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Add missing content from unused "m2/overview.m2" to docs
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@d-torrance
d-torrance committed May 26, 2024
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141 changes: 118 additions & 23 deletions M2/Macaulay2/packages/Macaulay2Doc/monomorderings.m2
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Original file line number Diff line number Diff line change
Expand Up @@ -4,13 +4,18 @@

document {
Key => "polynomial rings",
"Create a polynomial ring using the usual mathematical notation.",
EXAMPLE {
"R = QQ[x,y,z];",
"R",
},
"A polynomial ring can be created with the usual mathematical notation.",
EXAMPLE "ZZ[x,y,z]",
"If you try to construct this ring again, you will get a different
answer. We use the strict comparison operator ", TO "===", " to
demonstrate this.",
EXAMPLE "ZZ[x,y,z]===ZZ[x,y,z]",
"Thus it is a good idea to assign a new ring to a variable for
future reference.",
EXAMPLE "R = QQ[a,b,c,d,e,f]",
"Notice that after assignment to a global variable, Macaulay2
knows the ring's name, and this name is used when printing the ring.",
EXAMPLE "R",
"The original description of the ring can be recovered
with ", TO "describe", ".",
EXAMPLE "describe R",
Expand All @@ -24,6 +29,9 @@ document {
"Obtain the variables (generators) of the ring by subscripting the name of
the ring. As always in Macaulay2, indexing starts at 0.",
EXAMPLE "R_0^10+R_1^3+R_2",
"It is also possible to obtain the variables in a ring from strings
containing their names.",
EXAMPLE ///"a"_R^10+"b"_R^3+"c"_R///,
"The number of variables is provided by ", TO "numgens", ".",
EXAMPLE {
"numgens R",
Expand All @@ -43,6 +51,20 @@ document {
"The coefficient ring can be recovered with ", TO "coefficientRing", ".",
EXAMPLE "coefficientRing R",

"An element of the coefficient ring can be promoted to the polynomial ring.",
EXAMPLE "promote(11/2,R)",
"Conversely, an element of the polynomial ring that is known to be a scalar
can be lifted back to the coefficient ring.",
EXAMPLE {
"sc = (a-2)^2-a^2+4*a",
"lift(sc,QQ)",
},
"In programs, the function ", TO "liftable", " can be used to see whether
this is possible.",
EXAMPLE {
"liftable(sc,QQ)",
"liftable(c^3,QQ)",
},
"A random homogeneous element can be obtained with ", TO "random", ".",
EXAMPLE "random(2,R)",

Expand All @@ -51,12 +73,15 @@ document {
EXAMPLE "basis(2,R)",

"We may construct polynomial rings over polynomial rings.",
EXAMPLE "ZZ[a,b,c][d,e,f];",
EXAMPLE "R = ZZ[a,b,c][d,e,f];",
"When displaying an element of an iterated polynomial ring,
parentheses are used to organize the coefficients recursively, which
may themselves be polynomials.",
EXAMPLE "(a+d+1)^2",
"Internally, the polynomials in such towers are expressed in terms of a flattened monoid
containing all the variables, obtainable with the key ", TO "FlatMonoid", ".",

EXAMPLE "R.FlatMonoid",
"Variable names may be words.",
EXAMPLE {
"QQ[rho,sigma,tau];",
Expand Down Expand Up @@ -91,18 +116,23 @@ document {
"protect xx; protect yy; protect zz;",
"ZZ[ee_[xx],ee_[yy],ee_[zz]]",
},
"Polynomial rings over polynomial rings work:",
EXAMPLE lines ///
R = QQ[a,b][x]
(a+b+x)^3
///,
PARA {
"Internally, the polynomials in such towers are expressed in terms of a flattened monoid
containing all the variables, obtainable with the key ", TO "FlatMonoid", "."
},
EXAMPLE lines ///
R.FlatMonoid
///,
"A basis of the subspace of ring elements of a given degree can be obtained
in matrix form with ", TO "basis", ".",
EXAMPLE {
"R = QQ[a,b,c,d,e,f];",
"basis(2,R)"
},
"The Hilbert series of a polynomial ring can be obtained. Its power
series expansion is the generating function for the dimensions of the
degree ", TT "n", " parts.",
EXAMPLE "hilbertSeries R",
"We may use the option ", TO "Degrees", " to produce rings where the
generators have degrees other than 1.",
EXAMPLE {
"S = ZZ/101[a,b,c,d,Degrees=>{1,2,3,4}]",
"random(5,S)",
"hilbertSeries S"
},
"Some things to watch out for when using polynomial rings:",
UL {
LI ("Defining a ring twice gives different rings, as far as Macaulay2 is concerned:
Expand Down Expand Up @@ -133,16 +163,81 @@ document {
}

document {
Key => "monomial orderings",
Key => "monomial orderings",
"Every polynomial ring in Macaulay2 comes equipped with an ordering on
the monomials. See below for the definitions of all implemented
orderings. The default ordering is GRevLex, the graded reverse
lexicographic order.",
PARA{},
the monomials.",
"Polynomials are displayed by ordering the monomials in decreasing order.
The choice of monomial order can make a difference in the
time and space required for various computations,
especially Gröbner basis computations.",
"See below for the definitions of all implemented
orderings.",
PARA{},
"The default ordering is ", TO GRevLex, ", the graded reverse
lexicographic order.",
"This means that terms of higher total degree come first;
and for two terms of the same degree, the term with the higher
power of the last variable comes last; for terms with the same
power of the last variable, the exponent on the next to last
variable is consulted, and so on.",
EXAMPLE {
"R=ZZ/101[a,b,c]",
"(a+b+c+1)^2",
},
"Explicit comparison of monomials with respect to the chosen
ordering is possible.",
EXAMPLE "b^2 > a*c",
"The comparison operator ", TO "?", " returns a symbol indicating
the result of the comparison: the convention is that the larger
monomials are printed first (leftmost).",
EXAMPLE "b^2 ? a*c",
"The monomial ordering is also used when sorting lists with ", TO "sort", ".",
EXAMPLE "sort {1_R, a, a^2, b, b^2, a*b, a^3, b^3}",
PARA{},
"The next ring uses graded lexicographic ordering. This means that
terms of higher total degree come first; for two terms of the
same degree, the term with the higher power of the first variable comes
first: for terms with the same power of the first variable the
power of the second variable is consulted, and so on.",
EXAMPLE {
"R=ZZ/101[a,b,c,MonomialOrder=>GLex];",
"(a+b+c+1)^2",
},
"(Notice how similar the result above is to the one obtained when
graded reverse lexicographic ordering is used.)",
PARA{},
"The next ring uses lexicographic ordering. This means that
terms with the highest power of the first variable come
first: for two terms with the same power of the first variable the
power of the second variable is consulted, and so on.",
EXAMPLE {
"R=ZZ/101[a,b,c,MonomialOrder=>Lex];",
"(a+b+c+1)^2",
},
"The next ring uses an elimination order suitable for eliminating
the first two variables, ", TT "a", " and ", TT "b", ". In such an
ordering we want all terms in which either of the first two
variables appears to come before all of those terms in which
the first two variables don't appear. This particular ordering
accomplishes this by consulting first the graded reverse lexicographic
ordering ignoring all variables but the first two, and in case of
a tie, consulting the graded reverse lexicographic ordering of the
entire monomials.",
EXAMPLE {
"R=ZZ/101[a,b,c,MonomialOrder=>Eliminate 2];",
"(a+b+c+1)^2",
},
"The next ring uses the product ordering that segregates the
first variable from the next two. This means that terms come
first that would come first in the graded reverse lexicographic
ordering when their parts involving the
second two variables are ignored, and in case of equality,
the graded reverse lexicographic ordering of their parts involving
just the next two variables is consulted.",
EXAMPLE {
"R=ZZ/101[a,b,c,MonomialOrder=>ProductOrder{1,2}];",
"(a+b+c+1)^2"
},
PARA{},
Subnodes => {
TO "examples of specifying alternate monomial orders",
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