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--All rights reserved.
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)abbrev package IDECOMP IdealDecompositionPackage
++ Author: P. Gianni
++ Date Created: summer 1986
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: PolynomialIdeals
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package provides functions for the primary decomposition of
++ polynomial ideals over the rational numbers. The ideals are members
++ of the \spadtype{PolynomialIdeals} domain, and the polynomial generators are
++ required to be from the \spadtype{DistributedMultivariatePolynomial} domain.
IdealDecompositionPackage(vl,nv) : C == T -- take away nv, now doesn't
-- compile if it isn't there
where
vl : List Symbol
nv : NonNegativeInteger
Z ==> Integer -- substitute with PFE cat
Q ==> Fraction Z
F ==> Fraction P
P ==> Polynomial Z
UP ==> SparseUnivariatePolynomial P
Expon ==> DirectProduct(nv,NNI)
OV ==> OrderedVariableList(vl)
SE ==> Symbol
SUP ==> SparseUnivariatePolynomial(DPoly)
DPoly1 ==> DistributedMultivariatePolynomial(vl,Q)
DPoly ==> DistributedMultivariatePolynomial(vl,F)
NNI ==> NonNegativeInteger
Ideal == PolynomialIdeals(Q,Expon,OV,DPoly1)
FIdeal == PolynomialIdeals(F,Expon,OV,DPoly)
Fun0 == Union("zeroPrimDecomp","zeroRadComp")
GenPos == Record(changeval:List Z,genideal:FIdeal)
C == with
zeroDimPrime? : Ideal -> Boolean
++ zeroDimPrime?(I) tests if the ideal I is a 0-dimensional prime.
zeroDimPrimary? : Ideal -> Boolean
++ zeroDimPrimary?(I) tests if the ideal I is 0-dimensional primary.
prime? : Ideal -> Boolean
++ prime?(I) tests if the ideal I is prime.
radical : Ideal -> Ideal
++ radical(I) returns the radical of the ideal I.
primaryDecomp : Ideal -> List(Ideal)
++ primaryDecomp(I) returns a list of primary ideals such that their
++ intersection is the ideal I.
contract : (Ideal,List OV ) -> Ideal
++ contract(I,lvar) contracts the ideal I to the polynomial ring
++ \spad{F[lvar]}.
T == add
import MPolyCatRationalFunctionFactorizer(Expon,OV,Z,DPoly)
import GroebnerPackage(F,Expon,OV,DPoly)
import GroebnerPackage(Q,Expon,OV,DPoly1)
---- Local Functions -----
genPosLastVar : (FIdeal,List OV) -> GenPos
zeroPrimDecomp : (FIdeal,List OV) -> List(FIdeal)
zeroRadComp : (FIdeal,List OV) -> FIdeal
zerodimcase : (FIdeal,List OV) -> Boolean
is0dimprimary : (FIdeal,List OV) -> Boolean
backGenPos : (FIdeal,List Z,List OV) -> FIdeal
reduceDim : (Fun0,FIdeal,List OV) -> List FIdeal
findvar : (FIdeal,List OV) -> OV
testPower : (SUP,OV,FIdeal) -> Boolean
goodPower : (DPoly,FIdeal) -> Record(spol:DPoly,id:FIdeal)
pushdown : (DPoly,OV) -> DPoly
pushdterm : (DPoly,OV,Z) -> DPoly
pushup : (DPoly,OV) -> DPoly
pushuterm : (DPoly,SE,OV) -> DPoly
pushucoef : (UP,OV) -> DPoly
trueden : (P,SE) -> P
rearrange : (List OV) -> List OV
deleteunit : List FIdeal -> List FIdeal
ismonic : (DPoly,OV) -> Boolean
MPCFQF ==> MPolyCatFunctions2(OV,Expon,Expon,Q,F,DPoly1,DPoly)
MPCFFQ ==> MPolyCatFunctions2(OV,Expon,Expon,F,Q,DPoly,DPoly1)
convertQF(a:Q) : F == ((numer a):: F)/((denom a)::F)
convertFQ(a:F) : Q == (ground numer a)/(ground denom a)
internalForm(I:Ideal) : FIdeal ==
Id:=generators I
nId:=[map(convertQF,poly)$MPCFQF for poly in Id]
groebner? I => groebnerIdeal nId
ideal nId
externalForm(I:FIdeal) : Ideal ==
Id:=generators I
nId:=[map(convertFQ,poly)$MPCFFQ for poly in Id]
groebner? I => groebnerIdeal nId
ideal nId
lvint:=[variable(xx)::OV for xx in vl]
nvint1:=(#lvint-1)::NNI
deleteunit(lI: List FIdeal) : List FIdeal ==
[I for I in lI | not element?(1$DPoly,I)]
rearrange(vlist:List OV) :List OV ==
vlist=[] => vlist
sort(#1>#2,setDifference(lvint,setDifference(lvint,vlist)))
---- radical of a 0-dimensional ideal ----
zeroRadComp(I:FIdeal,truelist:List OV) : FIdeal ==
truelist=[] => I
Id:=generators I
x:OV:=truelist.last
#Id=1 =>
f:=Id.first
g:= (f exquo (gcd (f,differentiate(f,x))))::DPoly
groebnerIdeal([g])
y:=truelist.first
px:DPoly:=x::DPoly
py:DPoly:=y::DPoly
f:=Id.last
g:= (f exquo (gcd (f,differentiate(f,x))))::DPoly
Id:=groebner(cons(g,remove(f,Id)))
lf:=Id.first
pv:DPoly:=0
pw:DPoly:=0
while not one? degree(lf,y) repeat
val:=random()$Z rem 23
pv:=px+val*py
pw:=px-val*py
Id:=groebner([(univariate(h,x)).pv for h in Id])
lf:=Id.first
ris:= generators(zeroRadComp(groebnerIdeal(Id.rest),truelist.rest))
ris:=cons(lf,ris)
if not zero? pv then
ris:=[(univariate(h,x)).pw for h in ris]
groebnerIdeal(groebner ris)
---- find the power that stabilizes (I:s) ----
goodPower(s:DPoly,I:FIdeal) : Record(spol:DPoly,id:FIdeal) ==
f:DPoly:=s
I:=groebner I
J:=generators(JJ:= (saturate(I,s)))
while not in?(ideal([f*g for g in J]),I) repeat f:=s*f
[f,JJ]
---- is the ideal zerodimensional? ----
---- the "true variables" are in truelist ----
zerodimcase(J:FIdeal,truelist:List OV) : Boolean ==
element?(1,J) => true
truelist=[] => true
n:=#truelist
Jd:=groebner generators J
for x in truelist while Jd~=[] repeat
f := Jd.first
Jd:=Jd.rest
if ((y:=mainVariable f) case "failed") or (y::OV ~=x )
or not (ismonic (f,x)) then return false
while Jd~=[] and (mainVariable Jd.first)::OV=x repeat Jd:=Jd.rest
if Jd=[] and position(x,truelist)<n then return false
true
---- choose the variable for the reduction step ----
--- J groebnerner in gen pos ---
findvar(J:FIdeal,truelist:List OV) : OV ==
lmonicvar:List OV :=[]
for f in generators J repeat
t:=f - reductum f
vt:List OV :=variables t
if #vt=1 then lmonicvar:=setUnion(vt,lmonicvar)
badvar:=setDifference(truelist,lmonicvar)
badvar.first
---- function for the "reduction step ----
reduceDim(flag:Fun0,J:FIdeal,truelist:List OV) : List(FIdeal) ==
element?(1,J) => [J]
zerodimcase(J,truelist) =>
(flag case "zeroPrimDecomp") => zeroPrimDecomp(J,truelist)
(flag case "zeroRadComp") => [zeroRadComp(J,truelist)]
x:OV:=findvar(J,truelist)
Jnew:=[pushdown(f,x) for f in generators J]
Jc: List FIdeal :=[]
Jc:=reduceDim(flag,groebnerIdeal Jnew,remove(x,truelist))
res1:=[ideal([pushup(f,x) for f in generators idp]) for idp in Jc]
s:=pushup((_*/[leadingCoefficient f for f in Jnew])::DPoly,x)
degree(s,x)=0 => res1
res1:=[saturate(II,s) for II in res1]
good:=goodPower(s,J)
sideal := groebnerIdeal(groebner(cons(good.spol,generators J)))
in?(good.id, sideal) => res1
sresult:=reduceDim(flag,sideal,truelist)
for JJ in sresult repeat
if not(in?(good.id,JJ)) then res1:=cons(JJ,res1)
res1
---- Primary Decomposition for 0-dimensional ideals ----
zeroPrimDecomp(I:FIdeal,truelist:List OV): List(FIdeal) ==
truelist=[] => list I
newJ:=genPosLastVar(I,truelist);lval:=newJ.changeval;
J:=groebner newJ.genideal
x:=truelist.last
Jd:=generators J
g:=Jd.last
lfact:= factors factor(g)
ris:List FIdeal:=[]
for ef in lfact repeat
g:DPoly:=(ef.factor)**(ef.exponent::NNI)
J1:= groebnerIdeal(groebner cons(g,Jd))
if not (is0dimprimary (J1,truelist)) then
return zeroPrimDecomp(I,truelist)
ris:=cons(groebner backGenPos(J1,lval,truelist),ris)
ris
---- radical of an Ideal ----
radical(I:Ideal) : Ideal ==
J:=groebner(internalForm I)
truelist:=rearrange("setUnion"/[variables f for f in generators J])
truelist=[] => externalForm J
externalForm("intersect"/reduceDim("zeroRadComp",J,truelist))
-- the following functions are used to "push" x in the coefficient ring -
---- push x in the coefficient domain for a polynomial ----
pushdown(g:DPoly,x:OV) : DPoly ==
rf:DPoly:=0$DPoly
i:=position(x,lvint)
while not zero? g repeat
g1:=reductum g
rf:=rf+pushdterm(g-g1,x,i)
g := g1
rf
---- push x in the coefficient domain for a term ----
pushdterm(t:DPoly,x:OV,i:Z):DPoly ==
n:=degree(t,x)
xp:=convert(x)@SE
cf:=monomial(1,xp,n)$P :: F
newt := t exquo monomial(1,x,n)$DPoly
cf * newt::DPoly
---- push back the variable ----
pushup(f:DPoly,x:OV) :DPoly ==
h:=1$P
rf:DPoly:=0$DPoly
g := f
xp := convert(x)@SE
while not zero? g repeat
h:=lcm(trueden(denom leadingCoefficient g,xp),h)
g:=reductum g
f:=(h::F)*f
while not zero? f repeat
g:=reductum f
rf:=rf+pushuterm(f-g,xp,x)
f:=g
rf
trueden(c:P,x:SE) : P ==
degree(c,x) = 0 => 1
c
---- push x back from the coefficient domain for a term ----
pushuterm(t:DPoly,xp:SE,x:OV):DPoly ==
pushucoef((univariate(numer leadingCoefficient t,xp)$P), x)*
monomial(inv((denom leadingCoefficient t)::F),degree t)$DPoly
pushucoef(c:UP,x:OV):DPoly ==
c = 0 => 0
monomial((leadingCoefficient c)::F::DPoly,x,degree c) +
pushucoef(reductum c,x)
-- is the 0-dimensional ideal I primary ? --
---- internal function ----
is0dimprimary(J:FIdeal,truelist:List OV) : Boolean ==
element?(1,J) => true
Jd:=generators(groebner J)
not one?(#(factors factor Jd.last)) => return false
i:=subtractIfCan(#truelist,1)
(i case "failed") => return true
JR:=(reverse Jd);JM:=groebnerIdeal([JR.first]);JP:List(DPoly):=[]
for f in JR.rest repeat
if not ismonic(f,truelist.i) then
if not inRadical?(f,JM) then return false
JP:=cons(f,JP)
else
x:=truelist.i
i:=(i-1)::NNI
if not testPower(univariate(f,x),x,JM) then return false
JM :=groebnerIdeal(append(cons(f,JP),generators JM))
true
---- Functions for the General Position step ----
---- put the ideal in general position ----
genPosLastVar(J:FIdeal,truelist:List OV):GenPos ==
x := last truelist ;lv1:List OV :=remove(x,truelist)
ranvals:List(Z):=[(random()$Z rem 23) for vv in lv1]
val:=+/[rv*(vv::DPoly) for vv in lv1 for rv in ranvals]
val:=val+(x::DPoly)
[ranvals,groebnerIdeal(groebner([(univariate(p,x)).val
for p in generators J]))]$GenPos
---- convert back the ideal ----
backGenPos(I:FIdeal,lval:List Z,truelist:List OV) : FIdeal ==
lval=[] => I
x := last truelist ;lv1:List OV:=remove(x,truelist)
val:=-(+/[rv*(vv::DPoly) for vv in lv1 for rv in lval])
val:=val+(x::DPoly)
groebnerIdeal
(groebner([(univariate(p,x)).val for p in generators I ]))
ismonic(f:DPoly,x:OV) : Boolean == ground? leadingCoefficient(univariate(f,x))
---- test if f is power of a linear mod (rad J) ----
---- f is monic ----
testPower(uf:SUP,x:OV,J:FIdeal) : Boolean ==
df:=degree(uf)
trailp:DPoly := inv(df:Z ::F) *coefficient(uf,(df-1)::NNI)
linp:SUP:=(monomial(1$DPoly,1$NNI)$SUP +
monomial(trailp,0$NNI)$SUP)**df
g:DPoly:=multivariate(uf-linp,x)
inRadical?(g,J)
---- Exported Functions ----
-- is the 0-dimensional ideal I prime ? --
zeroDimPrime?(I:Ideal) : Boolean ==
J:=groebner((genPosLastVar(internalForm I,lvint)).genideal)
element?(1,J) => true
n:NNI:=#vl;i:NNI:=1
Jd:=generators J
#Jd~=n => false
for f in Jd repeat
if not ismonic(f,lvint.i) then return false
if i<n and not one?(degree univariate(f,lvint.i)) then return false
i:=i+1
g:=Jd.n
#(lfact:=factors(factor g)) >1 => false
lfact.1.exponent =1
-- is the 0-dimensional ideal I primary ? --
zeroDimPrimary?(J:Ideal):Boolean ==
is0dimprimary(internalForm J,lvint)
---- Primary Decomposition of I -----
primaryDecomp(I:Ideal) : List(Ideal) ==
J:=groebner(internalForm I)
truelist:=rearrange("setUnion"/[variables f for f in generators J])
truelist=[] => [externalForm J]
[externalForm II for II in reduceDim("zeroPrimDecomp",J,truelist)]
---- contract I to the ring with lvar variables ----
contract(I:Ideal,lvar: List OV) : Ideal ==
Id:= generators(groebner I)
empty?(Id) => I
fullVars:= "setUnion"/[variables g for g in Id]
fullVars = lvar => I
n:= # lvar
#fullVars < n => error "wrong vars"
n=0 => I
newVars:= append([vv for vv in fullVars
| not member?(vv,lvar)]$List(OV),lvar)
subsVars := [monomial(1,vv,1)$DPoly1 for vv in newVars]
lJ:= [eval(g,fullVars,subsVars) for g in Id]
J := groebner(lJ)
J=[1] => groebnerIdeal J
J=[0] => groebnerIdeal empty()
J:=[f for f in J| member?(mainVariable(f)::OV,newVars)]
fullPol :=[monomial(1,vv,1)$DPoly1 for vv in fullVars]
groebnerIdeal([eval(gg,newVars,fullPol) for gg in J])
|