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--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev domain INDE IndexedExponents
++ Author: James Davenport
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ IndexedExponents of an ordered set of variables gives a representation
++ for the degree of polynomials in commuting variables. It gives an ordered
++ pairing of non negative integer exponents with variables
IndexedExponents(Varset:OrderedSet): C == T where
C == Join(OrderedAbelianMonoidSup,
IndexedDirectProductCategory(NonNegativeInteger,Varset))
T == IndexedDirectProductOrderedAbelianMonoidSup(NonNegativeInteger,Varset) add
Term:= Record(k:Varset,c:NonNegativeInteger)
Rep:= List Term
coerceOF(t: Term):OutputForm == -- converts term to OutputForm
t.c = 1 => (t.k)::OutputForm
(t.k)::OutputForm ** (t.c)::OutputForm
coerce(x: %):OutputForm == -- converts entire exponents to OutputForm
null x => 1::Integer::OutputForm
null rest x => coerceOF(first x)
reduce("*",[coerceOF t for t in x])
)abbrev domain SMP SparseMultivariatePolynomial
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated: 30 November 1994
++ Fix History:
++ 30 Nov 94: added gcdPolynomial for float-type coefficients
++ Basic Functions: Ring, degree, eval, coefficient, monomial, differentiate,
++ resultant, gcd
++ Related Constructors: Polynomial, MultivariatePolynomial
++ Also See:
++ AMS Classifications:
++ Keywords: polynomial, multivariate
++ References:
++ Description:
++ This type is the basic representation of sparse recursive multivariate
++ polynomials. It is parameterized by the coefficient ring and the
++ variable set which may be infinite. The variable ordering is determined
++ by the variable set parameter. The coefficient ring may be non-commutative,
++ but the variables are assumed to commute.
SparseMultivariatePolynomial(R: Ring,VarSet: OrderedSet): C == T where
pgcd ==> PolynomialGcdPackage(IndexedExponents VarSet,VarSet,R,%)
C == PolynomialCategory(R,IndexedExponents(VarSet),VarSet)
SUP ==> SparseUnivariatePolynomial
T == add
--constants
--D := F(%) replaced by next line until compiler support completed
--representations
D := SparseUnivariatePolynomial(%)
VPoly:= Record(v:VarSet,ts:D)
Rep:= Union(R,VPoly)
--local function
--declarations
fn: R -> R
n: Integer
k: NonNegativeInteger
kp:PositiveInteger
k1:NonNegativeInteger
c: R
mvar: VarSet
val : R
var:VarSet
up: D
p,p1,p2,pval: %
Lval : List(R)
Lpval : List(%)
Lvar : List(VarSet)
--define
0 == 0$R::%
1 == 1$R::%
zero? p == p case R and zero?(p)$R
one? p == p case R and one?(p)$R
-- a local function
red(p:%):% ==
p case R => 0
if ground?(reductum p.ts) then leadingCoefficient(reductum p.ts) else [p.v,reductum p.ts]$VPoly
numberOfMonomials(p): NonNegativeInteger ==
p case R =>
zero?(p)$R => 0
1
+/[numberOfMonomials q for q in coefficients(p.ts)]
coerce(mvar):% == [mvar,monomial(1,1)$D]$VPoly
monomial? p ==
p case R => true
sup : D := p.ts
1 ~= numberOfMonomials(sup) => false
monomial? leadingCoefficient(sup)$D
-- local
moreThanOneVariable?: % -> Boolean
moreThanOneVariable? p ==
p case R => false
q:=p.ts
any?(not ground? #1 ,coefficients q) => true
false
-- if we already know we use this (slighlty) faster function
univariateKnown: % -> SparseUnivariatePolynomial R
univariateKnown p ==
p case R => (leadingCoefficient p) :: SparseUnivariatePolynomial(R)
monomial( leadingCoefficient p,degree p.ts)+ univariateKnown(red p)
univariate p ==
p case R =>(leadingCoefficient p) :: SparseUnivariatePolynomial(R)
moreThanOneVariable? p => error "not univariate"
monomial( leadingCoefficient p,degree p.ts)+ univariate(red p)
multivariate (u:SparseUnivariatePolynomial(R),var:VarSet) ==
ground? u => (leadingCoefficient u) ::%
[var,monomial(leadingCoefficient u,degree u)$D]$VPoly +
multivariate(reductum u,var)
univariate(p:%,mvar:VarSet):SparseUnivariatePolynomial(%) ==
p case R or mvar>p.v => monomial(p,0)$D
pt:=p.ts
mvar=p.v => pt
monomial(1,p.v,degree pt)*univariate(leadingCoefficient pt,mvar)+
univariate(red p,mvar)
-- a local functions, used in next definition
unlikeUnivReconstruct(u:SparseUnivariatePolynomial(%),mvar:VarSet):% ==
zero? (d:=degree u) => coefficient(u,0)
monomial(leadingCoefficient u,mvar,d)+
unlikeUnivReconstruct(reductum u,mvar)
multivariate(u:SparseUnivariatePolynomial(%),mvar:VarSet):% ==
ground? u => coefficient(u,0)
uu:=u
while not zero? uu repeat
cc:=leadingCoefficient uu
cc case R or mvar > cc.v => uu:=reductum uu
return unlikeUnivReconstruct(u,mvar)
[mvar,u]$VPoly
ground?(p:%):Boolean ==
p case R => true
false
-- const p ==
-- p case R => p
-- error "the polynomial is not a constant"
monomial(p,mvar,k1) ==
zero? k1 or zero? p => p
p case R or mvar>p.v => [mvar,monomial(p,k1)$D]$VPoly
p*[mvar,monomial(1,k1)$D]$VPoly
monomial(c:R,e:IndexedExponents(VarSet)):% ==
zero? e => (c::%)
monomial(1,leadingSupport e, leadingCoefficient e) *
monomial(c,reductum e)
coefficient(p:%, e:IndexedExponents(VarSet)) : R ==
zero? e =>
p case R => p::R
coefficient(coefficient(p.ts,0),e)
p case R => 0
ve := leadingSupport e
vp := p.v
ve < vp =>
coefficient(coefficient(p.ts,0),e)
ve > vp => 0
coefficient(coefficient(p.ts,leadingCoefficient e),reductum e)
-- coerce(e:IndexedExponents(VarSet)) : % ==
-- e = 0 => 1
-- monomial(1,leadingSupport e, leadingCoefficient e) *
-- (reductum e)::%
-- retract(p:%):IndexedExponents(VarSet) ==
-- q:Union(IndexedExponents(VarSet),"failed"):=retractIfCan p
-- q :: IndexedExponents(VarSet)
-- retractIfCan(p:%):Union(IndexedExponents(VarSet),"failed") ==
-- p = 0 => degree p
-- reductum(p)=0 and leadingCoefficient(p)=1 => degree p
-- "failed"
coerce(n) == n::R::%
coerce(c) == c::%
characteristic == characteristic$R
recip(p) ==
p case R => (uu:=recip(p::R);uu case "failed" => "failed"; uu::%)
"failed"
- p ==
p case R => -$R p
[p.v, - p.ts]$VPoly
n * p ==
p case R => n * p::R
mvar:=p.v
up:=n*p.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
c * p ==
c = 1 => p
p case R => c * p::R
mvar:=p.v
up:=c*p.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1 + p2 ==
p1 case R and p2 case R => p1 +$R p2
p1 case R => [p2.v, p1::D + p2.ts]$VPoly
p2 case R => [p1.v, p1.ts + p2::D]$VPoly
p1.v = p2.v =>
mvar:=p1.v
up:=p1.ts+p2.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1.v < p2.v =>
[p2.v, p1::D + p2.ts]$VPoly
[p1.v, p1.ts + p2::D]$VPoly
p1 - p2 ==
p1 case R and p2 case R => p1 -$R p2
p1 case R => [p2.v, p1::D - p2.ts]$VPoly
p2 case R => [p1.v, p1.ts - p2::D]$VPoly
p1.v = p2.v =>
mvar:=p1.v
up:=p1.ts-p2.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1.v < p2.v =>
[p2.v, p1::D - p2.ts]$VPoly
[p1.v, p1.ts - p2::D]$VPoly
p1 = p2 ==
p1 case R =>
p2 case R => p1 =$R p2
false
p2 case R => false
p1.v = p2.v => p1.ts = p2.ts
false
p1 * p2 ==
p1 case R => p1::R * p2
p2 case R =>
mvar:=p1.v
up:=p1.ts*p2
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1.v = p2.v =>
mvar:=p1.v
up:=p1.ts*p2.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1.v > p2.v =>
mvar:=p1.v
up:=p1.ts*p2
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
--- p1.v < p2.v
mvar:=p2.v
up:=p1*p2.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p ** kp == p ** (kp pretend NonNegativeInteger )
p ** k ==
p case R => p::R ** k
-- univariate special case
not moreThanOneVariable? p =>
multivariate( (univariateKnown p) ** k , p.v)
mvar:=p.v
up:=p.ts ** k
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
if R has IntegralDomain then
UnitCorrAssoc ==> Record(unit:%,canonical:%,associate:%)
unitNormal(p) ==
u,c,a:R
p case R =>
(u,c,a):= unitNormal(p::R)$R
[u::%,c::%,a::%]$UnitCorrAssoc
(u,c,a):= unitNormal(leadingCoefficient(p))$R
[u::%,(a*p)::%,a::%]$UnitCorrAssoc
unitCanonical(p) ==
p case R => unitCanonical(p::R)$R
(u,c,a):= unitNormal(leadingCoefficient(p))$R
a*p
unit? p ==
p case R => unit?(p::R)$R
false
associates?(p1,p2) ==
p1 case R => p2 case R and associates?(p1,p2)$R
p2 case VPoly and p1.v = p2.v and associates?(p1.ts,p2.ts)
if R has approximate then
p1 exquo p2 ==
p1 case R and p2 case R =>
a:= (p1::R exquo p2::R)
if a case "failed" then "failed" else a::%
zero? p1 => p1
one? p2 => p1
p1 case R or p2 case VPoly and p1.v < p2.v => "failed"
p2 case R or p1.v > p2.v =>
a:= (p1.ts exquo p2::D)
a case "failed" => "failed"
[p1.v,a]$VPoly::%
-- The next test is useful in the case that R has inexact
-- arithmetic (in particular when it is Interval(...)).
-- In the case where the test succeeds, empirical evidence
-- suggests that it can speed up the computation several times,
-- but in other cases where there are a lot of variables
-- and p1 and p2 differ only in the low order terms (e.g. p1=p2+1)
-- it slows exquo down by about 15-20%.
p1 = p2 => 1
a:= p1.ts exquo p2.ts
a case "failed" => "failed"
mvar:=p1.v
up:SUP %:=a
if ground? (up) then leadingCoefficient(up) else [mvar,up]$VPoly::%
else
p1 exquo p2 ==
p1 case R and p2 case R =>
a:= (p1::R exquo p2::R)
if a case "failed" then "failed" else a::%
zero? p1 => p1
one? p2 => p1
p1 case R or p2 case VPoly and p1.v < p2.v => "failed"
p2 case R or p1.v > p2.v =>
a:= (p1.ts exquo p2::D)
a case "failed" => "failed"
[p1.v,a]$VPoly::%
a:= p1.ts exquo p2.ts
a case "failed" => "failed"
mvar:=p1.v
up:SUP %:=a
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly::%
map(fn,p) ==
p case R => fn(p)
mvar:=p.v
up:=map(map(fn,#1),p.ts)
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
if R has Field then
(p : %) / (r : R) == inv(r) * p
if R has GcdDomain then
content(p) ==
p case R => p
c :R :=0
up:=p.ts
while not(zero? up) and not(one? c) repeat
c:=gcd(c,content leadingCoefficient(up))
up := reductum up
c
if R has EuclideanDomain and R has CharacteristicZero and not(R has FloatingPointSystem) then
content(p,mvar) ==
p case R => p
gcd(coefficients univariate(p,mvar))$pgcd
gcd(p1,p2) == gcd(p1,p2)$pgcd
gcd(lp:List %) == gcd(lp)$pgcd
gcdPolynomial(a:SUP $,b:SUP $):SUP $ == gcd(a,b)$pgcd
else if R has GcdDomain then
content(p,mvar) ==
p case R => p
content univariate(p,mvar)
gcd(p1,p2) ==
p1 case R =>
p2 case R => gcd(p1,p2)$R::%
zero? p1 => p2
gcd(p1, content(p2.ts))
p2 case R =>
zero? p2 => p1
gcd(p2, content(p1.ts))
p1.v < p2.v => gcd(p1, content(p2.ts))
p1.v > p2.v => gcd(content(p1.ts), p2)
mvar:=p1.v
up:=gcd(p1.ts, p2.ts)
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
if R has FloatingPointSystem then
-- eventually need a better notion of gcd's over floats
-- this essentially computes the gcds of the monomial contents
gcdPolynomial(a:SUP $,b:SUP $):SUP $ ==
ground? (a) =>
zero? a => b
gcd(leadingCoefficient a, content b)::SUP $
ground?(b) =>
zero? b => b
gcd(leadingCoefficient b, content a)::SUP $
conta := content a
mona:SUP $ := monomial(conta, minimumDegree a)
if not one? mona then
a := (a exquo mona)::SUP $
contb := content b
monb:SUP $ := monomial(contb, minimumDegree b)
if not one? monb then
b := (b exquo monb)::SUP $
mong:SUP $ := monomial(gcd(conta, contb),
min(degree mona, degree monb))
degree(a) >= degree b =>
not((a exquo b) case "failed") =>
mong * b
mong
not((b exquo a) case "failed") => mong * a
mong
coerce(p):OutputForm ==
p case R => (p::R)::OutputForm
outputForm(p.ts,p.v::OutputForm)
coefficients p ==
p case R => list(p :: R)$List(R)
"append"/[coefficients(p1)$% for p1 in coefficients(p.ts)]
retract(p:%):R ==
p case R => p :: R
error "cannot retract nonconstant polynomial"
retractIfCan(p:%):Union(R, "failed") ==
p case R => p::R
"failed"
-- leadingCoefficientRecursive(p:%):% ==
-- p case R => p
-- leadingCoefficient p.ts
mymerge:(List VarSet,List VarSet) ->List VarSet
mymerge(l:List VarSet,m:List VarSet):List VarSet ==
empty? l => m
empty? m => l
first l = first m =>
empty? rest l =>
setrest!(l,rest m)
l
empty? rest m => l
setrest!(l, mymerge(rest l, rest m))
l
first l > first m =>
empty? rest l =>
setrest!(l,m)
l
setrest!(l, mymerge(rest l, m))
l
empty? rest m =>
setrest!(m,l)
m
setrest!(m,mymerge(l,rest m))
m
variables p ==
p case R => empty()
lv:List VarSet:=empty()
q := p.ts
while not zero? q repeat
lv:=mymerge(lv,variables leadingCoefficient q)
q := reductum q
cons(p.v,lv)
mainVariable p ==
p case R => "failed"
p.v
eval(p,mvar,pval) == univariate(p,mvar)(pval)
eval(p,mvar,val) == univariate(p,mvar)(val)
evalSortedVarlist(p,Lvar,Lpval):% ==
p case R => p
empty? Lvar or empty? Lpval => p
mvar := Lvar.first
mvar > p.v => evalSortedVarlist(p,Lvar.rest,Lpval.rest)
pval := Lpval.first
pts := map(evalSortedVarlist(#1,Lvar,Lpval),p.ts)
mvar=p.v =>
pval case R => pts (pval::R)
pts pval
multivariate(pts,p.v)
eval(p,Lvar,Lpval) ==
empty? rest Lvar => evalSortedVarlist(p,Lvar,Lpval)
sorted?(#1 > #2, Lvar) => evalSortedVarlist(p,Lvar,Lpval)
nlvar := sort(#1 > #2,Lvar)
nlpval :=
Lvar = nlvar => Lpval
[Lpval.position(mvar,Lvar) for mvar in nlvar]
evalSortedVarlist(p,nlvar,nlpval)
eval(p,Lvar,Lval) ==
eval(p,Lvar,[val::% for val in Lval]$(List %)) -- kill?
degree(p,mvar) ==
p case R => 0
mvar= p.v => degree p.ts
mvar > p.v => 0 -- might as well take advantage of the order
max(degree(leadingCoefficient p.ts,mvar),degree(red p,mvar))
degree(p,Lvar) == [degree(p,mvar) for mvar in Lvar]
degree p ==
p case R => 0
degree(leadingCoefficient(p.ts)) + monomial(degree(p.ts), p.v)
minimumDegree p ==
p case R => 0
md := minimumDegree p.ts
minimumDegree(coefficient(p.ts,md)) + monomial(md, p.v)
minimumDegree(p,mvar) ==
p case R => 0
mvar = p.v => minimumDegree p.ts
md:=minimumDegree(leadingCoefficient p.ts,mvar)
zero? (p1:=red p) => md
min(md,minimumDegree(p1,mvar))
minimumDegree(p,Lvar) ==
[minimumDegree(p,mvar) for mvar in Lvar]
totalDegree(p, Lvar) ==
ground? p => 0
null setIntersection(Lvar, variables p) => 0
u := univariate(p, mv := mainVariable(p)::VarSet)
weight:NonNegativeInteger := (member?(mv,Lvar) => 1; 0)
tdeg:NonNegativeInteger := 0
while u ~= 0 repeat
termdeg:NonNegativeInteger := weight*degree u +
totalDegree(leadingCoefficient u, Lvar)
tdeg := max(tdeg, termdeg)
u := reductum u
tdeg
if R has CommutativeRing then
differentiate(p,mvar) ==
p case R => 0
mvar=p.v =>
up:=differentiate p.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
up:=map(differentiate(#1,mvar),p.ts)
if ground? up then leadingCoefficient(up) else [p.v,up]$VPoly
leadingCoefficient(p) ==
p case R => p
leadingCoefficient(leadingCoefficient(p.ts))
-- trailingCoef(p) ==
-- p case R => p
-- coef(p.ts,0) case R => coef(p.ts,0)
-- trailingCoef(red p)
-- TrailingCoef(p) == trailingCoef(p)
leadingMonomial p ==
p case R => p
monomial(leadingMonomial leadingCoefficient(p.ts),
p.v, degree(p.ts))
reductum(p) ==
p case R => 0
p - leadingMonomial p
-- if R is Integer then
-- pgcd := PolynomialGcdPackage(%,VarSet)
-- gcd(p1,p2) ==
-- gcd(p1,p2)$pgcd
--
-- else if R is RationalNumber then
-- gcd(p1,p2) ==
-- mrat:= MRationalFactorize(VarSet,%)
-- gcd(p1,p2)$mrat
--
-- else gcd(p1,p2) ==
-- p1 case R =>
-- p2 case R => gcd(p1,p2)$R::%
-- p1 = 0 => p2
-- gcd(p1, content(p2.ts))
-- p2 case R =>
-- p2 = 0 => p1
-- gcd(p2, content(p1.ts))
-- p1.v < p2.v => gcd(p1, content(p2.ts))
-- p1.v > p2.v => gcd(content(p1.ts), p2)
-- PSimp(p1.v, gcd(p1.ts, p2.ts))
)abbrev domain POLY Polynomial
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions: Ring, degree, eval, coefficient, monomial, differentiate,
++ resultant, gcd
++ Related Constructors: SparseMultivariatePolynomial, MultivariatePolynomial
++ Also See:
++ AMS Classifications:
++ Keywords: polynomial, multivariate
++ References:
++ Description:
++ This type is the basic representation of sparse recursive multivariate
++ polynomials whose variables are arbitrary symbols. The ordering
++ is alphabetic determined by the Symbol type.
++ The coefficient ring may be non commutative,
++ but the variables are assumed to commute.
Polynomial(R:Ring):
PolynomialCategory(R, IndexedExponents Symbol, Symbol) with
if R has Algebra Fraction Integer then
integrate: (%, Symbol) -> %
++ integrate(p,x) computes the integral of \spad{p*dx}, i.e.
++ integrates the polynomial p with respect to the variable x.
== SparseMultivariatePolynomial(R, Symbol) add
import UserDefinedPartialOrdering(Symbol)
coerce(p:%):OutputForm ==
(r:= retractIfCan(p)@Union(R,"failed")) case R => r::R::OutputForm
a :=
userOrdered?() => largest variables p
mainVariable(p)::Symbol
outputForm(univariate(p, a), a::OutputForm)
if R has Algebra Fraction Integer then
integrate(p, x) == (integrate univariate(p, x)) (x::%)
)abbrev package POLY2 PolynomialFunctions2
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package takes a mapping between coefficient rings, and lifts
++ it to a mapping between polynomials over those rings.
PolynomialFunctions2(R:Ring, S:Ring): with
map: (R -> S, Polynomial R) -> Polynomial S
++ map(f, p) produces a new polynomial as a result of applying
++ the function f to every coefficient of the polynomial p.
== add
map(f, p) == map(#1::Polynomial(S), f(#1)::Polynomial(S),
p)$PolynomialCategoryLifting(IndexedExponents Symbol,
Symbol, R, Polynomial R, Polynomial S)
)abbrev domain MPOLY MultivariatePolynomial
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions: Ring, degree, eval, coefficient, monomial, differentiate,
++ resultant, gcd
++ Related Constructors: SparseMultivariatePolynomial, Polynomial
++ Also See:
++ AMS Classifications:
++ Keywords: polynomial, multivariate
++ References:
++ Description:
++ This type is the basic representation of sparse recursive multivariate
++ polynomials whose variables are from a user specified list of symbols.
++ The ordering is specified by the position of the variable in the list.
++ The coefficient ring may be non commutative,
++ but the variables are assumed to commute.
MultivariatePolynomial(vl:List Symbol, R:Ring)
== SparseMultivariatePolynomial(--SparseUnivariatePolynomial,
R, OrderedVariableList vl)
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