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-- -*- coding: utf-8 -*-
--- status: TODO
--- author(s):
--- notes:
undocumented {
[kernel, DegreeLimit],
[kernel, Strategy],
}
doc ///
Node
Key
kernel
Headline
kernel of a map
SeeAlso
source
///
document {
Key => (kernel,RingMap),
Headline => "kernel of a ringmap",
Usage => "kernel f",
Inputs => {
"f" => {TT "R", " --> ", TT "S"},
SubringLimit => ZZ => "stop the computation after this many elements of the kernel have been found"
},
Outputs => {
Ideal => {"an ideal of ", TT "R"}
},
EXAMPLE lines ///
R = QQ[a..d];
S = QQ[s,t];
F = map(S,R,{s^3, s^2*t, s*t^2, t^3})
ker F
G = map(S,R,{s^5, s^3*t^2-t, s*t-s, t^5})
ker(G, SubringLimit=>1)
///,
"In the case when everything is homogeneous, Hilbert functions are
used to speed up the computations.",
Caveat => {"It should be possible to interrupt the computation and restart it, but this has
not yet been implemented."},
SeeAlso => {"substitution and maps between rings", "elimination of variables", monomialCurveIdeal},
Subnodes => { TO [kernel, SubringLimit] },
}
document {
Key => {(kernel,Matrix),
(kernel, RingElement)},
Headline => "kernel of a matrix",
Usage => "kernel f",
Inputs => {
"f" => {"a map of modules ", TT "M --> N"}
},
Outputs => {
Module => {"the kernel of f, a submodule of M"}
},
"If f is ", ofClass RingElement, ", then it will be interpreted as a one by one matrix.",
PARA{},
"The kernel is the submodule of M of all elements mapping to zero under ", TT "f", ".
Over polynomial rings, this is computed using a Gröbner basis computation.",
EXAMPLE lines ///
R = ZZ/32003[vars(0..10)]
M = genericSkewMatrix(R,a,5)
ker M
///,
SeeAlso => {syz,
-- Mike wanted this: "kernel, cokernel and image of a map of modules",
genericSkewMatrix}
}
document {
Key => [kernel, SubringLimit],
Headline => "stop after finding enough elements of a subring",
TT "SubringLimit => n", " -- an option for ", TO "kernel", " which
causes the computation of the kernel of a ring map to stop after ", TT "n", "
elements have been discovered."
}
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