Bar code for monomial ideals

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by  Michela Ceria

Bar code for monomial ideals
7
1 MichelaCeria
0
DepartmentofMathematics
2
UniversityofTrento
n
ViaSommarive14,38123,Trento
a
J [email protected]

7

]
O Abstract
C Aimofthispaperistocount0-dimensionalstableandstronglystableidealsin
. 2and3variables,giventheir(constant)affineHilbertpolynomial.
h
Todoso,wedefinetheBarCode,abidimensionalstructurerepresentingany
t
a finitesetoftermsMandallowingtodesumemanypropertiesofthecorresponding
m
monomial ideal I, if M is an order ideal. Then, we use it to give a connection
[ between(strongly)stablemonomialidealsandintegerpartitions,thusallowingto

countthemviaknowndeterminantalformulas.
1
v
1
1 Introduction
8
7
1 StronglystableidealsplayaspecialroleinthestudyofHilbertscheme,introducedfirst
0 byGrothendieck[22], since their escalier allowsto study the Hilbertfunctionofany
1. homogeneousideal,exploitingthetheoryofGroebnerbases,aspointedoutbyBayer
0 [5]andEisenbud[18].
7 The notion of generic initial ideal was introducedby Galligo [21] with the name
1
ofGrauertinvariant.Galligoprovedthatthegenericinitialidealofanyhomogeneous
:
v idealisclosedw.r.ttheactionoftheBorelgroupandgaveacombinatorialcharacter-
i izationofsuch ideals, providedthattheyare definedona field of characteristiczero.
X
AlsoEisenbudandPeeva[18,42],focusedonthatmonomialideals,labellingthem0-
r
a Borel-fixedideals. Later,Aramova-Herzog[2,3]renamedthemstronglystableideals.
Acombinatorialdescriptionoftheidealsclosedw.r.ttheactionoftheBorelgroup
overapolynomialringonafieldofcharacteristicp>0hasbeenprovidedbyParduein
hisThesis[41]andGalligo’sresulthasbeenextendedtothatsettingbyBayer-Stillman
[6].
ThenotionofstableidealhasbeenintroducedbyEliahou-Kervaire[19]asagen-
eralization of 0-Borel-fixed ideals. They were able to give a minimal resolution for
stableideals.
SuchminimalresolutionwasusedbyBigatti[10]andHulett[26]toextendMacaulay’s
result[37];theyprovedthatthelex-segmentidealhasmaximalBettinumbers,among
allidealssharingthesameHilbertfunction.
1
In connectionwith the studyofHilbertschemes[8, 9, 14, 33, 38, 45] ithasbeen
considered relevant to list all the stable ideals [7] and strongly stable ideals [15, 34]
withafixedHilbertpolynomial.
Aimofthispaperistocountzerodimensionalstableandstronglystableidealsin2
and3variables,giventheir(constant)affineHilbertpolynomial.
To doso, we first introducea bidimensionalstructure, calledBarCodewhichal-
lows, a priori,to representany(finite1) setofterms M and, if M is anorderideal, to
authomaticallydesume many propertiesof the correspondingmonomialideal I. For
example,aPommaretbasis[48,12]ofI canbeeasilydesumed.
TheBar Code isstrictly connectedto Felzeghy-Rath-Ronyay’sLexTrie[20, 35],
evenifourgoalandmethodsarecompletelydifferentfromtheirs.
Using the Bar Code, we providea connectionbetween stable and stronglystable
monomialidealsandintegerpartitions.
For the case of two variables, we see that there is a biunivocal correspondence
between (strongly) stable ideals with affine Hilbert polynomial p and partitions of p
withdistinctparts.
Thecase ofthreevariablesis morecomplicatedandsomemoretechnologyisre-
quired.ThankstotheBarCode,weprovideabijectionbetween(strongly)stableideals
andsomespecialplanepartitionsoftheirconstantaffineHilbertpolynomialp.
These plane partitions have been studied by Krattenthaler [31, 32], who proved
determinantalformulastofindtheirnormgeneratingfunctionsand-finally-tocount
them.
Asanexample,weconsiderthestablemonomialideal
I =(x3,x x ,x2,x2x ,x x ,x2)⊳k[x ,x ,x ],
1 1 1 2 2 1 3 2 3 3 1 2 3
whoseGroebnerescalierisN(I )= 1,x ,x2,x ,x ,x x .
1 { 1 1 2 3 1 3}
ItcanberepresentedbytheBarCodebelow
1 x1 x21 x2 x3 x1x3
x31 x1x2 x21x3
x22 x2x3
x2
3
anditcorrespondstotheplanepartition
3 1
2
ThecorrespondencecanbeseenobservingtherowsoftheBarCodeabove: sincethe
bottom row is composed by two segments, the plane partition has exactly two rows.
The numberof entriesin the i-th row of the partition, i = 1,2(i.e. 2 and 1 resp.), is
givenbythenumberofsegmentsinthemiddle-row,lyingoverthei-thsegmentofthe
bottomrow. Finally,theentriesarerepresentedbythenumberofsegmentsinthetop
row,lyingoverthesegmentsrepresentingthecorrespondingentry.
1Thereis alsothe possibility tohaveinfinite Bar Codes forinfinite sets ofterms, butitis outofthe
purposeofthispaper,sowewillonlyseeanexampleforcompleteness’sake.
2
Exploiting this bijection and the determinantalformulasby Krattenthaler, we are
finallyabletocountstableandstronglystableidealsinthreevariables.
Even if the Bar Code can easily represent finite sets of terms in any number of
variables, the generalization of our results to the case of 4 or more variables would
require the introduction of n-dimensionalpartitions, for which, in my knowledge, it
doesnotexista completestudyfromthe pointofview ofcountingthem2, so, in this
paper,wedonotextensivelydealwiththem.
2 Some algebraic notation
Throughoutthispaper,inconnectionwithmonomialideals,wemainlyfollowtheno-
tationof[39].
Wedenoteby := k[x ,...,x ]thegradedringofpolynomialsinnvariableswithco-
1 n
P
efficientsinthefieldk,assuming,onceforall,thatchar(k)=0.
Thesemigroupofterms,generatedbytheset x ,...,x is:
1 n
{ }
:= xγ := xγ1 xγn γ:=(γ ,...,γ ) Nn .
T { 1 ··· n | 1 n ∈ }
If τ = xγ1 xγn, thendeg(τ) = n γ is the degreeof τ and, for each h 1,...,n
1 ··· n i=1 i ∈ { }
degh(τ):=γhistheh-degreeofτ.P
Foreachd N, isthed-degreepartof ,i.e. := xγ deg(xγ) = d andit
d d
∈ T T T { ∈ T| }
iswellknownthat|Td| = n+dd−1 . Foreachsubset M ⊆ T weset Md = M∩Td. The
symbol (d)denotesthed(cid:16)egree(cid:17) dpartof ,namely (d)= xγ deg(xγ) d .
T ≤ T T { ∈T| ≤ }
Analogously, (d)denotesthedegree d partof andgivenanidealI of , I(d)is
P ≤ P P
itsdegree dpart,i.e. I(d)= I (d).
≤ ∩P
Wenoticethat (d)isthevectorspacegeneratedby (d)andweobservethatI(d)is
P T
avectorsubspaceof (d).
P
Asemigroupordering<on isatotalordering suchthatτ <τ ττ <ττ , τ,τ ,τ
1 2 1 2 1 2
T ⇒ ∀ ∈
. Foreach semigroupordering< on , we can representa polynomial f as a
T T ∈ P
linearcombinationoftermsarrangedw.r.t.<,withcoefficientsinthebasefieldk:
s
f = c(f,τ)τ= c(f,τ)τ : c(f,τ) k , τ , τ >...>τ ,
i i i ∗ i 1 s
∈ ∈T
Xτ Xi=1
∈T
withT(f):=τ theleadingtermof f,Lc(f):=c(f,τ )theleadingcoefficientof f and
1 1
tail(f):= f c(f,T(f))T(f)thetailof f.
−
A term orderingisa semigroupordering suchthat1 islower thaneveryvariableor,
equivalently,itisawellordering.
Unlessotherwisespecified,weconsiderthelexicographicalorderinginducedby
x <...< x ,i.e:
1 n
xγ11···xγnn <Lex xδ11···xδnn ⇔∃j|γj <δj, γi =δi, ∀i> j,
2In[1],Chapter11,theauthorobserves:
Surprisingly,thereismuchofinterestwhenthedimensionis1or2,andverylittlewhenthe
dimensionexceeds2.
3
whichisatermordering.
Sinceinallthepaperwewillconsiderthelexicographicalordering,noconfusion
mayariseandsowedropthesubscriptanddenoteitby<insteadof< .
Lex
Foreachtermτ and x τ,theonlyυ suchthatτ = x υiscalled j-thprede-
j j
∈ T | ∈ T
cessorofτ.
Given a term τ , we denote by min(τ) the smallest variable x, i 1,...,n , s.t.
i
∈ T ∈ { }
x τ.
i
|
For M ,wedenoteby M thelistobtainedbyorderingtheelementsof M increas-
⊂ T
inglyw.r.t. Lex.Forexample,ifM = x ,x2 k[x ,x ], x < x ,M = x2,x .
{ 2 1}⊂ 1 2 1 2 { 1 2}
Asubset J isasemigroupidealifτ J στ J, σ ;asubsetN
⊆ T ∈ ⇒ ∈ ∀ ∈ T ⊆ T
isanorderidealifτ N σ N στ. WehavethatN isanorderidealifand
∈ ⇒ ∈ ∀ | ⊆ T
onlyif N= J isasemigroupideal.
T \
Given a semigroup ideal J we define N(J) := J. The minimal set of
⊂ T T \
generatorsG(J)ofJ,calledthemonomialbasisofJ,satisfiestheconditionsbelow
G(J) := τ J eachpredecessorof τ N(J)
{ ∈ | ∈ }
= τ N(J) τ isanorderideal, τ<N(J) .
{ ∈T | ∪{ } }
For all subsets G , TG := T(g), g G and T(G) is the semigroup ideal of
⊂ P { } { ∈ }
leadingtermsdefinedasT(G):= τT(g), τ ,g G .
{ ∈T ∈ }
Fixed a term order <, for any ideal I ⊳ the monomialbasis of the semigroup ideal
P
T(I)=T I iscalledmonomialbasisofIanddenotedagainbyG(I),whereastheideal
{ }
In(I) := (T(I)) is called initial ideal and the order ideal N(I) := T(I) is called
T \
GroebnerescalierofI. ThebordersetofI isdefinedas:
B(I) := x τ, 1 h n, τ N(I) N(I)
h
{ ≤ ≤ ∈ }\
= T(I) ( 1 x τ, 1 h n, τ N(I) ).
h
∩ { }∪{ ≤ ≤ ∈ }
IfI⊳ isanideal,wedefineitsassociatedvarietyas
P
n
V(I)= P k , f(P)=0, f ,
{ ∈ ∀ ∈I}
wherekisthealgebraicclosureofk.
Definition1. LetI⊳ beanideal.TheaffineHilbertfunctionofI isthefunction
P
HF :N N
I
→
d dim( (d)/I(d)).
7→ P
Fordsufficientlylarge,theaffineHilbertfunctionofI canbewrittenas:
l
d
HF (d)= b ,
I i
l i!
Xi=0 −
4
wherel is the KrulldimensionofV(I), b are integerscalled Bettinumbersandb is
i 0
positive.
Definition 2. The polynomial which is equal to HF (d), for d sufficiently large, is
I
calledtheaffineHilbertpolynomialofIanddenotedH (d).
I
3 On the Integer Partitions
Inthissection,wegivesomedefinitionsandtheoremsfromthetheoryofintegerparti-
tionsthatwewilluseasatoolforourstudy,mainlyfollowing[1,31,32,49].
Letusstartgivingthedefinitionofintegerpartition.
Definition3([49]). Anintegerpartitionofp Nisak-tuple(λ ,...,λ ) Nksuchthat
1 k
k λ = pandλ ... λ . ∈ ∈
i=1 i 1 ≥ ≥ k
P
We regardtwo partitionsasidenticaliftheyonlydifferinthenumberofterminal
zeros.Forexample(3,2,1)=(3,2,1,0,0).
Thenonzerotermsarecalledpartsofλandwesaythatλhaskpartsifk= i, λ >0 .
i
|{ }|
Wewillmainlydealwiththespecialcaseλ > ... > λ > 0i.e. withintegerpartitions
1 k
of pintoknon-zerodistinctparts,denotingbyI thesetcontainingthem,i.e.
(p,k)
k
I := (λ ,...,λ ) Nk, λ >...>λ >0and λ = p .
(p,k) 1 k 1 k j
{ ∈ }
Xj=1
The number Q(p,i) of integer partitions of p into i distinct parts is well known in
literature.Forexample,wecanfindin[16]theformulasallowingtocomputeit:
i
p,i N, i,1, Q(p,i)= P p ,i , Q(p,1)=1
∀ ∈ − 2! !
whereP(n,k)denotesthenumberofintegerpartitionsofnwithlargestpartequaltok:
n,k N, P(n,k)= P(n 1,k 1)+P(n k,k),
∀ ∈ − − −
with
P(n,k)=0 for k>n
 P(n,n)=1
 P(n,0)=0
Wedefinenowthenotionofplanepartition.
Definition 4 ([31]). A plane partition π of a positive integer p N, is a partition of
∈
pinwhichthepartshavebeenarrangedina2-dimensionalarray,weaklydecreasing
acrossrowsanddowncolumns. Iftheinequalityisstrictacrossrows(resp. columns),
wesaythatthepartitionisrow-strict(respcolumn-strict).
Differentconfigurationsareregardedasdifferentplanepartitions.
Thenormofπisthesumn(π):= π ofallitsparts.
i,j i,j
P
5
We pointoutthatanintegerpartition(see Definition3)is a simpleandparticular
caseofplanepartition.
Example5. Anexampleofplanepartitionof p=6is
2 1 1
1 1
whichisdifferentfromtheplanepartition
2 1 1
1
1
♦
In sections6, 7, we will beinterestedin some particularplanepartitions, thatwe
defineinwhatfollows.
Definition6([31]). LetD denotethesetofallr-tuplesλ=(λ ,...,λ )ofintegerswith
r 1 r
λ ... λ .
1 r
≥ ≥
Forλ,µ D ,wewriteλ µifλ µ foralli= 1,2,...,r. Letc,darbitraryintegers
r i i
∈ ≥ ≥
andλ,µ D ,withλ µ. Wecallanarrayρofintegersoftheform
r
∈ ≥
ρ ρ ... ... ... ρ
1,µ1+1 1,µ1+2 1,λ1
ρ ... ... ... ... ... ρ
2,µ2+1 2,λ2
... ... ... ...
ρ ... ... ρ
r,µr+1 r,λr
a(c,d)-planepartitionofshapeλ/µif
ρ ρ +cfor1 i r, µ < j<λ,
i,j i,j+1 i i
≥ ≤ ≤
ρ ρ +dfor1 i r 1, µ < j λ .
i,j i+1,j i i+1
≥ ≤ ≤ − ≤
Inthecaseµ=0,weshortlysaythatρisofshapeλ.
Wedenoteby (c,d)thesetof(c,d)-planepartitionsofshapeλ.
λ
P
A (1,1)-plane partition containing only positive parts is a row and column-strict
plane partition; these partitions will be useful while dealing with stable ideals (see
section6).
Definition7([32]). Letc,dbearbitraryintegersandλbeapartitionwithλ r. We
r
≥
call“shifted(c,d)-planepartitionof shapeλ”anarrayπofintegersoftheform
π π ... ... ... ... ... ... π
1,1 1,2 1,λ1
π ... ... ... ... ... π
2,2 2,λ2
... ... ... ... ...
π ... ... π
r,r r,λr
6
andforwhich
π π +cfor1 i r, i j<λ,
i,j i,j+1 i
≥ ≤ ≤ ≤
π π +dfor1 i r 1, i< j λ .
i,j i+1,j i+1
≥ ≤ ≤ − ≤
Wepointoutthat,accordingtodefinition7,thereareλ i+1integersinthei-th
i
−
row.
We denote by (c,d) the set of shifted (c,d)-planepartitionsof shape λ. These
λ
S
partitionswillbeusefulinsection7,wherewewillcountstronglystableideals.
Example8. Theplanepartition
5 4 3
4 1
isa(1,1)-planepartitionwithshapeλ=(3,2)andnorm17.
Ontheotherhand,theplanepartition
5 4 3
4 1
isa shifted(1,0)-planepartitionofshapeλ = (3,3)andnorm17. Itcontainsλ = 3
1
elementsinthefirstrowandλ 1=2elementsinthesecondrow.
2
− ♦
Weintroducenowthenotionofnormgeneratingfunction,forcountingplanepar-
titions.
Definition9([31]). ThenormgeneratingfunctionforaclassC of(c,d)-planeparti-
tionsis
xn(π).
Xπ C
∈
Ifxisanindeterminate,weintroducethex-notations(see[31]):
[n]=1 xn
−
[n]!=[1][2] [n], [0]!=1
···
n [n]!
= , ifn k,0.
"k# [k]![n k]! ≥
−
Ifk=0, n =1;ifk,0andn<k,thenweset n =0.
k k
h i h i
Theorems10and12giveawaytocomputethenormgeneratingfunctionforplane
partitionsoftheformsintroducedinDefinitions6and7,undersomehypothesesonthe
sizeoftheirparts.
LetusstartwiththeplanepartitionsofDefinition6.
7
Theorem10(Krattenthaler,[31]). Letc,dbearbitraryintegers,λ,µ D andleta,b
r
∈
ber-tuplesofintegerssatisfying
a c(µ µ )+(1 d) a
i i i+1 i+1
− − − ≥
b +c(λ λ )+(1 d) b
i i i+1 i+1
− − ≥
fori=1,2,...,r 1.
Then,denotin−gN1(s,t)=bs(λs−s−µt+t)+(1−c−d) µt+2s−t − µ2t +c λs−s2−µt+t ,
thepolynomial h(cid:16) (cid:17) (cid:16) (cid:17)i (cid:16) (cid:17)
(1 c)(λ µ) d(s t)+a b +c
det1≤s,t≤r xN1(s,t)" − s−λst −s µ−t+t t− s #!,
− −
is the norm generating function for (c,d)-planepartitions of shape λ/µ in which the
firstpartinrowiisatmosta andthelastpartinrowiisatleastb.
i i
Example11. Letusconsiderthe(1,1)-planepartitionsofshapeλ = (2,1)(soµ = 0),
such thata = (4,3)and b = (1,1), i.e. row and columnstrict plane partitionsof the
form
ρ ρ
1,1 1,2
ρ2,1 0 !
with ρ 4, 1 ρ 3, ρ 1, With the notation introduced above, we have
1,1 2,1 1,2
≤ ≤ ≤ ≥
r=2.
Since
4=a c(µ µ )+(1 d) a =3
1 1 2 2
− − − ≥
2=b +c(λ λ )+(1 d) b =1,
1 1 2 2
− − ≥
wecanapplytheformulaofTheorem10,which,substitutingourdata,turnsouttobe
significantlysimplified:
(s t)+a b +1
det1≤s,t≤2 xN1(s,t)"− −λs st+−t s #!,
−
whereN1(s,t)=bs(λs−s+t)+(−1) s2−t + λs−2s+t .
Now, we have N(1,1) = (2 1 + 1)h(cid:16)+ (cid:17)2i =(cid:16) 2; N(cid:17)(1,2) = (2 1 + 2) + 3 = 5;
− 2 − 2
(cid:16) (cid:17) x3 4 x(cid:16)6(cid:17)4
N(2,1)= 0;N(2,2)= (1 2+2) = 1,sowehavetocomputedet 2 3 =
−  3h i xh3i 
det x3(1+x2)(1+x+x2) x5(1+x)(1+x2) = x10+2x9+3x8+3xh70i+3x6+h1xi5+x4
1 x(1+x+x2) !
Forexample,thereareexactly3partitionswithnorm8,namely
4 1 4 2 4 3
, ,
3 0 ! 2 0 ! 1 0 !
♦
We see now how to construct the norm generating function for the partitions of
Definition7.
8
Theorem 12 (Krattenthaler, [32]). Let c,d be arbitrary integers, λ a partition with
λ randleta,bber-tuplesofintegerssatisfying
r
≥
a c d a
i i+1
− − ≥
b +c(λ λ )+(1 d) b
i i i+1 i+1
− − ≥
fori=1,2,...,r−1. Then,denotingN1 = ri=1(bi(λi−i)+ai+c λi2−i ),thepolynomial
P (cid:16) (cid:17)
(λ s)(1 c)+(1 c d)(s t)+a b
xN1det1≤s,t≤r " s− − λ−s −s − t− s#!,
−
isthenormgeneratingfunctionforshifted(c,d)-planepartitionsofshapeλinwhich
thefirstpartinrowiisequaltoa andthelastpartinrowiisatleastb.
i i
Example13. Letusconsidertheshifted(1,0)-planepartitionsofshapeλ = (3,3,3),
suchthata=(6,3,1)andb=(1,1,1).Bydefinition,theyarematrices
π π π
1,1 1,2 1,3
 0 π2,2 π2,3 
 0 0 π3,3 
withπ =6,π =3,π =1. Moreover,π ,π 1.
1,1 2,2 3,3 1,3 2,3
≥
Wecomputethenormgeneratingfunctionforthesepartitions,viaTheorem12.
FirstofallN1 = ri=1(bi(λi−i)+ai+c λi2−i )=14.
tThheednewteermhainvaenttooPcfotmhepmutaetreiaxchMm=s,t(m=h((cid:16))λs−s(cid:17))(1−.c)+(1λ−s−c−sd)(s−t)+at−bsi,1≤ s,t ≤randthen
s,t 1 s,t r
≤ ≤
Wehave:
m1,1 = 52 = 2 (1Q5ix=i1)(1−3xi)(1 xi) =(x2+1)(x4+x3+x2+x+1)
h i i=1 − · i=1 −
m = 2 =1Q Q
1,2 2
m =h0i=0
1,3 2
m2,1 =h51i= 1 (1Q5ix=i1)(1−4xi)(1 xi) = x4+x3+x2+x+1
m2,2 =h21i= Qi1=1(1Q−2ix=i1)(·1Q−i1x=i1)(1−xi) = x+1
m =h0i=0Qi=1 − ·Qi=1 −
2,3 1
m =mh i =m =1.
3,1 3,2 3,3
Thisway
(x2+1)(x4+x3+x2+x+1) 1 0
M = x4+x3+x2+x+1 x+1 0 ,
so det(M) = x7 +2x6+3x5 +3x4 +31x3 +2x2 + x. The1gener1atingfunctionis then
x14det(M)= x15+2x16+3x17+3x18+3x19+2x20+x21.
Ifweconsider,forexample,n(π) = 17,thecoefficientof x17 intheabovepolynomial
is 3, so it tells us that there are exactly three shifted (1,0)-plane partitions of shape
9
λ=(3,3,3),suchthata=(6,3,1)andb=(1,1,1).
Wecanwritethemdownforcompleteness’sake:
6 5 1 6 4 2 6 3 2
 0 3 1 ,  0 3 1 ,  0 3 2 
 0 0 1   0 0 1   0 0 1 
♦
4 Bar Code associated to a finite set of terms
Inthissection,weprovidealanguageinordertorepresentzerodimensionalmonomial
ideals,whicharecharacterizedbyhavingaconstantaffineHilbertpolynomial.
Inthecaseoftwoorthreevariables,thiswillallowustoestablishaconnectionbetween
(strongly)stable ideals I ⊳ with constant affine Hilbert polynomial H (t) = p N
I
P ∈
andsomeparticularplanepartitionsoftheintegernumber p. Moreprecisely,wewill
giveacombinatorialrepresentationfortheassociated(finite)lexicographicalGroebner
escalierN(I).
Firstofall,wepointoutthat,since (cid:27) Nn,atermxγ = xγ1 xγn canberegardedas
T 1 ··· n
thepoint(γ ,...,γ )inthen-dimensionalspace.
1 n
Usingthisconvention,wecanrepresentN(I)withan-dimensionalpicture,calledtower
structureofI (formoredetailssee[11][39,II.33]).
Example14. ConsidertheradicalidealI =(x2 x ,x x ,x2 2x )⊳k[x ,x ],definedby
1− 1 1 2 2− 2 1 2
itslexicographicalreducedGroebnerbasis. Sincew.r.t.Lex3,wehaveT(x2 x )= x2,
1− 1 1
T(x x ) = x x ,T(x2 2x ) = x2,wecanconcludethatthelexicographicalGroebner
1 2 1 2 2− 2 2
escalierofI isN(I)= 1,x ,x ,soitcanberepresentedbythefollowingpicture:
1 2
{ }
x2
x2
1 x1 x1
♦
Foraradicalideal I,noticethatif N(I) < also V(I) < (and,moreprecisely,it
| | ∞ | | ∞
holds N(I) = V(I)),sotheassociatedvarietyconsistsofafinitesetofpoints.
| | | |
It has been proved by Cerlienco-Mureddu ([13]) that, in this case, any ordering on
thepointsinV(I)givesapreciseone-to-onecorrespondencebetweenthetermsinN(I)
and the points in V(I), so it is also possible to label the points in the tower structure
withthecorrespondingpointoftheorderedV(I).
3Since,inthispaper,weareworkingwiththelexicographicalorder,Iprecisedhere“w.r.t.”Lex.Anyway,
itcanbeeasilyobservedthatT(x21−x1)=x21,T(x1x2)=x1x2,T(x22−2x2)=x22triviallyholdsforeachterm
order.
10