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--- status: rewritten September 2018
--- author(s): Lily Silverstein
--- notes:
-*
-- TODO
part(InfiniteNumber,InfiniteNumber,Number)
part(InfiniteNumber,ZZ,Number)
part(ZZ,InfiniteNumber,Number)
part(ZZ,Number)
part(ZZ,ZZ,Number)
*-
doc///
Key
part
(part,List,RingElement)
(part,ZZ,ZZ,VisibleList,RingElement)
(part,InfiniteNumber,InfiniteNumber,RingElement)
(part,InfiniteNumber,InfiniteNumber,VisibleList,RingElement)
(part,InfiniteNumber,ZZ,RingElement)
(part,InfiniteNumber,ZZ,VisibleList,RingElement)
(part,Nothing,Nothing,RingElement)
(part,Nothing,Nothing,VisibleList,RingElement)
(part,Nothing,ZZ,RingElement)
(part,Nothing,ZZ,VisibleList,RingElement)
(part,ZZ,InfiniteNumber,RingElement)
(part,ZZ,InfiniteNumber,VisibleList,RingElement)
(part,ZZ,Nothing,RingElement)
(part,ZZ,Nothing,VisibleList,RingElement)
(part,ZZ,RingElement)
(part,ZZ,VisibleList,RingElement)
(part,ZZ,ZZ,RingElement)
Headline
select terms of a polynomial by degree(s) or weight(s)
Usage
part(deg, f)
part(lo, hi, f)
part(deg, f)
part(lo, hi, wt, f)
part(mdeg, f)
part_deg f
Inputs
f:RingElement
a polynomial
deg:ZZ
in which degree to select terms from {\tt f}
lo:ZZ
or @TO InfiniteNumber@, the lower bound of degrees to select
hi:ZZ
or @TO InfiniteNumber@, the upper bound of degrees to select
wt:List
(optionally) a list of integers to use as weights before selecting degrees
mdeg:List
a multidegree, in the case that {\tt f} is defined in a multigraded ring
Outputs
:RingElement
the sum of all terms of {\tt f} matching the specified degree or range of degrees (after weighting, if weights given)
Description
Text
To select terms of a single degree, use {\tt part(deg, f)}. An alternate syntax
uses an underscore.
Example
R = QQ[x,y];
f = (x+y+1)^4
part(2, f)
part_2 f
Text
To select terms within a range of degrees, use {\tt part(lo, hi, f)}.
Example
part(1, 2, f)
Text
In the next example, we apply the weights {\tt \{2,3\}} before selecting terms. In other words, the term
{\tt x^ay^b} is considered to have degree {\tt 2a+3b}.
Example
part(6, {2,3}, f)
part(6, 8, {2,3}, f)
Text
If the generators of the ring were defined to have non-unit degrees,
the weights {\em override} those degrees.
Example
R = QQ[x,y, Degrees=>{2,3}];
f = (x+y+1)^4
part(2, f)
part(2, {1,1}, f)
Text
By omitting {\tt lo} or {\tt hi}, but providing a comma indicating the omission, the range of degrees will
be unbounded in the appropriate direction.
Example
S = QQ[a,b,c]
g = (a - b*c + 2)^3
part(4, , g)
part(, 3, g)
part(, 3, 1..3, g)
Text
Infinite numbers may also be given for the bounds.
Example
part(4, infinity, g)
part(-infinity, 3, g)
part(-infinity, infinity, 1..3, g)
Text
For {\bf multigraded rings}, use a list to specify a single multidegree in the first argument.
The underscore syntax works here too.
Example
R = QQ[x,y,z, Degrees => {{1,0,0},{0,1,0},{0,0,1}}];
f = (x+y+z)^3
part({2,0,1}, f)
part_{2,0,1} f
Text
A range of degrees cannot be asked for in the multigraded case.
Polynomial rings over polynomial rings are multigraded, so either use a
multidegree or specify weights to avoid errors.
Example
R = QQ[a][x];
h = (1+a+x)^3
part(2, {1,0}, h)
part(2, {0,1}, h)
part({2,1}, h)
SeeAlso
degree
monomials
parts
select
someTerms
terms
"graded and multigraded polynomial rings"
"manipulating polynomials"
///
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