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++ Author: P. Gianni
++ Date Created: summer 1986
++ Date Last Updated: September 1996
++ Description:
++ This domain represents polynomial ideals with coefficients in any
++ field and supports the basic ideal operations, including intersection
++ sum and quotient.
++ An ideal is represented by a list of polynomials (the generators of
++ the ideal) and a boolean that is true if the generators are a Groebner
++ basis.
++ The algorithms used are based on Groebner basis computations. The
++ ordering is determined by the datatype of the input polynomials.
++ Users may use refinements of total degree orderings.
PolynomialIdeals(F,Expon,VarSet,DPoly) : SIG == CODE where
F : Field
Expon : OrderedAbelianMonoidSup
VarSet : OrderedSet
DPoly : PolynomialCategory(F,Expon,VarSet)
SUP ==> SparseUnivariatePolynomial(DPoly)
NNI ==> NonNegativeInteger
Z ==> Integer
P ==> Polynomial F
MF ==> Matrix(F)
ST ==> SuchThat(List P, List Equation P)
GenMPos ==> Record(mval:MF,invmval:MF,genIdeal:Ideal)
Ideal ==> %
SIG ==> SetCategory with
"*" : (Ideal,Ideal) -> Ideal
++ I*J computes the product of the ideal I and J.
"**" : (Ideal,NNI) -> Ideal
++ I**n computes the nth power of the ideal I.
"+" : (Ideal,Ideal) -> Ideal
++ I+J computes the ideal generated by the union of I and J.
one? : Ideal -> Boolean
++ one?(I) tests whether the ideal I is the unit ideal,
++ contains 1.
zero? : Ideal -> Boolean
++ zero?(I) tests whether the ideal I is the zero ideal
element? : (DPoly,Ideal) -> Boolean
++ element?(f,I) tests whether the polynomial f belongs to
++ the ideal I.
in? : (Ideal,Ideal) -> Boolean
++ in?(I,J) tests if the ideal I is contained in the ideal J.
inRadical? : (DPoly,Ideal) -> Boolean
++ inRadical?(f,I) tests if some power of the polynomial f
++ belongs to the ideal I.
zeroDim? : (Ideal,List VarSet) -> Boolean
++ zeroDim?(I,lvar) tests if the ideal I is zero dimensional,
++ all its associated primes are maximal,
++ in the ring \spad{F[lvar]}
zeroDim? : Ideal -> Boolean
++ zeroDim?(I) tests if the ideal I is zero dimensional,
++ all its associated primes are maximal,
++ in the ring \spad{F[lvar]},
++ where lvar are the variables appearing in I
intersect : (Ideal,Ideal) -> Ideal
++ intersect(I,J) computes the intersection of the ideals I and J.
intersect : List(Ideal) -> Ideal
++ intersect(LI) computes the intersection of the list of ideals LI.
quotient : (Ideal,Ideal) -> Ideal
++ quotient(I,J) computes the quotient of the ideals I and J,
++ \spad{(I:J)}.
quotient : (Ideal,DPoly) -> Ideal
++ quotient(I,f) computes the quotient of the ideal I by the principal
++ ideal generated by the polynomial f, \spad{(I:(f))}.
groebner : Ideal -> Ideal
++ groebner(I) returns a set of generators of I that are a
++ Groebner basis for I.
generalPosition : (Ideal,List VarSet) -> GenMPos
++ generalPosition(I,listvar) perform a random linear
++ transformation on the variables in listvar and returns
++ the transformed ideal along with the change of basis matrix.
backOldPos : GenMPos -> Ideal
++ backOldPos(genPos) takes the result
++ produced by generalPosition from PolynomialIdeals
++ and performs the inverse transformation, returning the original ideal
++ \spad{backOldPos(generalPosition(I,listvar))} = I.
dimension : (Ideal,List VarSet) -> Z
++ dimension(I,lvar) gives the dimension of the ideal I,
++ in the ring \spad{F[lvar]}
dimension : Ideal -> Z
++ dimension(I) gives the dimension of the ideal I.
++ in the ring \spad{F[lvar]}, where lvar are the variables
++ appearing in I
leadingIdeal : Ideal -> Ideal
++ leadingIdeal(I) is the ideal generated by the
++ leading terms of the elements of the ideal I.
ideal : List DPoly -> Ideal
++ ideal(polyList) constructs the ideal generated by the list
++ of polynomials polyList.
groebnerIdeal : List DPoly -> Ideal
++ groebnerIdeal(polyList) constructs the ideal generated by the list
++ of polynomials polyList which are assumed to be a Groebner
++ basis.
++ Note: this operation avoids a Groebner basis computation.
groebner? : Ideal -> Boolean
++ groebner?(I) tests if the generators of the ideal I are a
++ Groebner basis.
generators : Ideal -> List DPoly
++ generators(I) returns a list of generators for the ideal I.
coerce : List DPoly -> Ideal
++ coerce(polyList) converts the list of polynomials polyList
++ to an ideal.
saturate : (Ideal,DPoly) -> Ideal
++ saturate(I,f) is the saturation of the ideal I
++ with respect to the multiplicative
++ set generated by the polynomial f.
saturate : (Ideal,DPoly,List VarSet) -> Ideal
++ saturate(I,f,lvar) is the saturation with respect to the prime
++ principal ideal which is generated by f in the polynomial ring
++ \spad{F[lvar]}.
if VarSet has ConvertibleTo Symbol then
relationsIdeal : List DPoly -> ST
++ relationsIdeal(polyList) returns the ideal of relations among the
++ polynomials in polyList.
CODE ==> add
--- Representation ---
Rep := Record(idl:List DPoly,isGr:Boolean)
---- Local Functions ----
contractGrob : newIdeal -> Ideal
npoly : DPoly -> newPoly
oldpoly : newPoly -> Union(DPoly,"failed")
leadterm : (DPoly,VarSet) -> DPoly
choosel : (DPoly,DPoly) -> DPoly
isMonic? : (DPoly,VarSet) -> Boolean
randomat : List Z -> Record(mM:MF,imM:MF)
monomDim : (Ideal,List VarSet) -> NNI
variables : Ideal -> List VarSet
subset : List VarSet -> List List VarSet
makeleast : (List VarSet,List VarSet) -> List VarSet
newExpon: OrderedAbelianMonoidSup
newExpon:= Product(NNI,Expon)
newPoly := PolynomialRing(F,newExpon)
import GaloisGroupFactorizer(SparseUnivariatePolynomial Z)
import GroebnerPackage(F,Expon,VarSet,DPoly)
import GroebnerPackage(F,newExpon,VarSet,newPoly)
newIdeal ==> List(newPoly)
npoly(f:DPoly) : newPoly ==
f=0$DPoly => 0$newPoly
monomial(leadingCoefficient f,makeprod(0,degree f))$newPoly +
npoly(reductum f)
oldpoly(q:newPoly) : Union(DPoly,"failed") ==
q=0$newPoly => 0$DPoly
dq:newExpon:=degree q
n:NNI:=selectfirst (dq)
n^=0 => "failed"
((g:=oldpoly reductum q) case "failed") => "failed"
monomial(leadingCoefficient q,selectsecond dq)$DPoly + (g::DPoly)
leadterm(f:DPoly,lvar:List VarSet) : DPoly ==
empty?(lf:=variables f) or lf=lvar => f
leadterm(leadingCoefficient univariate(f,lf.first),lvar)
choosel(f:DPoly,g:DPoly) : DPoly ==
g=0 => f
(f1:=f exquo g) case "failed" => f
choosel(f1::DPoly,g)
contractGrob(I1:newIdeal) : Ideal ==
J1:List(newPoly):=groebner(I1)
while (oldpoly J1.first) case "failed" repeat J1:=J1.rest
[[(oldpoly f)::DPoly for f in J1],true]
makeleast(fullVars: List VarSet,leastVars:List VarSet) : List VarSet ==
n:= # leastVars
#fullVars < n => error "wrong vars"
n=0 => fullVars
append([vv for vv in fullVars| ^member?(vv,leastVars)],leastVars)
isMonic?(f:DPoly,x:VarSet) : Boolean ==
ground? leadingCoefficient univariate(f,x)
subset(lv : List VarSet) : List List VarSet ==
#lv =1 => [lv,empty()]
v:=lv.1
ll:=subset(rest lv)
l1:=[concat(v,set) for set in ll]
concat(l1,ll)
monomDim(listm:Ideal,lv:List VarSet) : NNI ==
monvar: List List VarSet := []
for f in generators listm repeat
mvset := variables f
#mvset > 1 => monvar:=concat(mvset,monvar)
lv:=delete(lv,position(mvset.1,lv))
empty? lv => 0
lsubset : List List VarSet := sort((a,b)+->#a > #b ,subset(lv))
for subs in lsubset repeat
ldif:List VarSet:= lv
for mvset in monvar while ldif ^=[] repeat
ldif:=setDifference(mvset,subs)
if ^(empty? ldif) then return #subs
0
-- Exported Functions ----
---- is I = J ? ----
(I:Ideal = J:Ideal) == in?(I,J) and in?(J,I)
---- check if f is in I ----
element?(f:DPoly,I:Ideal) : Boolean ==
Id:=(groebner I).idl
empty? Id => f = 0
normalForm(f,Id) = 0
---- check if I is contained in J ----
in?(I:Ideal,J:Ideal):Boolean ==
J:= groebner J
empty?(I.idl) => true
"and"/[element?(f,J) for f in I.idl ]
---- groebner base for an Ideal ----
groebner(I:Ideal) : Ideal ==
I.isGr =>
"or"/[^zero? f for f in I.idl] => I
[empty(),true]
[groebner I.idl ,true]
---- Intersection of two ideals ----
intersect(I:Ideal,J:Ideal) : Ideal ==
empty?(Id:=I.idl) => I
empty?(Jd:=J.idl) => J
tp:newPoly := monomial(1,makeprod(1,0$Expon))$newPoly
tp1:newPoly:= tp-1
contractGrob(concat([tp*npoly f for f in Id],
[tp1*npoly f for f in Jd]))
---- intersection for a list of ideals ----
intersect(lid:List(Ideal)) : Ideal == "intersect"/[l for l in lid]
---- quotient by an element ----
quotient(I:Ideal,f:DPoly) : Ideal ==
--[[(g exquo f)::DPoly for g in (intersect(I,[f]::%)).idl ],true]
import GroebnerInternalPackage(F,Expon,VarSet,DPoly)
[minGbasis [(g exquo f)::DPoly
for g in (intersect(I,[f]::%)).idl ],true]
---- quotient of two ideals ----
quotient(I:Ideal,J:Ideal) : Ideal ==
Jdl := J.idl
empty?(Jdl) => ideal [1]
[("intersect"/[quotient(I,f) for f in Jdl ]).idl ,true]
---- sum of two ideals ----
(I:Ideal + J:Ideal) : Ideal == [groebner(concat(I.idl ,J.idl )),true]
---- product of two ideals ----
(I:Ideal * J:Ideal):Ideal ==
[groebner([:[f*g for f in I.idl ] for g in J.idl ]),true]
---- power of an ideal ----
(I:Ideal ** n:NNI) : Ideal ==
n=0 => [[1$DPoly],true]
(I * (I**(n-1):NNI))
---- saturation with respect to the multiplicative set f**n ----
saturate(I:Ideal,f:DPoly) : Ideal ==
f=0 => error "f is zero"
tp:newPoly := (monomial(1,makeprod(1,0$Expon))$newPoly * npoly f)-1
contractGrob(concat(tp,[npoly g for g in I.idl ]))
---- saturation with respect to a prime principal ideal in lvar ---
saturate(I:Ideal,f:DPoly,lvar:List(VarSet)) : Ideal ==
Id := I.idl
fullVars := "setUnion"/[variables g for g in Id]
newVars:=makeleast(fullVars,lvar)
subVars := [monomial(1,vv,1) for vv in newVars]
J:List DPoly:=groebner([eval(g,fullVars,subVars) for g in Id])
ltJ:=[leadterm(g,lvar) for g in J]
s:DPoly:=_*/[choosel(ltg,f) for ltg in ltJ]
fullPol:=[monomial(1,vv,1) for vv in fullVars]
[[eval(g,newVars,fullPol) for g in (saturate(J::%,s)).idl],true]
---- is the ideal zero dimensional? ----
---- in the ring F[lvar]? ----
zeroDim?(I:Ideal,lvar:List VarSet) : Boolean ==
J:=(groebner I).idl
empty? J => false
J = [1] => false
n:NNI := # lvar
#J < n => false
for f in J while ^empty?(lvar) repeat
x:=(mainVariable f)::VarSet
if isMonic?(f,x) then lvar:=delete(lvar,position(x,lvar))
empty?(lvar)
---- is the ideal zero dimensional? ----
zeroDim?(I:Ideal):Boolean ==
zeroDim?(I,"setUnion"/[variables g for g in I.idl])
---- test if f is in the radical of I ----
inRadical?(f:DPoly,I:Ideal) : Boolean ==
f=0$DPoly => true
tp:newPoly :=(monomial(1,makeprod(1,0$Expon))$newPoly * npoly f)-1
Id:=I.idl
normalForm(1$newPoly,groebner concat(tp,[npoly g for g in Id])) = 0
---- dimension of an ideal ----
---- in the ring F[lvar] ----
dimension(I:Ideal,lvar:List VarSet) : Z ==
I:=groebner I
empty?(I.idl) => # lvar
element?(1,I) => -1
truelist:="setUnion"/[variables f for f in I.idl]
"or"/[^member?(vv,lvar) for vv in truelist] => error "wrong variables"
truelist:=setDifference(lvar,setDifference(lvar,truelist))
ed:Z:=#lvar - #truelist
leadid:=leadingIdeal(I)
n1:Z:=monomDim(leadid,truelist)::Z
ed+n1
dimension(I:Ideal) : Z ==
dimension(I,"setUnion"/[variables g for g in I.idl])
-- leading term ideal --
leadingIdeal(I : Ideal) : Ideal ==
Idl:= (groebner I).idl
[[(f-reductum f) for f in Idl],true]
---- ideal of relations among the fi ----
if VarSet has ConvertibleTo Symbol then
monompol(df:List NNI,lcf:F,lv:List VarSet) : P ==
g:P:=lcf::P
for dd in df for v in lv repeat
g:= monomial(g,convert v,dd)
g
relationsIdeal(listf : List DPoly): ST ==
empty? listf => [empty(),empty()]$ST
nf:=#listf
lvint := "setUnion"/[variables g for g in listf]
vl: List Symbol := [convert vv for vv in lvint]
nvar:List Symbol:=[new() for i in 1..nf]
VarSet1:=OrderedVariableList(concat(vl,nvar))
lv1:=[variable(vv)$VarSet1::VarSet1 for vv in nvar]
DirP:=DirectProduct(nf,NNI)
nExponent:=Product(Expon,DirP)
nPoly := PolynomialRing(F,nExponent)
gp:=GroebnerPackage(F,nExponent,VarSet1,nPoly)
lf:List nPoly :=[]
lp:List P:=[]
for f in listf for i in 1.. repeat
vec2:Vector(NNI):=new(nf,0$NNI)
vec2.i:=1
g:nPoly:=0$nPoly
pol:=0$P
while f^=0 repeat
df:=degree(f-reductum f,lvint)
lcf:=leadingCoefficient f
pol:=pol+monompol(df,lcf,lvint)
g:=g+monomial(lcf,makeprod(degree f,0))$nPoly
f:=reductum f
lp:=concat(pol,lp)
lf:=concat(monomial(1,makeprod(0,directProduct vec2))-g,lf)
npol:List P :=[v::P for v in nvar]
leq : List Equation P :=
[p = pol for p in npol for pol in reverse lp ]
lf:=(groebner lf)$gp
while lf^=[] repeat
q:=lf.first
dq:nExponent:=degree q
n:=selectfirst (dq)
if n=0 then leave "done"
lf:=lf.rest
solsn:List P:=[]
for q in lf repeat
g:Polynomial F :=0
while q^=0 repeat
dq:=degree q
lcq:=leadingCoefficient q
q:=reductum q
vdq:=(selectsecond dq):Vector NNI
g:=g+ lcq*
_*/[p**vdq.j for p in npol for j in 1..]
solsn:=concat(g,solsn)
[solsn,leq]$ST
coerce(Id:List DPoly) : Ideal == [Id,false]
coerce(I:Ideal) : OutputForm ==
Idl := I.idl
empty? Idl => [0$DPoly] :: OutputForm
Idl :: OutputForm
ideal(Id:List DPoly) :Ideal == [[f for f in Id|f^=0],false]
groebnerIdeal(Id:List DPoly) : Ideal == [Id,true]
generators(I:Ideal) : List DPoly == I.idl
groebner?(I:Ideal) : Boolean == I.isGr
one?(I:Ideal) : Boolean == element?(1, I)
zero?(I:Ideal) : Boolean == empty? (groebner I).idl
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