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--- status: Draft
--- author(s): MES
--- notes:
undocumented{(numgens, InexactField)}
document {
Key => numgens,
Headline => "the number of generators",
SeeAlso => {}
}
document {
Key => (numgens,Module),
Headline => "number of generators of a module",
Usage => "numgens M",
Inputs => {
"M",
},
Outputs => {
ZZ => "number of generators of M"
},
"In Macaulay2, each module comes equipped with a matrix of
generators. It is the number of columns of this matrix
which is returned. If the module is graded, this may or
may not be the number of minimal generators.",
EXAMPLE {
"R = QQ[a..d];",
"M = ker vars R",
"generators M",
"numgens M"
},
"The number of generators of a free module is its rank.",
EXAMPLE {
"numgens R^10"
},
SeeAlso => {generators, trim, prune, mingens, ker,vars}
}
document {
Key => (numgens, Monoid), -- TODO: combine with (numgens, Ring) below
Headline => "number of generators of a monoid",
Usage => "numgens M",
Inputs => {
"M"
},
Outputs => {
ZZ => "number of generators of M"
},
EXAMPLE {
"M = monoid[x_1..x_10];",
"numgens M"
},
SeeAlso => {monoid}
}
document {
Key => (numgens,Ideal),
Headline => "number of generators of an ideal",
Usage => "numgens I",
Inputs => {
"I"
},
Outputs => {
ZZ => "number of generators of I"
},
"In Macaulay2, each ideal comes equipped with a matrix of generators. It is the
number of columns of this matrix that is returned. If the ideal is homogeneous,
this may or may not be the number of minimal generators.",
EXAMPLE {
"R = QQ[a..d];",
"I = ideal(a^2-b*d, a^2-b*d, c^2, d^2);",
"numgens I"
},
PARA{},
"In order to find a more efficient set of generators, use ",
TO mingens, " or ", TO trim, ".",
EXAMPLE {
"mingens I",
"numgens trim I"
},
SeeAlso => {mingens, trim, generators}
}
document {
Key => (numgens,CoherentSheaf),
Headline => "the number of generators of the underlying module",
Usage => "numgens F",
Inputs => {
"F"
},
Outputs => {
ZZ => {"number of generators of the underlying module ", TT "M", " of ", TT "F"}
},
"In Macaulay2, each coherent sheaf comes equipped with a module over
the coordinate ring. In the homogeneous case, this is not
necessarily the number of generators of the sum of twists ",
TT "H^0(F(d))", ", summed over all d, which in fact could be infinitely
generated.",
EXAMPLE {
"R = QQ[a..d]/(a^3+b^3+c^3+d^3)",
"X = Proj R;",
"T' = cotangentSheaf X",
"numgens T'",
"module T'"
},
SeeAlso => {(module,CoherentSheaf),tangentSheaf}
}
document {
Key => {(numgens,Ring),(numgens, EngineRing),(numgens, FractionField),(numgens, MonomialIdeal),(numgens, PolynomialRing),(numgens, QuotientRing)},
Headline => "number of generators of a polynomial ring",
Usage => "numgens R",
Inputs => {
"R"
},
Outputs => {
ZZ => "number of generators of R over the coefficient ring"
},
"If the ring ", TT "R", " is a fraction ring or a (quotient of a) polynomial ring, the number
returned is the number of generators of ", TT "R", " over the coefficient ring.
In all other cases, the number of generators is zero.",
EXAMPLE {
"numgens ZZ",
"A = ZZ[a,b,c];",
"numgens A",
"KA = frac A",
"numgens KA"
},
"If the ring is polynomial ring over another polynomial ring, then only the
outermost variables are counted.",
EXAMPLE {
"B = A[x,y];",
"numgens B",
"C = KA[x,y];",
"numgens C",
},
"In this case, use the ", TO "CoefficientRing", " option to ", TO generators, " to obtain the complete set of generators.",
EXAMPLE {
"g = generators(B, CoefficientRing=>ZZ)",
"#g"
},
"Galois fields created using ", TO GF, " have zero generators, but their underlying
polynomial ring has one generators.",
EXAMPLE {
"K = GF(9,Variable=>a)",
"numgens K",
"R = ambient K",
"numgens R"
},
SeeAlso => {generators, minPres, GF, ambient}
}
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