ProjectiveHilbertPolynomial -- the class of all Hilbert polynomials

Description

For convenience, these polynomials are expressed in terms of the Hilbert polynomials of projective space.

The functions degree and dim are designed so they correspond the degree and dimension of the algebraic variety that may have been used to produce the Hilbert polynomial.

i1 : Z = Proj(QQ[x_0..x_12]/(x_0^3+x_12^3))

o1 = Z

o1 : ProjectiveVariety

i2 : hilbertPolynomial Z

o2 = P  - 3*P   + 3*P
      9      10      11

o2 : ProjectiveHilbertPolynomial

Functions and methods returning a projective Hilbert polynomial :

ProjectiveHilbertPolynomial * ZZ -- see * -- a binary operator, usually used for multiplication
ZZ * ProjectiveHilbertPolynomial -- see * -- a binary operator, usually used for multiplication
ProjectiveHilbertPolynomial + ProjectiveHilbertPolynomial -- see + -- a unary or binary operator, usually used for addition
ProjectiveHilbertPolynomial + ZZ -- see + -- a unary or binary operator, usually used for addition
ZZ + ProjectiveHilbertPolynomial -- see + -- a unary or binary operator, usually used for addition
- ProjectiveHilbertPolynomial -- see - -- a unary or binary operator, usually used for negation or subtraction
ProjectiveHilbertPolynomial - ProjectiveHilbertPolynomial -- see - -- a unary or binary operator, usually used for negation or subtraction
ProjectiveHilbertPolynomial - ZZ -- see - -- a unary or binary operator, usually used for negation or subtraction
ZZ - ProjectiveHilbertPolynomial -- see - -- a unary or binary operator, usually used for negation or subtraction
diff(ProjectiveHilbertPolynomial)
diff(ProjectiveHilbertPolynomial,ZZ)
hilbertPolynomial -- compute the Hilbert polynomial
projectiveHilbertPolynomial -- Hilbert polynomial of projective space

Methods that use a projective Hilbert polynomial :

ProjectiveHilbertPolynomial == ProjectiveHilbertPolynomial -- see == -- equality
ProjectiveHilbertPolynomial == ZZ -- see == -- equality
ZZ == ProjectiveHilbertPolynomial -- see == -- equality
degree(ProjectiveHilbertPolynomial)
dim(ProjectiveHilbertPolynomial) -- the degree of the Hilbert polynomial
euler(ProjectiveHilbertPolynomial) -- constant term of the Hilbert polynomial
hilbertSeries(ProjectiveHilbertPolynomial) -- compute the Hilbert series of a projective Hilbert polynomial
ProjectiveHilbertPolynomial ZZ -- value of polynomial

For the programmer

The object ProjectiveHilbertPolynomial is a type, with ancestor classes HashTable < Thing.