Buchberger-Zacharias Theory of Multivariate Ore Extensions

0.43 MB·

by  Michela Ceria

Buchberger-Zacharias Theory of Multivariate Ore
7 Extensions
1
0
2
MichelaCeria TeoMora

n
DipartimentodiMatematica DIMA
a
J Universita`diTrento Universita`diGenova

8 [email protected] [email protected]

] January 10,2017
A
R
.
h Abstract
t
a WepresentBuchbergerTheoryandAlgorithmofGro¨bnerbasesformultivari-
m ate Oreextensions of ringspresented as modules over a principal ideal domain.
[ ThealgorithmsarebasedonMo¨llerLiftingTheorem.

1 Inher1978Bachelor’sthesis[53]ZachariasdiscussedhowtoextendBuchberger
v Theory[7,8,10]fromthecaseofpolynomialringsoverafieldtothatofpolynomials
9 overaNoetherianringwithsuitableeffectivenessconditions.
9
Inthemeantimeasimilartaskwasperformedinaseriesofpapers—Kandri-Rody–
8
Kapur[20]mergedtherewritingrulesbehindEuclideanAlgorithmwithBuchberger’s
1
0 rewriting, proposing a Buchberger Theory for polynomial rings over Euclidean do-
. mains;Pan[39]studiedBuchbergerTheoryforpolynomialringsoverdomainsintro-
1
ducingthenotionsofstrong/weakGro¨bnerbases—whichculminatedwith[34].
0
7 Such unsorpassed paper, reformulating and improving Zacharias’ intuition, gave
1 efficientsolutionstocomputebothweakandstrongGro¨bnerbasesforpolynomialrings
: overeachZachariasring,withparticolarattentiontothePIRcase.Itsmaincontribution
v
i isthereformulationofBuchbergertest/completion(“abasis F isGro¨bnerifandonly
X
if eachS-polynomialbetweentwo elementsof F, reducesto 0”)in the moreflexible
r liftingtheorem(“abasisF isGro¨bnerifandonlyifeachelementinaminimalbasisof
a
thesyzygiesamongtheleadingmonomialslifts,viaBuchbergerreduction,toasyzygy
amongtheelementsofF”). Theonlyfurthercontributiontothisultimatepaperisthe
survey [6] of Mo¨ller’s results which reformulated them in terms of Szekeres Theory
[50].
The suggestion of extendingBuchbergerTheory to non-commuativerings which
satisfy Poincare´-Birkhoff-WittTheoremwas putforwardby Bergman[5], effectively
appliedbyApel–Lassner[3,4]toLiealgebrasandfurtherextendedtosolvablepoly-
nomialrings[21,22],skewpolynomialrings[15,16,17]andtootheralgebras[1,11,
25,26]whichsatisfyPoincare´-Birkhoff-WittTheoremandthus,underthestandardin-
tepretationofBuchbergerTheoryintermsoffilration/graduations[2,29,32,49,28],
havetheclassicalpolynomialringasassociatedgradedrings.
1
In particularWeispfenning [52] adapted his results to a generalizationof the Ore
extension [38] proposed by Tamari [51] and its construction was generalized by his
studentPesch[40,41]thusintroducingthenotionofiterativeOreextensionwithcom-
mutingvariables
R:=R[Y ;α ,δ ][Y ;α ,δ ] [Y ;α ,δ ], Radomain;
1 1 1 2 2 2 n n n
···
the concepthas been called Ore algebra in [12] and is renamed here as multivariate
Oreextension(foradifferentandpromisingapproachtoOrealgebrassee[23]).
Bergman’sapproachandmostofallextensionsareformulatedforringswhichare
vector spaces over a field; in our knowledge the only instances in which the coeffi-
cientring R is presentedas a D-moduleovera domainD (or at least as a Z-module)
are Pritchard’s [42, 43] extension of Mo¨ller Lifting Theory to non-commutativefree
algebrasandReinert’s[44,45]deepstudyofBuchbergerTheoryonFunctionRings.
Following the recent survey on Buchberger-Zacharias Theory for monoid rings
R[S]overaunitaryeffectiveringRandaneffectivemonoidS[31],weproposeherea
Mo¨ller–PritchardliftingtheorempresentationofBuchberger-ZachariasTheoryandre-
latedGro¨bnerbasiscomputationalgorithmsformultivariateOreextensions.Thetwist
w.r.t. [31] is thatthereR[S]coincideswith itsassociated gradedring; hereR and its
associatedgradedring
G(R):=R[Y ;α ][Y ;α ] [Y ;α ]
1 1 2 2 n n
···
coincideassetsandasleftR-modulesbetweenthemselvesandwiththecommutative
polynomialringR[Y , ,Y ],butasringshavedifferentmultiplications.
1 n
···
WebeginbyrecallingOre’soriginaltheory[38]ofnon-commutativepolynomials
R[Y],relaxingtheoriginalassumptionthatRisafieldtothecaseinwhichRisadomain
(Section 1.1) and Pesch’s constructions of multivariate Ore extensions (Section 1.2)
and graded multivariate Ore extensions (Section 1.3), focusing on the arithmetics of
themainExample14
R:=R[Y1;α1][Y2;α2] [Yn;αn],R:=Z[x] αi(x):=cixei,ci Z 0 ,ei N 0 .
··· ∈ \{ } ∈ \{ }
NextweintroduceBuchbergerTheoryinmultivariateOreextensionsrecallingthe
notionofterm-orderings(Section2.1),definitionandmainpropertiesofleft,right,bi-
lateralandrestrictedGro¨bnerbases(Section2.2)andBuchbergerAlgorithmforcom-
putingcanonicalformsinthecaseinwhichRisafield(Section2.3).
WeadapttooursettingSzekeresTheory[50](Section3),Zachariascanonicalrep-
resentationwithrelatedalgorithm(Section4)andMo¨llerLiftingTheorem(Section5).
ThenextSectionsarethealgorithmiccoreofthepaper: wereformulateformulti-
variateOreextensionsoveraZachariasring
– Mo¨ller’s algorithm for computing the required Gebauer–Mo¨ller set (id est the
minimalbasisofthemoduleofthesyzygiesamongtheleadingmonomials)for
Buchbergertest/completionofleftweakbases(Section6.1);
– hisreformulationofitwhichrequiresonlyl.c.m. computationinRforthecase
inwhichRisaPIR(Section6.2);
2
– stillinthecaseinwhichRisaPIR,Mo¨ller’scompletionofaleftweakbasisto
aleftstrongone(Section6.3);
– Gebauer–Mo¨llercriteria[18]forproducingaGebauer–Mo¨llerset(Section6.4);
– Kandri-Rody–Weispfenningcompletion[21]ofaleftweakbasesforproducing
abilateralone(Section7.1);
– Weispfenning’s[52]restrictedcompletion(Section7.2);
– asatechnicaltoolrequiredbyWeispfenning’srestrictedcompletion,howtopro-
ducerightGebauer–Mo¨llersets(Section7.3).
Finally we reverseto a theoreticalsurveysummarizingthe structuraltheoremfor
the case in which R is a Zacharias ring over a PID (Section 8), specializing to our
setting Spear’s Theorem [48, 28] (Section 9) and extending to it Lazard’s Structural
Theorem[24](Section10).
In an appendixwe discuss, as far as it is possible, how to extend this theory and
algorithmstothecaseinwhichRisaPIR(SectionA).
1 Recalls on Ore Theory
1.1 OreExtensions
LetRbeanotnecessarilycommutativedomain;Ore[38]investigatedunderwhichcon-
ditionstheR-moduleR:=R[Y]ofalltheformalpolynomialsismadearingunderthe
assumptionthatthemultiplicationofpolynomialsshallbeassociativeandboth-sided
distributive and the limitation imposed by the postulate that the degree of a product
shallbeequaltothesumofthedegreeofthefactors.
Itis clear that, due to the distributiveproperty,itsufficesto define the productof
twomonomialsbYr aYs orevenmorespecifically,todefinetheproductY r,r R;
· · ∈
thisnecessarilyrequirestheexistenceofmapsα,δ:R Rsuchthat
→
Y r=α(r)Y+δ(r)foreachr R;
· ∈
Orecallsα(r)theconjugateandδ(r)thederivativeofr.
Undertherequiredpostulateclearlywehave
1. foreachr R,α(r)=0 = r =0,
∈ ⇒
sothatαisinjective.
Itismoreoversufficienttoconsider,foreachr,r R,therelations
′
∈
α(r+r )Y+δ(r+r )=Y (r+r )=Y r+Y r = α(r)+α(r ) Y+δ(r)+δ(r ),
′ ′ ′ ′ ′ ′
· · ·
(cid:0) (cid:1)
α(rr )Y+δ(rr )=Y (rr )=(Y r) r =α(r)α(r )Y+α(r)δ(r )+δ(r)r ,
′ ′ ′ ′ ′ ′ ′
· · ·
and,ifRisaskewfield,andr,0,
Y =(Y r) r 1 =α(r)α(r 1)Y+α(r)δ(r 1)+δ(r)r 1,
− − − −
· ·
todeducethat
3
2. αisaringendomorphism;
3. thefollowingconditionsareequivalent:
(a) foreachd R 0 existsc R 0 :Y c=dY+δ(c),α(c)=d;
∈ \{ } ∈ \{ } ·
(b) αisaringautomorphism;
4. δisanα-derivationofRidestanadditivemapsatisfying
δ(rr )=α(r)δ(r)+δ(r)r foreachr,r R;
′ ′ ′ ′
∈
5. ifRisaskewfield,theneachr R 0 satisfies
∈ \{ }
α(r 1)=(α(r)) 1, δ(r 1)= (α(r)) 1δ(r)r 1;
− − − − −
−
6. Im(α) Risasubring,whichisanisomorphismcopyofR;
⊂
7. R := r R:r=α(r) Risaring,theinvariantringofR;
1
{ ∈ }⊂
8. R := r R:δ(r)=0 Risaring,theconstantringofR;
0
{ ∈ }⊂
9. r R:Y r =rY =R R .
0 1
{ ∈ · } ∩
10. IfRisaskewfield,sucharealsoIm(α),R andR .
1 0
11. DenotingZ := z R:zr =rzforeachr R thecenterofR,wehave
{ ∈ ∈ }
r R: f r=rf foreach f R =R R Z.
0 1
{ ∈ · ∈ } ∩ ∩
Moreover,ifweconsidertwopolynomials f(Y),g(Y) R 0 ,
∈ \{ }
f =aYm+f ,g=bYn+g ,a,b R 0 ,m,n N, f ,g R,deg(f )<m,deg(g )<n,
0 0 0 0 0 0
∈ \{ } ∈ ∈
wehave
f g=aαm(b)Ym+n+h(Y),deg(h)<m+n;
·
sinceαisanendomorphism,b , 0 = α(b) , 0 = αm(b) , 0andsinceRisa
⇒ ⇒
domainitholdsαm(b),0,a = aαm(b),0 = f g,0. Asaconsequence
⇒ ⇒ ·
12. Risadomain.
Definition1. RwiththeringstructuredescribedaboveiscalledanOreextensionand
isdenotedR[Y;α,δ].
Remark2(Ore). InanOreextensionR[Y;α,δ],denoting = α,δ thefreesemigroup
S h i
overthealphabet α,δ and,foreachd Nandi N,0 i d, thesetofthe d
{ } ∈ ∈ ≤ ≤ Sd,i i
wordsin of lengthd in which occuriinstancesof αand d i instancesofδ in(cid:16)an(cid:17)
S −
arbitraryorder,wehave
d
Yd r= τ(r)Yi
·
Xi=0 τX∈Sd,i
4
foreachd N;forinstance
∈
Y3 r = α3(r)Y3+δ3(r)
·
+ α2δ(r)+αδα(r)+δα2(r) Y2
(cid:16) (cid:17)
+ αδ2(r)+δαδ(r)+δ2α(r) Y.
(cid:16) (cid:17)
Inparticular,for f(Y)= n aYn iandg(Y)= m bYm i inRwehave
i=0 i − i=0 i −
P P
n+m i i a
g(Y)f(Y)= cYn+m i withc =b αm(a )andc = b − τ(a ).
i − 0 0 0 i a b
Xi=0 Xa=0 Xb=0τ∈SXm−a,i−a−b
Remark3(Ore). Undertheassumptionthat(cf. 3.) αisanautomorphism,eachpoly-
nomial n aYi Rcanbeuniquelyrepresentedas n Yia¯ forpropervaluesa¯ R.
i=1 i ∈ i=1 i i ∈
In factwe haveax = xα 1(a) δ(α 1(a))fromwhichwe candeduceinductively
P − − P
−
properexpressions
n
axn = xnα n(a)+ ( 1)ixn iσ (a).
− − in
−
Xi=1
⊓⊔
1.2 MultivariateOreExtensions
Definition4. AniterativeOreextensionisaring(whosemultiplicationwedenote⋆)
definedas
R:=R[Y ;α ,δ ][Y ;α ,δ ] [Y ;α ,δ ]
1 1 1 2 2 2 n n n
···
where, for eachi > 1, α is an endomorphismandδ an α-derivationof the iterative
i i i
Oreextension
R :=R[Y ;α ,δ ] [Y ;α ,δ ].
i 1 1 1 1 i 1 i 1 i 1
− ··· − − −
AmultivariateOreextension(or:Orealgebra[12];or:iterativeOreextensionwith
commutingvariables[40,41])isaniterativeOreextensionwhichsatisfies
– α δ =δα ,foreachi, j,i, j,
j i i j
– αα =α α,δδ =δ δ for j>i,
i j j i i j j i
– α (Y)=Y,δ (Y)=0for j>i.
j i i j i
⊓⊔
Lemma5(Pesch). InaniterativeOreextension,foreachi< jitholds
Y ⋆Y =YY α (Y)=Y,δ (Y)=0.
j i i j j i i j i
⇐⇒
Proof. Foreachi< j,wehaveY ⋆Y =α (Y)Y +δ (Y).
j i j i j j i
⊓⊔
Lemma6(Pesch). AniterativeOreextensionisamultivariateOreextensioniffY ⋆
j
Y =YY foreachi< j.
i i j
5
Proof. Infact,usingLemma5foreachr R,wehave
∈
Y ⋆Y ⋆r = Y ⋆(α(r)Y +δ(r))
j i j i i i
= α (α(r)Y +δ(r))Y +δ (α(r)Y +δ(r))
j i i i j j i i i
= α α(r)YY +α δ(r)Y +δ (α(r)Y)+δ δ(r)
j i i j j i j j i i j i
= α α(r)YY +α δ(r)Y +δ α(r)Y +δ δ(r)
j i i j j i j j i i j i
and(bysymmetry)
YY ⋆r = Y ⋆(α (r)Y +δ(r))
i j i j j i
= αα (r)YY +δα (r)Y +αδ (r)Y +δδ (r).
i j i j i j j i j i i j
⊓⊔
ThustheR-modulestructureofamultivariateOreextensioncanbeidentifiedwith
thatofthepolynomialringR[Y ,...,Y ]overitsnaturalR-basis
1 n
T :={Y1a1···Ynan :(a1,...,an)∈Nn}, R(cid:27)R[T]=SpanR{T}.
Wecanthereforedenoteα :=α,δ :=δ foreachiand,iteratively,
Yi i Yi i
α :=α α,δ :=δ δ, foreachτ .
τYi τ i τYi τ i ∈T
RemarkthatamultivariateOreextensionisnotanalgebra;infact,ifwedefine,for
τ=Yd1 Ydn andt=Ye1 Yen suchthatτ t
1 ··· n 1 ··· n |
t e e
:= 1 n ,
τ! d1!··· dn!
wehave
t
t⋆r= δ α (r)τ, foreacht andr R.
t τ
τ! τ ∈T ∈
Xτ
τ∈Tt
|
Wecandefine,foreacht ,amap
∈T
t
θt :R→R, θt(r)= τ!δτtατ(r)τ,
Xτ
τt∈,τT,t
|
sothatt⋆r=α(r)t+θ(r)foreacht andeachr R.
t t
∈T ∈
Such mapsα andθ satisfy propertiesanalogousofthose ofOre’sconjugateand
t t
derivative:
Lemma7. Withthepresentnotation,foreacht ,wehave
∈T
1. foreachr R,α(r)=0 = r =0,
t
∈ ⇒
2. α isaringendomorphism;
t
6
3. thefollowingconditionsareequivalent:
(a) foreachd R 0 existsc R 0 :Y⋆c=dY+θ(c),α(c)=d;
t t
∈ \{ } ∈ \{ }
(b) α isaringautomorphism;
t
4. θ isanα-derivationofR;
t t
5. ifRisaskewfield,theneachr R 0 satisfies
∈ \{ }
α(r 1)=(α(r)) 1, θ(r 1)= (α(r)) 1θ(r)r 1;
t − t − t − t − t −
−
6. Im(α) Risasubring,whichisanisomorphismcopyofR.
t
⊂
Wefurtherhave
7. ifeachα isanautomorphism,alsoeachα,t ,issuch.
i t
∈T ⊓⊔
1.3 AssociatedgradedOreExtension
Definition8. AmultivariateOreextension
R[Y ;α ,δ ][Y ;α ,δ ] [Y ;α ,δ ]
1 1 1 2 2 2 n n n
···
where each δ is zero, will be called a graded Ore extension (or: Ore extension with
i
zeroderivations[40,41])andwillbedenoted
R:=R[Y ;α ][Y ;α ] [Y ;α ].
1 1 2 2 n n
···
⊓⊔
Lemma9. InamultivariategradedOreextension,
– sinceitisanOrealgebra,theαscommute,
– andt⋆r=α(r)t =:M(t⋆r) foreacht andr R.
t
∈T ∈
h i
Remark 10. Notethat, sincemultivariateOreextensionscoincide,asleftR-modules,
withtheclassicalpolynomialringsR[Y ,...,Y ]andsohavethesameR-basis,namely
1 n
,theycansharewiththepolynomialringstheirstandard -valuation[50,29,2,32]
T T
[30, 24.4,24.6].Thisjustifiesthedefinitionbelow.
§
Definition11. GivenanOreextensionR:=R[Y ;α ,δ ][Y ;α ,δ ] [Y ;α ,δ ]the
1 1 1 2 2 2 n n n
···
correspondinggradedOreextensionG(R) := R[Y ;α ][Y ;α ] [Y ;α ]iscalledits
1 1 2 2 n n
···
associatedgradedOreextension.
⊓⊔
Example12.
1. ThefirstnonobviousexampleofOreextensionwasproposedin1948byD.Ta-
mari[51]inconnectionwiththenotionof“orderofirregularity”introducedby
Orein[37];itconsistsofthegradedOreextension.
R:=R[Y;α], R=Q[x]whereα:R R: x x2.
→ 7→
7
2. Such constructionwas generalizedbyWeispfenning[52] whichintroducedthe
rings
R:=R[Y;α], R=Q[x]whereα:R R: x xe, e N
i
→ 7→ ∈
3. andextendedbyPesch[40]tohisiteratedOreextensionswithpowersubstitution
R:=R[Y ;α ][Y ;α ] [Y ;α ], R=Q[x]
1 1 2 2 n n
···
whereαi :R R: x xei, ei N.
→ 7→ ∈
4. AnOreextensionwhereαisinvertibleisdiscussedin[27]:
R:=R[S;α],R=Q[D ,D ,D ]
1 2 3
where
α:R R: f(D ,D ,D ) f(D +2D ,D , D )
1 2 3 2 1 3 1
→ 7→ −
whoseinverseis
α 1 :R R: f(D ,D ,D ) f( D ,D +2D ,D ).
− 1 2 3 3 1 3 2
→ 7→ −
⊓⊔
Notethat,whileasR-modulesRandG(R)coincidebothwiththepolynomialring
=R[Y ,...,Y ],thethreeringshave,ingeneral,differentarithmetics;wewilldenote
1 n
P
⋆themultiplicationofRand thoseofG(R).
∗
Example13. TheringofExample12,1.
R:=R[Y;α], R=Q[x]whereα:R R: x x2
→ 7→
isanOreextensionwhichisgraded.
Themap
2i 1
−
δ:Q[x] Q[x]: xi xh
→ 7→
Xh=i
isanα-derivationsince
2i+2j 1
−
δ(xi xj) = xh
·
hX=i+j
2i+2j 1 2i+j 1
− −
= xh+ xh
h=X2i+j hX=i+j
2j 1 2i 1
− −
= x2i xh+xj xh
Xh=j Xh=i
= α(xi)δ(xj)+δ(xi)xj;
thusS:=R[Y;α,δ]isanOreextensionofwhichRistheassociatedgradedOreexten-
sion.
⊓⊔
8
Example14. SinceinBuchberger-ZachariasTheory,fromanalgorithmicpointofview,
theinterestpointsareassociatedgradedringsandthustheroˆleofderivatesisirrelevant,
weillustratetheresultsfortheOreextensionswiththezero-derivatives
R:=R[Y ;α ][Y ;α ] [Y ;α ],R=Z[x],
1 1 2 2 n n
···
withαi(x):=cixei,ci Z 0 ,ei N 0 .
∈ \{ } ∈ \{ }
Ifwedenoteγthemap
γ:N N 0 N,(a,e) a−1ei = 1−ea
× \{ }→ 7→ 1 e
Xi=0 −
wherethelastequalityholdsfore,1,wehaveYia∗xb =cibγ(a,ei)xeaibYia.
Notethat
b 1 a 1 a+b 1
γ(b,e)+ebγ(a,e)= − ei+ − eb+i = − ei =γ(a+b,e). (1-a)
Xi=0 Xi=0 Xi=0
Sinceαj(αi(x)) = ciαj(xei) = cicejixeiej andαi(αj(x)) = cjαi(xej) = cjceijxeiej, then
RisagradedOreextensionifandonlyif
cicejixeiej =αj(αi(x))=αi(αj(x))=cjαi(xej)=cjceijxeiej
idest
ceji−1 =ciej−1. (1-b)
Wethushave n relationsamongthencoefficientsc. Inparticularweneedtopartition
2 i
theindicesas(cid:16) (cid:17)
1,...,n = E O S,E = i:2 e ,O= i:2∤e >1 ,S = i:e =1 .
i i i
{ } ⊔ ⊔ { | } { } { }
If I := O E = theneachsuchequationsarethetrivialequality1 = 1andthus
⊔ ∅
allc arefree.ThesituationiscompletelydifferentwhenI :=O E , ;infact,
i
⊔ ∅
– fori S necessarilyc = 1;
i
∈ ±
– ifaprime pdividesatleastac , j I,thenitdivideseachc,i I.
j i
∈ ∈
Asregardsthesignofc wecansaythat
i
– ifE , then
∅
– c ispositiveforeachi S O,
i
∈ ∪
– the sign of c,i E, is undeterminedbut all the c,i E, have the same
i i
∈ ∈
sign.
– ifE = thenthesignofc,i S Oisundetermined.
i
∅ ∈ ∪
Forinstance
9
– fore =e =1,e =5,e =3wehaveS = 1,4 ,O= 2,3 ,E = and
1 4 2 3
{ } { } ∅
c4 =c0,c2 =c0,c0 =c0,c2 =c4,c0 =c4,c0 =c2;
1 2 1 3 1 4 2 3 2 4 3 4
whencec = 1,c = 1,c = c2;
1 ± 4 ± 2 ± 3
– fore =e =1,e =2,e =3wehaveS = 1,4 ,O= 3 ,E = 2 ,and
1 4 2 3
{ } { } { }
c =c0,c2 =c0,c0 =c0,c2 =c ,c0 =c ,c0 =c2;
1 2 1 3 1 4 2 3 2 4 3 4
whencec =c =1,c =c2 >0;
1 4 3 2
– fore = 1, e = 2,e = 3,S = 1 E = 2 , O = 3 . Supposec = 6,soboth
1 2 3 2
{ } { } { }
theprimes2and3dividec . Fromc = c0,c2 = c0,c2 = c wegetc = 1and
2 1 2 1 3 2 3 1
c =36. Wenoticethat2 c and3 c ,butneither2nor3dividec ;
3 3 3 1
| |
– fore =e =1,e =4,e =8wehaveS = 1,4 ,E = 2,3 ,O= and
1 4 2 3
{ } { } ∅
c3 =c0,c7 =c0,c0 =c0,c7 =c3,c0 =c3,c0 =c7.
1 2 1 3 1 4 2 3 2 4 3 4
whencec =c =1,c =χ3,c =χ7,c c >0.
1 4 2 3 2 3
Asregardsthevalues c ,1 i n,setting
i
| | ≤ ≤
n n
ρ:= (e 1)= (e 1),χ:= ρ c ,
j− j− vt | j|
Xj=1 Xj I Yj=1
∈
wehave
cj =χej−1foreach j 1,....,n . (1-c)
| | ∈{ }
Infact,sinceifaprimepdividesatleastac , j I,thenitdivideseachc,i I,we
j i
canexpresseach c ,i I,as c = pai,1 pai,h whe∈re p , ,p aretheprime∈factors
| i| ∈ | i| 1 ··· h 1 ··· h
ofthesquarefreeassociate √χ= p1 phofχ.
···
Wehave
ciej−1 = cjei−1 = pai(ej−1) = paj(ei−1) = ai(ej 1)=aj(ei 1)
| | | | ⇒ ⇒ − −
whencea =a e =e anda >a e <e.
i j i j i j j i
⇐⇒ ⇐⇒
Thusthecswithminimale minimalizealsoalla ,1 l h.
i i i,l
≤ ≤
WTheemreoforereovernj=h1a|vcej|a=j,l =j∈aIi(,ie|(cie−jj−1|)1=),1≤hl=1lp≤lajh,l.= hl=1plai,lPnej=i−11(ej−1) = hl=1pleaii−,lρ1 whence
Q Q Q Q Q
n h ai,l h aj,l
χ:= vtρ |cj|= plei−1 = plej−1
Yj=1 Yl=1 Yl=1
and(1-c).
Theformula(1-c)allowstoreformulate(1-b)as
cjei−1 = ciej−1 =χ(ei−1)(ej−1). (1-d)
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