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-- -*- coding: utf-8 -*-
--- status: Draft
--- author(s): Smith
--- notes:
document {
Key => {standardPairs, (standardPairs,MonomialIdeal), (standardPairs, MonomialIdeal, List)},
Headline => "find the standard pairs of a monomial ideal",
Usage => "standardPairs I",
Inputs => {
"I" => MonomialIdeal
},
Outputs => {
{"a ", TO List, " of standard pairs for ", TT "I"}
},
"The standard monomials of a monomial ideal ", TT "I", " (those monomials that are not
in ", TT "I", ") can be enumerated as follows. Given a monomial ",
TT "m", " and a subset ", TT "F", " of the variables, the pair ", TT "(m,F)", " indexes
the set of monomials of the form ", TT {"m", "m'"},
" where the monomial ", TT "m'", " is supported on ", TT "F", ". A ", TO List, " of pairs ",
TT "(m, F)", " form ", EM "standard pairs", " for the monomial ideal ", TT "I",
" if it satisfies the following three conditions:",
UL {
{"for each pair, the monomial ", TT {"m"}, " is supported on the complement of ",
TT "F", ";"},
{"all of the monomials represented by a pair are standard;"},
{"the pairs index disjoint sets of monomials."}
},
EXAMPLE {
"R = QQ[x,y,z];",
"I = monomialIdeal(x*y^3*z, x*y^2*z^2, y^3*z^2, y^2*z^3)",
"standardPairs I"
},
PARA{},
"The standard pairs are computed with Algorithm 3.2.5 in
Gröbner Deformations of Hypergeometric Differential
Equations, by Mutsumi Saito, Bernd Sturmfels and Nobuki Takayama;
Algorithms and Computation in Mathematics 6, Springer-Verlag, 2000.
Implemented by Gregory G. Smith.",
PARA{},
"For more information, see the ", EM "Monomial ideals", " chapter in Computations
in algebraic geometry with Macaulay2, edited by David Eisenbud, Daniel R. Grayson,
Michael E. Stillman, and Bernd Sturmfels, Algorithms and Computations in Mathematics 8,
Springer-Verlag, 2001.",
SeeAlso => {}
}
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