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--- status: DRAFT
--- author(s): L. Gold
--- notes: show ex of eventually = HF
document {
Key => hilbertPolynomial,
Headline => "compute the Hilbert polynomial",
"In a singly graded ambient ring, the ",
TO2(hilbertFunction, "Hilbert function"),
" eventually is a polynomial called the Hilbert polynomial. By
default this polynomial is written in terms of the Hilbert
polynomials of projective spaces because it is a good form for
extracting geometric information from the polynomial. The Hilbert
polynomial of ", TT "P^i", " is ", TT "z |--> binomial(z + i, i).",
SeeAlso => {"degreesRing", "reduceHilbert", "poincare", "poincareN",
"hilbertSeries", "hilbertFunction"}
}
document {
Key => (hilbertPolynomial,Ring),
Headline => "compute the Hilbert polynomial of the ring",
Usage => "hilbertPolynomial R",
Inputs => {
"R" => Ring
},
Outputs => {
ProjectiveHilbertPolynomial => "unless the option Projective is false"
},
"We compute the ", TO2(hilbertPolynomial, "Hilbert polynomial"),
" of a coordinate ring of the rational quartic curve in ",
TT "P^3.",
EXAMPLE {
"R = ZZ/101[a..d];",
"S = coimage map(R, R, {a^4, a^3*b, a*b^3, b^4});",
"presentation S",
"h = hilbertPolynomial S",
"hilbertPolynomial(S, Projective=>false)"
},
"The rational quartic curve in ", TT "P^3", " is therefore 'like'
4 copies of ", TT "P^1", ", with three points missing. One can
see this by noticing that there is a deformation of the rational
quartic to the union of 4 lines, or 'sticks', which intersect in
three successive points.",
PARA{},
"These Hilbert polynomials can serve as ",
TO2 (hilbertFunction,"Hilbert functions"),
" too since the values of the Hilbert polynomial eventually are
the same as the Hilbert function. ",
EXAMPLE {
"apply(5, k-> h(k))",
"apply(5, k-> hilbertFunction(k,S))"
}
}
document {
Key => (hilbertPolynomial,Module),
Headline => "compute the Hilbert polynomial of the module",
Usage => "hilbertPolynomial M",
Inputs => {
"M" => Module
},
Outputs => {
ProjectiveHilbertPolynomial => "unless the option Projective is false"
},
"We compute the ", TO2(hilbertPolynomial, "Hilbert polynomial"),
" of a module.",
EXAMPLE {
"R = QQ[a..d];",
"M = module monomialCurveIdeal(R, {1,3,4});",
"h = hilbertPolynomial M",
"hilbertPolynomial(M, Projective=>false)"
},
PARA{},
-- "These Hilbert polynomials can serve as ",
-- TO2 (hilbertFunction,"Hilbert functions"),
-- " too since the values of the Hilbert polynomial eventually are
-- the same as the Hilbert function. ",
-- EXAMPLE {
-- "apply(5, k-> h(k))",
-- "apply(5, k-> hilbertFunction(k,M))"
-- }
}
document {
Key => (hilbertPolynomial,CoherentSheaf),
Headline => "compute the Hilbert polynomial of the coherent sheaf",
Usage => "hilbertPolynomial S",
Inputs => {
"S"
},
Outputs => {
ProjectiveHilbertPolynomial => "unless the option Projective is false"
},
"We compute the ", TO2(hilbertPolynomial, "Hilbert polynomial"),
" of a coherent sheaf.",
EXAMPLE {
"R = ZZ/101[x_0..x_2];",
"V = Proj R;",
"S = sheaf(image matrix {{x_0^3+x_1^3+x_2^3}})",
"h = hilbertPolynomial S",
"hilbertPolynomial(S, Projective=>false)"
}
}
document {
Key => (hilbertPolynomial,Ideal),
Headline => "compute the Hilbert polynomial of the quotient of
the ambient ring by the ideal",
Usage => "hilbertPolynomial I",
Inputs => {
"I" => Ideal
},
Outputs => {
ProjectiveHilbertPolynomial => "unless the option Projective is false"
},
"We compute the ", TO2(hilbertPolynomial, "Hilbert polynomial"),
" of the quotient of the ambient ring by an ideal.",
EXAMPLE {
"R = QQ[a..d];",
"I = monomialCurveIdeal(R, {1,3,4});",
"h = hilbertPolynomial I",
"hilbertPolynomial (R/I)",
"hilbertPolynomial(I, Projective=>false)"
},
PARA{},
"These Hilbert polynomials can serve as ",
TO2 (hilbertFunction,"Hilbert functions"),
" too since the values of the Hilbert polynomial eventually are
the same as the Hilbert function.",
EXAMPLE {
"apply(10, k-> h(k))",
"apply(10, k-> hilbertFunction(k,I))"
},
Caveat => {
"As is often the case, calling this function on an ideal ",
TT "I", " actually computes it for ", TT "R/I", " where ",
TT "R", " is the ring of ", TT "I", "."
}
}
document {
Key => (hilbertPolynomial,ProjectiveVariety),
Headline => "compute the Hilbert polynomial of the projective variety",
Usage => "hilbertPolynomial V",
Inputs => {
"V" => ProjectiveVariety
},
Outputs => {
ProjectiveHilbertPolynomial => "unless the option Projective is false"
},
"We compute an example of the ",
TO2(hilbertPolynomial, "Hilbert polynomial"), " of a projective
Hilbert variety. This is the same as the Hilbert polynomial of
its coordinate ring.",
EXAMPLE {
"R = QQ[a..d];",
"I = monomialCurveIdeal(R, {1,3,4});",
"V = Proj(R/I)",
"h = hilbertPolynomial V",
"hilbertPolynomial(V, Projective=>false)"
},
PARA{},
"These Hilbert polynomials can serve as ",
TO2 (hilbertFunction,"Hilbert functions"),
" too since the values of the Hilbert polynomial eventually are
the same as the Hilbert function of the sheaf of rings or of the underlying ring.",
EXAMPLE {
"apply(5, k-> h(k))",
"apply(5, k-> hilbertFunction(k,ring V))"
}
}
document {
Key => [hilbertPolynomial, Projective],
Headline => "choose how to display the Hilbert polynomial",
Usage => "hilbertPolynomial(...,Projective => b",
Inputs => {
"b" => Boolean => "either true or false"
},
TT "Projective => true", " is an option to ", TO "hilbertPolynomial",
" which specifies that the Hilbert polynomial produced should be
expressed in terms of the Hilbert polynomials of projective
spaces. This is the default.",
PARA{},
TT "Projective => false", " is an option to ", TO "hilbertPolynomial",
" which specifies that the Hilbert polynomial produced should be
expressed as a polynomial in the variable ", TT "i", ".",
PARA{},
"We compute the ", TO2(hilbertPolynomial, "Hilbert polynomial"),
" of a coordinate ring of the rational quartic curve in ",
TT "P^3.",
EXAMPLE {
"R = ZZ/101[a..d];",
"S = coimage map(R, R, {a^4, a^3*b, a*b^3, b^4});",
"hilbertPolynomial S",
"hilbertPolynomial(S, Projective=>false)"
},
"When the option Projective is false, the variable ",
TT "i", " is a local variable. The command ",
TT "use ring", "will make ", TT "i",
"into a global variable.",
SeeAlso => "ProjectiveHilbertPolynomial"
}
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