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OrderedQQn doc plus bug fixes
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ollieclarke8787 committed Dec 14, 2023
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174 changes: 171 additions & 3 deletions Valuations/Valuations.m2
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Original file line number Diff line number Diff line change
Expand Up @@ -35,6 +35,8 @@ export{"valuation",
"getMExponent",
"domain",
"codomain",
"OrderedQQn",
"OrderedQQVector",
"orderedQQn"
}

Expand Down Expand Up @@ -134,7 +136,7 @@ OrderedQQVector ? OrderedQQVector := (a, b) -> (
bScaled := d*b;

R := M.cache.Ring;
c := for i from 0 to numgens R-1 list min(a_i, b_i, 0);
c := for i from 0 to numgens R-1 list min(aScaled_i, bScaled_i, 0);

aMonomial := product for i from 0 to numgens R-1 list (R_i)^(sub(aScaled_i - c_i,ZZ));
bMonomial := product for i from 0 to numgens R-1 list (R_i)^(sub(bScaled_i - c_i,ZZ));
Expand All @@ -145,7 +147,7 @@ OrderedQQVector ? OrderedQQVector := (a, b) -> (
)

OrderedQQVector == InfiniteNumber := (a, b) -> false
InfiniteNumber == OrderedQQn := (a, b) -> false
InfiniteNumber == OrderedQQVector := (a, b) -> false

--
-- monomialToOrderedQQVector
Expand Down Expand Up @@ -468,9 +470,10 @@ doc ///
Description
Text
Many standard valuations take values in a totally ordered subgroup $\Gamma \subseteq \QQ^n$.
These standard valuations implement the \textit{OrderedQQn} object, whose order is based on the
These standard valuations implement @ofClass OrderedQQn@, whose order is based on the
monomial order of a given ring $R$.
The values in $\QQ^n$ are compared using the monomial order of $R$.
By default, our valuations use the min convention, that is $v(a + b) \ge \min(v(a), v(b))$.
Example
R = QQ[x,y];
I = ideal(x,y);
Expand All @@ -487,10 +490,175 @@ doc ///
SeeAlso
leadTermValuation
lowestTermValuation
OrderedQQn
orderedQQn

///

doc ///
Key
OrderedQQn
Headline
The class of all ordered modules $\QQ^n$
Description
Text
For an introduction see @TO "Ordered modules"@. Every element of
an ordered $\QQ^n$ module is @ofClass OrderedQQVector@. A new
ordered $\QQ^n$ module is created with the function @TO "orderedQQn"@.
SeeAlso
"Ordered modules"
orderedQQn
OrderedQQVector

///


doc ///
Key
orderedQQn
(orderedQQn, PolynomialRing)
(orderedQQn, ZZ, List)
Headline
Construct an ordered module $\QQ^n$
Usage
M = orderedQQn R
M = orderedQQn(n, monomialOrders)
Inputs
R:PolynomialRing
polynomial ring for the construction
n:ZZ
rank of the module
monomialOrder:List
monomial order for comparison
Outputs
M:OrderedQQn
Description
Text
For an overview see @TO "Ordered modules"@.
Let $R$ be @ofClass PolynomialRing@ with $n$ variables $x_1 \dots x_n$.
Then the corresponding ordered $\QQ^n$ module has the following
ordering. Suppose that $v, w \in QQ^n$.
Let $d \in \ZZ$ be a positive integer and $c \in \ZZ^n_{\ge 0}$
be a vector such that $dv + c$ and $dw + c$ have non-negative
entries. Then we say $v < w$ if and only if $x^{dv + c} > x^{dw + c}$
in $R$. Note that this property does not depend on the choice of $d$
$c$ so we obtain a well-defined order on $\QQ^n$.

Example
R = QQ[x_1 .. x_3, MonomialOrder => Lex]
M = orderedQQn R
v = 1/2 * M_0 - 1/3 * M_1
w = 1/2 * M_0 + 1/4 * M_2
v < w

Text
Instead of supplying @ofClass PolynomialRing@, we may supply directly
give the rank $n$ of the module along with a monomial order.
The constructor creates the ring $R$ with $n$ variables and the
given monomial order and uses this for the comparison operations.

Example
N = orderedQQn(3, {Lex})
R = N.cache.Ring
N' = orderedQQn R
N == N'

Text
In the above example, $N$ and $N'$ are the same module
because they are built from the same ring. See @TO (symbol ==, OrderedQQn, OrderedQQn)@.

SeeAlso
"Ordered modules"
orderedQQn
///


doc ///
Key
"OrderedQQVector == InfiniteNumber"
"InfiniteNumber == OrderedQQVector"
"(symbol ==, OrderedQQVector, InfiniteNumber)"
"(symbol ==, InfiniteNumber, OrderedQQVector)"
Headline
Ordered $\QQ^n$ vectors that are infinte
Description
Text
The image of $0$ under a valuation is $\infty$ or $-\infty$,
depending on the choice of min or max convention. So it may be
necessary to test whether an element of an ordered module $\QQ^n$
is equal to the valuation of $0$.
Example
M = orderedQQn(3, {Lex})
M_0 < infinity
M_0 == infinity
SeeAlso
"Ordered modules"
OrderedQQn
orderedQQn
///

doc ///
Key
(symbol ==, OrderedQQn, OrderedQQn)
Headline
Equality of ordered modules $\QQ^n$
Description
Text
Two ordered $\QQ^n$ modules are considered equal if they are
built from the same ring. Note that isomorphic rings with the
same term order may not be equal.
Example
M1 = orderedQQn(3, {Lex})
R = M1.cache.Ring
M2 = orderedQQn R
M1 == M2
S = QQ[x_1 .. x_3, MonomialOrder => {Lex}]
M3 = orderedQQn S
M1 == M3
SeeAlso
"Ordered modules"
OrderedQQn
orderedQQn
///


doc ///
Key
"OrderedQQVector ? OrderedQQVector"
Headline
Comparison of vectors of an ordered module $\QQ^n$
Description
Text
See @TO "Ordered modules"@.
-- TODO add more explanation ...
SeeAlso
"Ordered modules"
OrderedQQn
orderedQQn
///


doc ///
Key
OrderedQQVector
Headline
The class of all vectors of an ordered module $\QQ^n$
Description
Text
For an introduction see @TO "Ordered modules"@. Every ordered $\QQ^n$
vector belongs to @ofClass OrderedQQn@. The ordered $\QQ^n$ vectors
are most easily accessed though the original module.
Example
M = orderedQQn(3, {Lex})
M_0 + 2 * M_1 + 3 * M_2
SeeAlso
"Ordered modules"
orderedQQn
///




--------------------------------------------------------------------------------
------------------------------------ Tests -------------------------------------
--------------------------------------------------------------------------------
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