine schemes which produces objects with the typical

Definition

The Hilbert polynomial of a graded commutative algebra

S = ⊕S_n

over a field K that is generated by the finite dimensional space S₁ 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 Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations 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. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure

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. In Mathematics, a numerical polynomial is a Polynomial with rational coefficients that takes Integer values on integers 41).

Similarly, one can define the Hilbert polynomial H_M of a finitely generated graded module M, at least, when the grading is positive. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure

The Hilbert polynomial of a projective variety V in Pⁿ is defined as the Hilbert polynomial of the homogeneous coordinate ring of V. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety

Examples

The Hilbert polynomial of the polynomial ring in k+1 variables, S = K[x₀, x₁,…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. In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x k term in the Polynomial

References

Eisenbud, David (1995), Commutative algebra. David Eisenbud (born 8 April, 1947) is an American Mathematician. With a view toward algebraic geometry, vol. 150, Graduate Texts in Mathematics, New York: Springer-Verlag, MR 1322960, ISBN 0-387-94268-8 . Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many
Schenck, Hal (2003), Computational Algebraic Geometry, Cambridge: Cambridge University Press, MR 011360, ISBN 978-0-521-53650-9
Stanley, Richard (1978), “Hilbert functions of graded algebras”, Advances in Math. The city of Cambridge (ˈkeɪmbrɪdʒ is a university town and the administrative centre of the county of Cambridgeshire, England Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Richard Stanley may refer to Richard Stanley (film director (born 1966 South African-born film director and screenwriter Richard Stanley 28 (1): 57–83, MR 0485835 . Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many

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