integral domains residue class rings of polynomial rings over On Gröbner bases and Krull dimension of Journal of Symbolic Computation

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Journal of Symbolic Computation

www.elsevier.com/locate/jsc

On Gröbner bases and Krull dimension of residue class rings of polynomial rings over integral domains

Maria Francis, Ambedkar Dukkipati

Dept.ofComputerScience&Automation,IndianInstituteofScience,Bangalore560012,India

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received20July2016 Accepted3March2017 Availableonline24March2017

Keywords:

Gröbnerbasesovercommutativerings Krulldimensionofresidueclassringsof polynomialringsoverrings

Independentsetsmoduloanideal

GivenanidealainA[^x1,. . . ,xn]^{where AisaNoetherianintegral} domain,weproposeanapproachtocomputetheKrulldimension ofA[^x1,. . . ,xn]/a^{,whentheresidueclassringisafree A-module.}

WhenAisafield,theKrulldimensionofA[^x1,. . . ,xn]/ahassev- eralequivalent algorithmic definitions by whichit can be com- puted.ButthisisnottrueinthecaseofarbitraryNoetherianrings.

Fora Noetherian integral domain A weintroduce the notion of combinatorialdimensionofA[^x1,. . . ,xn]/aandgiveaGröbnerba- sismethodtocomputeitforresidue classringsthat haveafree A-modulerepresentationw.r.t. alexicographicordering. Forsuch A-algebras,wederivearelationbetweenKrulldimensionandcom- binatorialdimensionof A[^x1,. . . ,xn]/a.Animmediateapplication ofthisrelationis thatit givesauniformmethod,the firstofits kind,to computethe dimension of A[^x1,. . . ,xn]/a withouthav- ingtoconsider individualproperties ofthe ideal.For A-algebras thathaveafree A-modulerepresentationw.r.t.degreecompatible monomialorderings,weintroducetheconceptsofHilbertfunction, HilbertseriesandHilbertpolynomialsandshowthatGröbnerbasis methodscan beusedtocomputethesequantities.We thenpro- ceedtoshowthatthecombinatorialdimensionofsuchA-algebras isequaltothedegree oftheHilbertpolynomial.Thisenablesus toextendtherelationbetweenKrulldimensionandcombinatorial dimensiontoA-algebraswithafreeA-modulerepresentationw.r.t.

adegreecompatibleorderingaswell.

©^{2017ElsevierLtd.Allrightsreserved.}

E-mailaddresses:mariaf@csa.iisc.ernet.in(M. Francis),ad@csa.iisc.ernet.in(A. Dukkipati).

http://dx.doi.org/10.1016/j.jsc.2017.03.003 0747-7171/©^{2017ElsevierLtd.Allrightsreserved.}

1. Introduction

One of the fundamental problems in computational ideal theory is determining the dimension of an ideal, i.e. the Krull dimension of the k^-algebra, k[^x1

, . . . ,

xn

] /a

. The dimension of an affine variety associated with an ideal a

⊆

k[^x1

, . . . ,

xn

]

^{is the Krull dimension of the affine} k^-algebra k[^x1

, . . . ,

xn

] /a

foran algebraically closedfield k^{. Since the definition ofKrull dimension does not} leadtoan algorithmicmethodtocomputeit,variousalternateequivalent definitionshavebeenpro- posed. The Krull dimension ofan affine k^{-algebra isequal to its} transcendencedegree, the degree oftheHilbertpolynomialofaandthelargestnumberofelements amongthemaximalsetofinde- terminatesindependentmod a(calledthecombinatorialdimensionofk[^x1

, . . . ,

x_n

]/a

^)(Kreuzerand Robbiano, 2005).Gröbner basisbasedalgorithms havebeenproposed tocompute thedegreeofthe Hilbertpolynomialofaandthecombinatorialdimensionofk[^x1

, . . . ,

xn

] /a

(MoraandMöller,1983;

KredelandWeispfenning,1988),thusprovidingan algorithmicframeworkfordeterminingtheKrull dimension ofthe affine variety associated witha. This paperstudies the question ofwhether one can give Gröbner basis methods to compute the Krull dimension of A

[

^x1

, . . . ,

xn

] /a

, where A is a Noetherianintegral domain,giventhat ithasa free A-modulerepresentationw.r.t.some monomial order.

ForanyNoetheriancommutativering A,anecessaryandsufficientconditionforafinitelygener- ated A-module A

[

^x1

, . . . ,

x_n

]/a

^{to haveafree A-module} representationw.r.t.a monomialorder has been studied inFrancis andDukkipati (2014). Here, we show that thischaracterization can be ex- tendedto A

[

^x1

, . . . ,

x_n

]/a

^{thatneednotbefinitelygeneratedasan A-module.}

Given an integral domain A andan A-algebra witha free A-module representationw.r.t. some monomial order,we studyalternate algorithmicdefinitions forKrull dimension.Wefirstextendthe conceptofcombinatorialdimensiontoA-algebras.Foran A-algebrawithafreeA-modulerepresenta- tionw.r.t. alexicographicordering,wegiveaGröbnerbasisalgorithmforcomputingitscombinatorial dimension.Inaffinek^{-algebras,the}combinatorialdimensionisequaltotheKrulldimension.Wede- rivearelationbetweenKrulldimensionandcombinatorialdimensionfor A-algebrasthathaveafree A-modulerepresentationw.r.t.alexicographicorder.WealsoshowthattheconceptsofHilbertfunc- tions,HilbertseriesandHilbertpolynomialcanbeextendedto A-algebrasthathaveafree A-module representation w.r.t.a degreecompatibleordering. Wealso givea Gröbner basis algorithmto com- putethesequantities.Fordegreecompatibleorderings,weshowthatthecombinatorialdimensionof A

[

^x1

, . . . ,

x_n

]/a

^{isequalto thedegreeoftheHilbertpolynomial of}aandthereforewehaveaGröb- ner basis algorithmto compute the combinatorial dimension.We also show how thiscan be used to derive a relation between the Krull dimension of A

[

^x1

, . . . ,

xn

]/a

^{andthe degree of the Hilbert} polynomial ofa.TheconceptsofcombinatorialdimensionandHilbertpolynomialareimportantbe- causetheygiveusauniformmethod,independentoftheideal,todeterminetheKrulldimensionof A

[

^x1

, . . . ,

xn

] /a

thathasafree A-modulerepresentationw.r.t.eitheralexicographicoradegreecom- patible monomialordering.Moreimportantly,theseconcepts allowforan algorithmicinterpretation ofthealgebraicconceptofKrulldimensionforcertain A-algebras.

The rest of the paper is organized as follows. In Section 2, we discuss the notations used in the paper. In Section 3, we extend the necessary and sufficient condition for the quotient ring A

[

^x1

, . . . ,

x_n

]/a

^{,where A isaNoetherian}commutativering,tohavea free A-modulerepresentation w.r.t.amonomialordertotheinfinitecase.Afterthissection,thepaperrestrictsitsstudytoresidue class ringsofpolynomial ringsover Noetherianintegral domains.In Section 4,we define combina- torial dimension for A-algebras, where A is a Noetherianintegral domain. In Section 4.2, we give a Gröbner basis methodtocompute the combinatorialdimension of A

[

^x1

, . . . ,

xn

] /a

that hasa free A-modulerepresentationw.r.t.alexicographicordering.Forsuch A-algebras,wederivearelationbe- tween combinatorialdimension andKrulldimension inSection 5.In Section 5.2,we illustrate with examples how thisrelationgivesan algorithmic methodtodetermine theKrull dimension.We de- fineHilbertfunction,HilbertseriesandHilbertpolynomialforA-algebrasthathaveafree A-module representationw.r.t.adegreecompatiblemonomialorderinSection6.Wealsogive aGröbnerbasis algorithmtocomputethesequantities.WethenshowinSection6.2thatthecombinatorialdimension of A-algebraswithafreeA-modulerepresentationw.r.t.adegreecompatiblemonomialorderisequal

tothedegreeoftheHilbertpolynomial.Thisenablesustogivearelationbetweenthedegreeofthe HilbertpolynomialandtheKrulldimensionofthecorrespondingresidueclassringinSection6.3.

2. Preliminaries

Throughoutthispaper,k ^{denotesafield,} Z^{theringofintegersand}N^{thesetofpositiveintegers} includingzero.We use A to denotea Noetheriancommutative ring.From Section 4onwards, A is restrictedto Noetherianintegral domains. Apolynomial ring in indeterminatesx1

, . . . ,

xn over A is denotedas A

[

^x1

, . . . ,

xn

]

^{.At times,we representthe}indeterminatescollectively asa set X andthe correspondingpolynomialringasA

[

^X

]

^{.Werepresentamonomialinx}1

, . . . ,

x_n asxα where

α ∈

Zⁿ_≥0^. The monoid isomorphism between the set of all monomials in indeterminates x1

, . . . ,

xn and Zⁿ_≥0 allows us to denote the set of all monomials as Zⁿ_≥0^{.A nonzero} polynomial, f in x1

, . . . ,

xn with coefficientsfromA isgivenby

f

=

α∈_f a_αx^α

,

where

_f Zⁿ_≥0 ^{is a finiteset anda}α

∈

^A

\ {

⁰

}

^{.We denote all the monomialsof a polynomial f} asMon

(

f

)

.We assumethat there isa monomialorder

≺

^{on themonomialsinthe} indeterminates x₁

, . . . ,

x_n.Withrespecttothismonomialorder,wehavetheleadingmonomial(lm_≺),leadingcoeffi- cient(lc_≺),leading term(lt_≺) andmultidegree(multideg_≺)ofapolynomial f

∈

^A

[

^x1

, . . . ,

xn

]

^,where multideg_≺

(

f

) =

^max≺

{ α ∈

f

}

^andlt≺

(

f

) =

^lc≺

(

f

)

lm_≺

(

f

)

in A

[

^x1

, . . . ,

xn

]

^{.Incertain scenarios,we} alsoconsider another concept ofdegree ofa polynomial which we will denoteasdeg

(

f

)

.The de- greeof amonomial xα, deg

(

xα

)

isthe sumof itsexponents. The degree ofa polynomial, f is the maximumdegreeofthemonomialsin f,i.e.

deg

(

f

) =

^max

{

^deg

(

x^α

) :

^xα

∈

^Mon

(

f

) } .

Adegree compatible monomial ordering

≺

^{is amonomial ordering on A}

[

^x1

, . . . ,

x_n

]

^{such that two} monomialsxα, xα withxα

≺

^{xα satisfy deg}

(

xα

) ≤

^deg

(

xα

)

.Fora degree compatiblemonomial or- dering,theleadingmonomialwillbeamonomialwithmaximumdegree.Theleadingtermideal(or initial ideal) ofa set S

⊆

A

[

x1

, . . . ,

xn

]

, is

lt_≺

(

S

) = {

lt_≺

(

f

) |

f

∈

S

}

.When there is no confusion regardingwhichmonomialordertoconsiderweomitthemonomialordersubscript

≺

^fromtheno- tations.Forafree A-moduleM,theminimumcardinalityofabasisof Aiscalleditsfreerankandis denotedbyFreeRankA

(

M

)

.

3. CharacterizationofafreeresidueclassringofA[^x1,. . . ,xn]

ConsideranidealainA

[

^x1

, . . . ,

xn

]

^andletG

= {

^gi

:

ⁱ

=

¹

, . . . ,

t

}

^{beitsGröbnerbasisw.r.t.amono-} mialorder

≺

^{.Foreachmonomialxα,let J}x^α

= {

ⁱ

:

^lm

(

g_i

) |

^xα

,

g_i

∈

^G

}

^andIJ_xα

= {

^lc

(

g_i

) :

ⁱ

∈

^Jx^α

}

^.We refertoIJ_xα astheleadingcoefficientidealw.r.t. G.LetCJ_xα representasetofcosetrepresentatives oftheequivalenceclassesin A

/

IJ_xα.WeusethesamedefinitionsasgiveninAdamsandLoustaunau (1994).Givenapolynomial f

∈

^A

[

^x1

, . . . ,

xn

]

^{,let f}

=

^m

i=¹

aixαi mod

^{G ,wherea}i

∈

^A

,

i

=

¹

, . . . ,

m.If A

[

^x1

, . . . ,

xn

]/

^{G isan A-modulefinitelygeneratedbym elements,then}corresponding tothecoset representatives,CJ_xα₁

, . . . ,

CJ_xαm,thereexistsan A-moduleisomorphism,

φ :

A

[

x1

, . . . ,

xn

]/

G

−→

A

/

IJ_xα₁

× · · · ×

A

/

IJ_xαm

m i=¹

a_ix^αi

+

^G

−→ (

c₁

+

^IJ_xα₁

, · · · ,

c_m

+

^IJ_xα_m

),

⁽¹⁾

whereci

=

^aimod IJ_xα_i andci

∈

^CJ_xα_i.

Wereferto A

/

IJ_xα₁

× · · · ×

^A

/

IJ_xαm asthe A-modulerepresentationof A

[

^x1

, . . . ,

xn

] /a

w.r.t.G (or w.r.t.

≺

^{).If I}J_xα_i

= {

⁰

}

^,wehaveCJ_xα_i

=

^{A,for alli}

=

¹

, . . . ,

m.Thisimplies A

[

^x1

, . . . ,

x_n

]/a ∼ =

^Am,i.e.

A

[

^x1

, . . . ,

xn

] /a

hasan A-modulebasisanditisfree.Wesaythat A

[

^x1

, . . . ,

xn

] /a

hasafree A-module representation w.r.t. G (or w.r.t.

≺

^{). If the A-module is infinitely generated, we say that it has a} free A-modulerepresentationw.r.t. G (orequivalently w.r.t.

≺

^{)if I}J_xα

= {

⁰

}

^{forall xα}

∈ /

^lm

(a)

^and I_Jxα

=

^{Aforall xα}

∈

^lm

(a)

^.

The necessaryandsufficientcondition foran A-module A

[

^x1

, . . . ,

xn

]/a

^{tohaveafree A-module} representationw.r.t.G (orw.r.t.

≺

^{) makesuseof}the conceptof‘shortreducedGröbnerbasis’intro- ducedinFrancisandDukkipati (2014)whichwebrieflydescribebelow.

Definition3.1.Leta

⊆

^A

[

^x1

, . . . ,

xn

]

^{beanideal.Considerthe}isomorphismin(1).AreducedGröbner basis(asdefinedinPauer,2007),GofaiscalledashortreducedGröbnerbasisifforeachxα

∈

^lm

(

G

)

, the number of elements in the generating set of the leading coefficient idealof xα, I_Jxα in (1)is minimal.

One candefinereducedGröbnerbasesoverringsexactlyasthat offieldsbutitmaynotexistin all thecases.ThedefinitionofreducedGröbnerbasisgivenby Pauer (2007)ensurestheexistence of suchabasisforeveryidealinthepolynomialring.

A necessaryandsufficient condition forafinitely generated A-module A

[

^x1

, . . . ,

x_n

]/a

^{to havea} free A-modulerepresentationw.r.t.G (orw.r.t.

≺

^{)isgiveninFrancisand}Dukkipati (2014).Onecan easilyextendthistoresidueclassringsthatarenotfinitelygeneratedasshownbelow.

Lemma3.2.Leta

⊆

^A

[

^x1

, . . . ,

xn

]

^{beanon-zeroidealandletG beashortreducedGröbnerbasisfor}a.Allthe leadingcoefficientidealsassociatedwithG areeithertrivialortheentireringA ifandonlyifG ismonic.

Proof. TheproofisalongthelinesofFrancisandDukkipati (2014,Lemma3.10). 2 Wenowprovethenecessarycondition.

Theorem3.3.Leta

⊆

^A

[

^x1

, . . . ,

xn

]

^{beanon-zeroidealandletG beashortreducedGröbnerbasisof}a.If A

[

^x1

, . . . ,

xn

] /a

hasafreeA-modulerepresentationw.r.t.G (orw.r.t.

≺

^{)thenG ismonic.}

Proof. Bydefinition,ifA

[

^x1

, . . . ,

x_n

]/a

^{hasafree A-module}representationw.r.t.GthenI_Jxα

= {

⁰

}

^for all xα

∈ /

^lm

(a)

^andIJ_xα

= {

¹

}

^forallxα

∈

^lm

(a)

^{.Thatis,alltheleadingcoefficientidealsassociated} withGareeithertrivialortheentirering, A.ThereforebyLemma 3.2,G ismonic. 2

The sufficientconditionissubsumedby FrancisandDukkipati (2014,Theorem3.8).Westatethe characterizationresultasfollows.

Proposition3.4.Leta

⊆

^A

[

^x1

, . . . ,

x_n

]

^{beanonzeroideal.LetG beashortreducedGröbnerbasisfor}aw.r.t.

somemonomialordering

≺

^.Then,A

[

^x1

, . . . ,

xn

] /a

hasafreeA-modulerepresentationw.r.t.G (or

≺

^)ifand onlyifG ismonic.

Example3.5. Let G

= {

^3x

,

5x

}

^{be the Gröbner basis of an ideal} a in Z[^x

,

y

]

^{w.r.t. a} lexicographic ordering

≺

^suchthatx

≺

^y.WehaveIJx

=

³

,

5

=

^1.Therefore,Z[^x

,

y

] /

^3x

,

5x hasafreeA-module representationw.r.t.

≺

^{.TheshortreducedGröbnerbasisof}aisgivenby G_red

= {

^x

}

^{.Itismonicanda} Z^{-modulebasisof}Z[^x

,

y

]/a

^isgivenby

{

¹

+

a,yⁿ

+

a,n

∈

N

\ {

⁰

}}

^.

Note that ifa Gröbner basis or reducedGröbner basis of an idealw.r.t. a monomial order

≺

^is monic its shortreduced Gröbner basis w.r.t.

≺

^{will also be monic, but not} vice versa. Throughout thispaper,wewillassumethat A

[

^x1

, . . . ,

xn

] /a

hasafree A-modulerepresentationw.r.t.amonomial orderorequivalently,w.r.t.anyGröbnerbasis correspondingtothatmonomialorder.

4. CombinatorialdimensionofA[^x1,. . . ,xn]/a

Intherestofthepaperweassume that A isaNoetherianintegraldomain.TheKrulldimension ofaring isdefinedasthesupremumofthelengthsofall thechainsofprimeidealsinit.Thereare manyalternate algorithmicdefinitions forthedimension ofan affine k^{-algebra.All ofthemcan be} showntobeequivalent.Ontheotherhand,for A-algebrasthesedefinitionsareeithernotequivalent ornotvalid.

WedefinecombinatorialdimensionofA

[

^x1

, . . . ,

xn

] /a

,denotedbycdim

(

A

[

^x1

, . . . ,

xn

] /a)

,inaman- neranalogoustothedefinitionofcombinatorialdimensionofk[^x1

, . . . ,

xn

] /a

(KreuzerandRobbiano, 2005).

Definition4.1. Given a Noetherian integral domain A, let a

⊆

^A

[

^x1

, . . . ,

xn

]

^{be an ideal. Let X}

⊆ {

^x1

, . . . ,

x_n

}

^{be a set of} indeterminates. The set X is said tobe independent modulo a oran inde- pendentsetofindeterminatesmoduloaifa

∩

^A

[

^X

] = {

⁰

}

^{.ThesetX iscalledamaximal}independent setmodulo aif X is independentmodulo aandthereis noset Y

⊆ {

^x1

, . . . ,

x_n

}

independent mod- uloawith X^{Y.Thelargestnumberofelements ofamaximal} independentsetofindeterminates moduloaiscalledthecombinatorialdimensionof A

[

^x1

, . . . ,

xn

] /a

,denotedascdim

(

A

[

^x1

, . . . ,

xn

] /a)

. 4.1. Somepropertiesofcombinatorialdimension

The Krull dimension of A

[

^x1

, . . . ,

xn

] /a

, for an ideal a

⊆

^A

[

^x1

, . . . ,

xn

]

^{, is the maximal Krull di-} mension of an isolated prime ideal associated with a. Below we show that this result holds for combinatorialdimensionaswell.

Lemma 4.2. Let A be a Noetherian integral domain and a be an ideal in A

[

^x1

, . . . ,

x_n

]

^{. Then} cdim

(

A

[

^x1

, . . . ,

xn

] /a)

isthemaximumofcdim

(

A

[

^x1

, . . . ,

xn

] /p)

,wherepisanisolatedprimeidealasso- ciatedwitha.

Proof. Wewill denotecdim

(

A

[

^x1

, . . . ,

xn

] /a)

asd.Let pbe an isolated primeideal associatedwith a and S

⊆

^{X denotethe maximal set of} indeterminatesthat are independent modulo p. Theyare independent modulo a and therefore, d

≥ |

^S

|

^. Conversely, let S

⊆

^{X be a maximal} independent set of indeterminates modulo a such that

|

^S

| =

^{d. Then M}

=

^A

[

^S

] \ {

⁰

}

^is multiplicatively closed and disjoint from a. There exists a prime ideal P, that contains a and does not meet M. Let p

⊆

P be the isolated prime ideal associated with a. S is independent modulo p. This implies cdim

(

A

[

^x1

, . . . ,

x_n

]/p

) ≥

^d. 2

Fora subsetofindeterminatesS,theset S representsthesetofresidueclassesof S modulo the ideala.

Proposition4.3.GivenaNoetherianintegraldomain A,leta

⊆

^A

[

^x1

, . . . ,

xn

]

^{beaprimeideal.Then,all} maximalsetsofindeterminatesindependentmoduloahavethesamecardinality.

Proof. Sinceaisaprimeideal, A

[

^x1

, . . . ,

xn

] /a

isan integraldomain.LetQuot

(

A

[

^x1

, . . . ,

xn

] /a)

rep- resentthequotientfieldof A

[

^x1

, . . . ,

xn

] /a

.LetX

= {

^x1

, . . . ,

xn

}

^andS

⊆

^{X beasetof}indeterminates.

S is independent modulo a if and only if S is algebraically independent in Quot

(

A

[

^x1

, . . . ,

x_n

]/a)

over A.Assumethattherearemaximalindependentsetsmodulo aofdifferentcardinalities.Letthe twosetsthataremaximalindependentmoduloabe S

∪ {

^a

}

^andS

∪ {

^b1

,

b₂

}

^{.Thisimplies S}

∪ {

^a

,

b₁

}

andS

∪{

^a

,

b2

}

^{aredependentsetsof}indeterminatesmoduloa.Therefore,wehaveb1isalgebraicover Quot

(

A

[

^S

] )(

a

)

andaisalgebraicoverQuot

(

A

[

^S

] )(

b2

)

.Therefore,b1 isalgebraic overQuot

(

A

[

^S

] )(

b2

)

, whichisacontradictiontotheindependenceof S

∪ {

^b1

,

b2

}

^moduloa. 2

4.2. Gröbnerbasismethodforcomputingcombinatorialdimensionforlexicographicorderings

WeextendtheconceptofstronglyindependentindeterminatesmoduloaintroducedinKredeland Weispfenning (1988)foridealsink[^x1

, . . . ,

xn

]

^{,topolynomialringsover A.}

Definition4.4.Let S

⊆

^{X be asetof}indeterminatesand

≺

^{a monomialorderin A}

[

^x1

, . . . ,

xn

]

^.Then, A

[

^S

/(

X

\

^S

) ]

^{denotesthefollowingset,}

A

[

^S

/(

X

\

^S

) ] = {

^f

∈

^A

[

^x1

, . . . ,

xn

] :

⁰

=

^{fand lt}

(

f

) ∈

^A

[

^S

]} .

Wesaythat S isstronglyindependentmoduloaifA

[

^S

/(

X

\

^S

) ] ∩

a

= ∅

^.

Clearly,if Sisstronglyindependentmoduloa,thenitisindependentmoduloa.Buttheconverse isnottrue.

Lemma4.5.Leta

⊆

^A

[

^x1

, . . . ,

xn

]

^{beaproperidealand}

≺

^{beamonomialorderin A}

[

^x1

, . . . ,

xn

]

^.LetS

⊆

^X beasetofindeterminates.

(i) IfS isstronglyindependentmoduloaw.r.t.

≺

^{,thenthereexistsanisolatedprimeideal}passociatedwith asuchthatS isalsostronglyindependentmodulopw.r.t.

≺

^.

(ii) LetU

= {

^S

⊆

^X

:

S is strongly independent moduloaw.r.t.

≺}

^andU

= {

^S

⊆

^X

:

there exists an isolated prime idealpassociated withasuch that S is strongly independent modulopw.r.t.

≺}

^,thenU

=

^U. Proof.

(i) Let S bestronglyindependentmoduloa.LetM

=

^A

[

^S

/(

X

\

^S

) ] \ {

⁰

}

^beamultiplicativelyclosed subset of A

[

^x1

, . . . ,

xn

]

^{disjoint to} a. Then there exists a prime ideal P such that a

⊆

P and disjoint from M. Letp

⊆

P be an isolated prime ideal associated witha. Then S is strongly independentmodulo p.Also, ifS ismaximalstronglyindependentmodulo a,then forany S

⊆

S

⊆

^{X,where S is strongly}independent modulo p, S is strongly independent modulo a, so S

=

^S.

(ii) Clearly,U

⊆

^{U andby(i),U}

⊆

^U. 2

We recalltheconcept ofinessentialsetofindeterminatesfromKredelandWeispfenning (1988).

Let S

⊆

^{X beasetof}indeterminates, f

∈

^A

[

^x1

, . . . ,

xn

]

^{beapolynomialand}

≺

^{beamonomialorder} in A

[

^x1

, . . . ,

xn

]

^{.Wedenote fS asthepolynomial resultingfrom f by} substituting1 forallindeter- minatesfromS in f.Wesaythat Sisinessentialfor f ifforalltermst occurringin f,t^Slt

(

f

)

^S.

Theorem4.6.Let S

⊆

^{X ,} abeaprimeideal in A

[

^x1

, . . . ,

xn

]

^andlet

≺

^{bea monomialordersuchthat} A

[

^x1

, . . . ,

xn

] /a

hasafree A-modulerepresentationw.r.t.

≺

^{.AssumethatS is}independentmoduloaand thatforanyx

∈

^X

\

^{S,thereexistsapolynomial f}x

∈

^A

[

^S

∪ {

^x

}/

^X

\ (

S

∪ {

^x

})] ∩

asuchthatS isinessential for fx.ThenS ismaximalindependentmoduloaand

|

^S

| =

^cdim

(

A

[

^x1

, . . . ,

xn

] /a)

.

Proof. Forx

∈

^X

\

^S,letdx bethedegreeoflt

(

f_x

)

inx.Thend_x^{0,forotherwiselt}

(

f_x

)

^S

=

^{1 andso} t^S

=

^{1 foralltermst occurringin f}x,whichimplies fx

∈

^A

[

^S

]

^whichcontradictstheindependenceof S moduloa.LetT bethesetofallt

∈

^Mon

(

A

[

^x1

, . . . ,

xn

] )

suchthatforeveryx

∈

^X

\

^{S,thedegreeof} xint is^dx.

Claim1.Foreveryt

∈

^Mon

(

A

[

^x1

, . . . ,

xn

] ) \

^{T ,thereexists0}

=

^p

,

p1

, . . . ,

pm

∈

^A

[

^S

]

^,t1

, . . . ,

tm

∈

^{T and} f

∈

asuchthatpt

=

^p1t₁

+ · · · +

^pmt_m

+

^{f .}

Proofoftheclaim. Assumethecontradiction,thattheclaimfailsforsomet

∈

^Mon

(

A

[

^x1

, . . . ,

xn

] ) \

^T and thatt is

≺

^{-minimal amongthe monomialswith thisproperty. Choosex}

∈

^X

\

^{S such that the}

degree doft in x

≥

^dx andlet u

=

^tx−d

∈

^Mon

(

A

[

^x1

, . . . ,

xn

] )

. fx can be writtenas px^dx

− (

p1t1

+

· · · +

^pmtm

)

with0

=

^p

,

p1

, . . . ,

pm

∈

^A

[

^S

]

^,ti

∈

^Mon

(

A

[

^X

\

^S

] )

,ti

≺

^xdx,1

≤

ⁱ

≤

^m.Bymultiplyingwith x^d−dxu,weget

pt

=

x^d−dxu fx

− (

p1t1x^d−dxu

+ · · · +

pmtmx^d−dxu

).

Wehavex^d−dxu fx

∈

aandtix^d−dxu

≺

x^dxx^d−dxu

=

t for1

≤

i

≤

m.(Notethatherewehavetwocom- parisons, one isthe lessthan comparison,

<

based onthe degreesof avariable inthe monomials andtheother isthecomparisonbasedonthemonomialorder,

≺

^{.)Sincet is}

≺

^{-minimalamongthe} monomialsthat violatetheclaim,theclaimisvalidforalltix^d−dxu

,

1

≤

ⁱ

≤

^{m andthereforeClaim 1} isvalidfort aswell,a contradiction.

Let Quot

(

A

)

representthe quotient field ofthe integral domain, A. Since A

[

^x1

, . . . ,

xn

] /a

hasa free A-module representation w.r.t.

≺

^{we have A}

∩

a

= {

⁰

}

^{. This} implies Quot

(

A

) ⊆

^Quot

(

A

)(

S

)

Quot

(

A

[

^x1

, . . . ,

xn

] /a)

.ByClaim 1,Quot

(

A

) [

^x1

, . . . ,

xn

] /a

isfinitelygeneratedasa Quot

(

A

)(

S

)

-vector spacebyT.Eachx,x

∈

^X

\

^{SisalgebraicoverQuot}

(

A

)(

S

)

.Since A

∩

a

= {

⁰

}

^{,thisimpliesthatforeach} x

∈

^X

\

^{S wecandeterminea f}

∈

^A

[

^S

∪ {

^x

}] ∩

a.Therefore, S ismaximalindependentmoduloaand sinceaisaprimeideal,cdim

(

A

[

^x1

, . . . ,

x_n

]/a) = |

^S

|

^. 2

Definition4.7(LeftBasicSet(LBS)).Let

≺

^{bea monomialorderin A}

[

^x1

, . . . ,

xn

]

^andabean idealin A

[

^x1

, . . . ,

x_n

]

^{.Giventhesetof}indeterminates X,wedefineS_k

⊆

^X,0

≤

^k

≤

ⁿinductivelyas

S0

= ∅

Sk+¹

=

⎧ ⎪

⎨

⎪ ⎩

S_k

∪ {

^xk

}

^ifSk

∪ {

^xk

}

is strongly independent modulo

a

w.r.t.

≺

S_k otherwise

.

ThesetSn iscalledtheleftbasicsetofaw.r.t.

≺

^.

S_n is maximal strongly independentmodulo aw.r.t.

≺

^{. For}lexicographic orderings, asa conse- quenceofTheorem 4.6wehavethefollowingresultforprimeideals.

Corollary 4.8. Let a be a prime ideal in A

[

^x1

, . . . ,

x_n

]

^and

≺

^{be a} lexicographic ordering such that A

[

^x1

, . . . ,

xn

] /a

hasafree A-modulerepresentationw.r.t.

≺

^{.If S istheleftbasicsetof}aw.r.t.

≺

^,thenS ismaximalindependentmoduloaandso

|

^S

| =

^cdim

(

A

[

^x1

, . . . ,

xn

] /a)

.

Proof. SinceS ismaximalstronglyindependentmodulo a,foreveryx

∈

^X

\

^{S,thereexistsapolyno-} mial fx

∈

^A

[

^S

∪ {

^x

} /

X

\ (

S

∪ {

^x

} ) ] ∩

a. fx containsno y

∈

^{X suchthatx}

≺

^{y since}

≺

^isalexicographic order.Alsoforeverymonomialt

∈

^Mon

(

f_x

)

,thedegreeoftinxislessthanorequaltothedegreeof theleadingtermof fx in x.Therefore,lt

(

fx

)

^S

≥

^{tS foralltermsin f}x.Therefore, S isinessentialfor

fx andwecanapplyTheorem 4.6and

|

^S

| =

^cdim

(

A

[

^x1

, . . . ,

xn

] /a)

. 2 TheideacanbeextendedtootherproperidealsinA

[

^x1

, . . . ,

xn

]

^.

Theorem4.9.Letabeaproperidealin A

[

^x1

, . . . ,

x_n

]

^and

≺

^bealexicographicmonomialordersuchthat A

[

^x1

, . . . ,

xn

]/a

^{hasafreeA-module}representationw.r.t.

≺

^.Let

d

=

max

{|

S

| :

S

⊆

X

,

S is maximal strongly independent modulo

a

w.r.t.

≺}.

Then,d

=

^cdim

(

A

[

^x1

, . . . ,

x_n

]/a)

^.

Proof. Since each S that is maximal stronglyindependent modulo a isindependent modulo awe havecdim

(

A

[

^x1

, . . . ,

x_n

]/a) ≥

^{d.Pickanisolatedprimeideal}passociatedwithasuchthat

cdim

(

A

[

^x1

, . . . ,

x_n

]/p) =

^cdim

(

A

[

^x1

, . . . ,

x_n

]/a).

Let SbetheLBSofp.Then,

|

^S

| =

^cdim

(

A

[

^x1

, . . . ,

x_n

]/p) =

^cdim

(

A

[

^x1

, . . . ,

x_n

]/a)

and Sisstronglyindependentmodulopandthereforeaandsod

≥

^cdim

(

A

[

^x1

, . . . ,

x_n

]/a)

^. 2

Since strongly independent modulo a dependson the leading terms ofan ideal,we explore its connectionswithGröbnerbasis.

Theorem4.10.Let

≺

^{beamonomialorderinginA}

[

^x1

, . . . ,

xn

]

^andS

⊆

^{X beasetof}indeterminates.LetG be aGröbnerbasisofanideal,a

⊆

^A

[

^x1

, . . . ,

xn

]

^w.r.t.

≺

^{.ThenS isstrongly}independentmoduloaw.r.t.

≺

^ifand onlyifA

[

^S

] ∩

^lt

(

G

) = ∅

^.

Proof. If for some g

∈

^{G, lt}

(

g

) ∈

^A

[

^S

]

^{, then g}

∈

^A

[

^S

/(

X

\

^S

) ] ∩

a and therefore S is not strongly independentmoduloa.Conversely,assumethereexists f

∈

^A

[

^S

/(

X

\

^S

) ] ∩

a,thenthereexistsatleast one g

∈

^{G suchthatlm}

(

g

) |

^lm

(

f

)

.Sincelt

(

f

) ∈

^A

[

^S

]

^,lm

(

g

) ∈

^A

[

^S

]

^. 2

We canconstructtheLBSofaw.r.t.

≺

^{fromG bythefollowingalgorithm whichisanalogousto} KredelandWeispfenning (1988,Corollary 2.2).

Corollary4.11.Let

≺

^{beamonomialorderin A}

[

^x1

, . . . ,

x_n

]

^{andG beaGröbnerbasisw.r.t.}

≺

^foranideal a

⊆

^A

[

^x1

, . . . ,

xn

]

^.Algorithm 1determinestheleftbasicsetofaw.r.t.

≺

^.

Algorithm1FindingtheLeftBasicSetofanidealainA

[

^x1

, . . . ,

xn

]

^.

InputG,Gröbnerbasisofa⊆A[x1,. . . ,xn]w.r.t.≺ OutputS,LeftBasicSetofaw.r.t.≺.

S= ∅,U= {x1,. . . ,xn} whileU= ∅^do

SelectxfromU.

U=^U\ {^x}

ifMon(A[^S]∪ {^x})∩^lt(G)= ∅^then S=S∪ {x}

end if end while

Corollary4.12.Leta

⊆

^A

[

^x1

, . . . ,

x_n

]

^{beanidealsuchthatA}

[

^x1

, . . . ,

x_n

]/a

^{hasafreeA-module}representa- tionw.r.t.somelexicographicordering,

≺

^{andG beitsmonicshortreducedGröbnerbasisw.r.t.}

≺

^.LetS

⊆

^X beasetofindeterminatessuchthat

Mon

(

A

[

^S

] ) ∩

^lt

(

G

) = ∅ ,

andS hasthelargestnumberofelementsamongallsubsetsof X thatsatisfytheaboveequation.ThenS is maximalindependentmoduloaand

|

^S

| =

^cdim

(

A

[

^x1

, . . . ,

xn

] /a)

.

Proof. ThisresultisadirectconsequenceofTheorem 4.10andTheorem 4.9. 2

The above result gives us an algorithmic technique to determine the combinatorial dimension of A

[

^x1

, . . . ,

x_n

]/a

^{. It involves computing a Gröbner basis w.r.t. a} lexicographic ordering. Given a Noetherian integral domain A, we give belowan explicit description of the algorithm to compute thecombinatorialdimensionof A-algebras A

[

^x1

, . . . ,

xn

] /a

,thathaveafree A-modulerepresentation w.r.t.alexicographicordering.ThecorrectnessofthealgorithmfollowsfromCorollary 4.12.Itconsists oftworoutines,Algorithm 2andAlgorithm 3,thelatterofwhichisrecursive.Algorithm 3alsodeter- minesthemaximalindependentsetofindeterminatesmodulo a.Thisalgorithmisalongthelinesof thealgorithmdescribedinKredelandWeispfenning (1988,Section 3).

Algorithm2 Algorithmforfinding the combinatorialdimension of A

[

^x1

, . . . ,

xn

] /a

forlexicographic orderings.

InputG,shortreducedGröbnerbasisofa⊆^A[^x1,. . . ,xn]^w.r.t.alexicographicordering≺^, X= {^x1,. . . ,xn}

Outputc,combinatorialdimensionofA[x1,. . . ,xn]/a S,themaximalsetofindeterminatesindependentmoduloa. ifGisnotmonicthen

Exit end if

c=^0,S= ∅^,U=^X,M= ∅ {Callstherecursivealgorithm}

M=Algorithm 3(G,S,U,M) S=M

whileM= ∅^do SelectanyMfromM M=M\ {^M} ifc≤ |^M|^then

c= |M| end if end while

Algorithm3Recursivealgorithmforfindingthemaximalsetofindeterminatesindependentmodulo theideala

⊆

^A

[

^x1

, . . . ,

xn

]

^forlexicographicorderings.

InputG,Gröbnerbasisofa⊆A[x1,. . . ,xn]w.r.t.alexicographicordering≺, S,setofindeterminatessuchthatMon(S)∩^lt(G)= ∅^,

U,asubsetoftheindeterminatessetX,

M^{,asetofalreadycomputedmaximalsetsSwithMon}(S)∩^lt(G)= ∅^.

OutputM^{,theupdatedsetofmaximalsetof}indeterminatesSwithMon(S)∩^lt(G)= ∅^. {Findingthemaximalindependentsetsofindeterminates}

M=M whileU= ∅do

SelectufromU U=^U\ {^u}

ifMon(S∪ {^u})∩^lt(G)= ∅^then M=Algorithm 3(G,S∪ {^u},U,M) end if

end while

{TestingifSisalreadycontainedinsomeelementofM^} M=M^,t=¹

whileM= ∅^andt=^1do

SelectMfromM^,M=M\ {^M}^. ifS⊆^Mthen

t=0 end if end while ift=^1then

M=M∪ {^S} end if

Therunningtimeofthealgorithmisexactlyasthatofcomputingthecombinatorialdimensionfor fieldsexceptforthecomputationofshortreducedGröbner basis.Thecomputationofshortreduced Gröbner basisdependson thecoefficientring, A. When A

=

k ^orZ^{,the time complexityis doubly} exponential(computationofa singleGröbner basis)andwhen A

=

k[^y1

, . . . ,

ym

]

^{,thecomplexity is} stilldoublyexponentialbutinvolvestwoGröbnerbasiscomputations,firstink[^y1

, . . . ,

ym

]

^andthen in A

[

^x1

, . . . ,

xn

]

^.

5. RelationbetweenKrulldimensionandcombinatorialdimensionofA[^x1,. . . ,xn]/a

Theresultswe derivein thissection willalsohelp usderivea relationbetweenthedegreeofa HilbertpolynomialandKrulldimension(Section6.3).

Lemma5.1.Leta

⊆

^A

[

^x1

, . . . ,

xn

]

^{beanidealandM bean A-algebra.Thenthereexistsa}homomorphism fromA

[

x1

, . . . ,

xn

]

toM

[

x1

, . . . ,

xn

]

.Leta^erepresenttheextensionoftheidealainM

[

x1

, . . . ,

xn

]

underthe homomorphism.Wehave

M

⊗

AA

[

^x1

, . . . ,

x_n

]/a ∼ =

^M

[

^x1

, . . . ,

x_n

]/a

^e

.

Proof. Clearly, M

[

^x1

, . . . ,

xn

] /a

^e is an A-module.Consider thefollowing operation: for any f

+

a

∈

A

[

^x1

, . . . ,

x_n

]/a

^andg

+

a^e

∈

^M

[

^x1

, . . . ,

x_n

]/a

^e,let

(

f

+

a)(g

+

a^e

) =

^{f g}

+

a^e.Itiswelldefinedbecause a

⊆

a^e. Thisimplies M

[

^x1

, . . . ,

xn

]/a

^{e isan A}

[

^x1

, . . . ,

xn

]/a

^{-module aswell. We define thefollowing} homomorphism,

φ :

^{A(A[x1,...,xn]/a×M)}

→

^M

[

^x1

, . . . ,

x_n

]/a

^e

φ (

i∈

(

a_ix^αi

+ a,

m_i

)) =

i∈

(

a_im_ix^αi

+ a

^e

).

Notethat

φ

is A-multilinear.Therefore,thereexistthefollowinghomomorphisms,

ψ :

A

[

x1

, . . . ,

xn

]/a ⊗

AM

→

M

[

x1

, . . . ,

xn

]/a

^e

and

π :

A^{(A[x1,...,xn]/a×M)}

→

^A

[

^x1

, . . . ,

xn

] /a ⊗

AM

suchthat

φ = ψ ◦ π

.Since

φ

issurjective,

ψ

issurjectivetoo.Consider

ψ (

i∈

(

a_ix^αi

+ a ⊗

Am_i

)) =

⁰

.

Wehave

i∈

(

aix^αi

+ a ⊗

Am