A fast algorithm for constructing Arf closure and a conjecture

Contents lists available atScienceDirect

Journal

of

Algebra

www.elsevier.com/locate/jalgebra

A

fast

algorithm

for

constructing

Arf

closure

and

a conjecture

Feza Arslana,∗_{, Nil Şahin}b

a _{DepartmentofMathematics,MimarSinanFineArtsUniversity,Istanbul,34349,}

Turkey

b

DepartmentofIndustrialEngineering,BilkentUniversity,Ankara,06800,Turkey

a r t i c l e i n f o a bs t r a c t

Article history:

Received18September2013 Availableonline19July2014 CommunicatedbySethSullivant

Keywords: Branch Arfring Arfclosure Arfsemigroup Hilbertfunction

In this article, we give a fast and an easily implementable algorithm for computing the Arf closure of an irreducible algebroidcurve(orabranch).Moreover,westudytherelation betweenthebrancheshavingthesameArfclosureandtheir regularity indices. We give some results and a conjecture, whicharestepstowardstheinterpretationofArfclosureasa specificwayoftamingthesingularity.

© 2014ElsevierInc.All rights reserved.

1. Introduction

Canonical closureof alocal ringconstructed byCahitArfsolvestheproblem of de-terminingthecharactersofaspacecurvesingularity[1].Thecharactersofaplanecurve singularity,introducedfirstbyDuValin1942,arespecialintegers,whichdeterminethe multiplicity sequence of the plane curve singularity, [15]. Contrary to the well-known plane case, inwhich the characteristic exponents,the multiplicity sequence, the semi-groupofthesingularityandthecharactersdetermineeachother,itwasnotknownhow

* Correspondingauthor.

E-mailaddresses:sefa.feza.arslan@msgsu.edu.tr(F. Arslan),nilsahin@bilkent.edu.tr(N. Şahin). http://dx.doi.org/10.1016/j.jalgebra.2014.06.023

to obtain the characters in the space case, until Cahit Arf developed his theory [1]. In1946, Arf showedthatthe characters of aspace branchcould be obtainedfrom the completionof the local ring corresponding to the branchby constructing itscanonical closure, laterknownas Arfclosure[1]. Sincethen, manyalgebraic geometersand alge-braistshaveworkedonArfringsandArfclosure,[3,8,9,13,19].Moreover,Arfsemigroups andtheirapplicationsincodingtheoryhavebeenarecentareaofinterest[5,6,10,14,20]. Foragoodsurveyandaquickintroductionto Arftheory,see [22].

InspiteofallthisinterestinArfringsandArfclosure,thereisnotafastimplementable algorithmforthecomputationofArfclosureintheliterature.Theconstructionmethod givenbyArfcannot beimplementedasanalgorithm,withoutfindingabound,that de-terminesuptowhichdegreeadivisionseriesshouldbeexpanded,andfindinganefficient bound isnoteasyat all. Theconstructionof Arfclosureby usingHamburger–Noether matrices presented byCastellanos doesnot givean answerto this problem either[12]. Theonly implementedalgorithm isgiven by Arslan [2]. The algorithm usesArf’s con-structionmethodandstartswithdeterminingthesemigroupofvaluesofthebranchand itsconductor, but determining thesemigroup of valuesand the conductorof a branch is a difficult problem, which has been studied by many mathematicians and different algorithms havebeen given [11,18]. Notingthatthe specialcase ofthis problem is the famousFobeniusproblem(orcoinproblem)makesitclear,whythisproblemisdifficult, anditisunnecessarytomentionthatthereisavastliteratureontheFrobeniusproblem. Our main objects of interest in this article are space curve singularities. Following Castellanos and Castellanos [12] and using their notation, we consider a space curve singularityasanalgebroidcurveC = Spec(R), where(R,m,k) isalocalring,complete for the m-adic topology, with Krull dimension 1, and having k as a coefficient field. Wework withirreducible algebroid curves (orbranches), inother words R will always be a domain.In this case, it can be shown thatR ∼= kJϕ1(t),...,ϕn(t)K ⊂ kJtK, where

ϕ1(t),...,ϕn(t) are power series in t, see [12] or [18]. Hence, we will be working with

subringsofkJtK.Here,theset{ϕ1(t),...,ϕn(t)} is aparametrizationofthecurveC and

theminimumpossiblen iscalleditsembeddingdimension.Itisdenotedbyembdim(C) andisalsoequaltodimkm/m2.Theminimumorderoftheseriesofanyparametrization

ofthecurveC iscalleditsmultiplicity,whichisalsoequaltothemultiplicityofthelocal ringR.

By using the notation in [22], we denote the semigroup of orders of the local ring

R = kJϕ1(t),...,ϕn(t)K byW (R).For n∈ N, In ={r ∈ R | ord(r) ≥ n} and In/Sn =

{r · S−1

n | ord(r)≥ n}, whereSn ∈ R hasorder n. Ingeneral,In/Sn isnot aring. The

ringgeneratedbyIn/Sn isdenotedby[In]. AringiscalledanArfring,ifIn/Sn = [In]

for any n in its semigroup of orders. For a local ring R ⊂ kJtK, the smallest Arf ring containingR iscalledtheArfclosureofR.In[1],ArfnotonlydefinestheArfclosure,but alsogives amethodforits construction:The Arfclosureof R = kJϕ1(t),...,ϕn(t)K can

bepresentedasR∗= k +kF0+kF0F1+...+kF0...Fl−2+kJtKF0...Fl−1,whereR0= R,Fi

isasmallestorderedelementofRiwithai= ord(Fi) andRi = [Iai−1] fori= 1,...,l and

it isshownin[1]thatthe multiplicity sequenceof C is(a0,a1,...,al−1,1,1,...). In[16], aninstructive anddetailedexampleisgivenwithadiscussiononthegeometricaspects of theproblem.

Recallingthatdeterminingthesemigroupofvaluesofabranchisadifficultproblem, the construction of Arf closure by computing Ri’s in each step is not efficient at all.

Inthenextsection,we giveaneasily implementablealgorithm forconstructingtheArf closure byavoidingthese difficultandtimeconsumingcomputations.

2. AnalgorithmforthecomputationofArf closure

Let C be the branch with the corresponding local ring R = kJϕ1(t),...,ϕn(t)K and

W (R) ={i0,i1,...,ih−1,ih+N},where i0= 0 and i1 = ord(ϕ1(t)). Thering R canbe presented as

R = k + kSi1+ kSi2+ ... + kSih−1+ kJT KSih

where Sij’s are elements of R of order ij chosen such that ϕ1(t),...,ϕn(t) are among them.Hence,Si1= ϕ1(t).Byusingthisnotation,wehavethefollowingobviouslemma:

Lemma 2.1.[Ii1]= kJϕ1, ϕ2 ϕ1,..., ϕn ϕ1K. Proof. [Ii1] =  k  Si2 ϕ1 α2 Si3 ϕ1 α3 ...  Sih−1 ϕ1 αh−1 + kJT KSih ϕ1 where thesumis overallα2,α3,...,αh−1 satisfying

α2(i2− i1) + α3(i3− i1) + ... + αh−1(ih−1− i1) < (ih− i1) and Sir is an element of R of order ir. It is obvious that kJϕ1,

ϕ2 ϕ1,..., ϕn ϕ1K ⊂ [Ii1]. To prove that [Ii1] ⊂ kJϕ1, ϕ2 ϕ1,..., ϕn ϕ1K, it is enough to show S_ij ϕ1 is contained in

kJϕ1,ϕ_ϕ2₁,...,ϕ_ϕn₁K for any ij ∈ W (R). Since Sij is an element of R, it can written as

Sij =  aα1α2...αn(ϕ α1 1 ...ϕαnn) and Sij ϕ1 =aα1α2...αn(ϕ1) α1+α2+...+αn−1  ϕ2 ϕ1 α2 ...  ϕn ϕ1 αn

Since eachsummand intheexpressionof S_ϕij

1 isanelementoftheringkJϕ1,

ϕ2

ϕ1,...,

ϕn

ϕ1K,

S_ij

ϕ1 is also an element of the same ring for any ij ∈ W (R), showing that [Ii1] ⊂

Remark2.2.Notethat[Ii1] isthelocal ringcorrespondingto theblowupofthebranch

ofthecurveC attheorigin.

Asaconsequence,theparametrizationcorrespondingtoRicanbeobtainedfromthe

parametrization corresponding to Ri−1 by doing power series divisions, and these pa-rameterizationscanbeusedtodetermineFi’stoconstructtheArfclosure.Theproblem

thatshouldbesolvedisuptowhichdegreethedivisionseriesmustbeexpandedsothat noinformationislost.Tosolvethisproblem,wefirstrecallthefollowingimportant the-orem,showing thateverypowerseries parametrizationofabranchcanbe interchanged withapolynomialparametrization.

Theorem 2.3. (See [11].) Let C be a branch with the parametrization {ϕ1(t),ϕ2(t),...,

ϕn(t)}.Letc betheconductorofthesemigroupofvaluesofC.Thenanyparametrization

{φ1(t),φ2(t),...,φn(t)} withφi≡ ϕi (mod tc) for1≤ i≤ n gives thebranchC.

Byusing this theorem, and observing that the conductor of Ri is smaller than the

conductor of Ri−1 where R0 = R, we can immediately propose a bound for expand-ingthe divisionseries: theconductor c ofW (R).Unfortunately, as we havementioned above, determining c from the parametrization is a problem on its own. We aim to propose a bound without determining the conductor c of W (R). To do this, we first construct a blow up schema, which not only summarizes the construction of the Arf closure of the ring R = kJϕ1,...,ϕnK, but is also essential in the proof of our main

theorem.

Column 1 Column 2 Column n smallest ordered element ϕ(0)₁ (t), ϕ(0)₂ (t), . . . , ϕ(0)n (t), −→ F0= ta0+ higher degree terms

ϕ(1)₁ (t), ϕ(1)₂ (t), . . . , ϕ(1)n (t), −→ F1= ta1+ higher degree terms ..

. ... ... ...

ϕ(l)₁ (t), ϕ(l)₂ (t), . . . , ϕ(l)n (t), −→ Fl= t + higher degree terms

(2.1) Hereϕ(0)_i = ϕi,and ϕ(j)_i (t) =  ϕ(j_i −1)(t), if F_j−1= ϕ(j_i −1)(t) ϕ(j_i−1)(t) Fj−1 − cij, if Fj−1 = ϕ (j−1) i (t) (2.2)

cij ∈ k andcij = 0 ifandonlyiford(ϕi(j−1)(t))= ord(Fj−1).Notealsothatord(Fl−1)≥ 2

witha0≥ a1≥ ...≥ al−1 ≥ 2.Then Rj = kJϕ(j)1 (t),ϕ (j)

2 (t),...,ϕ (j)

s (t)K andtheArf

R∗= k + kF0+ kF0F1+ ... + kF0F1...Fl−2+ F0...Fl−1kJtK.

Letc∗ denotetheconductorofthesemigroupW (R∗) oftheArfclosure.AsW (R∗)=

{0,a0,a0 + a1,...,a0+ ...+ al−2,a0 + ...+ al−1+N}, c∗ = a0+ ...+ al−1, and thus

R∗= k + kF0+ kF0F1+ ...+ kF0F1...Fl−2+ tc∗kJtK,whereFj = Fj (mod tc

∗

).Hence,it is enoughtofindFj’sinsteadof theexactFj’sfor constructingtheArfclosureR∗. We

cannowstateourmaintheorem.

Theorem2.4. Therings kJϕ1,...,ϕnK andkJφ1,...,φnK have thesameArf closure,where

φj ≡ ϕj (mod tc

∗₊₁

) for1≤ j ≤ n.

Proof. We have to show that we do not lose any significant data by expanding the division series up to degree c∗+ 1, while determining the series ϕ(j)_i (t). We prove our claim intwo partsbyconsideringtheconstants ‘cij’sandbyusing theblowupschema

(2.1) and the notationthere. Note that,ifa columninthe blow upschema (2.1) does notenterthealgorithm,thismeansthatthealgorithmisthesamewithoutthatcolumn. Therefore,losingmonomialsinthatcolumnhasnoeffectonthecomputationoftheArf closure. Hence, without loss of generalization, we can assume that all of the columns enter the algorithm at least once. Using the schema (2.1) again, this is equivalent to sayingthatforalli,ϕ(j)_i = Fj forat leastonej.

We first consider the case with zero constants. In other words, recalling Eq. (2.2),

cij = 0 foralli,j, so thatallthe importantmonomials arethe smallestordered terms

of ϕ(0)_i ’s.

Inthissituation,foralli,ord(ϕ(0)_i )≥ ord(ϕ(1)_i )≥ ...≥ ord(ϕ(k)_i ) and,

ϕ(j)_i (t) = ϕ(k)_i (t)  l∈Iij Fl where Iij=  l : j≤ l < k and Fl = ϕ(l)i (t)

Then, ord(ϕ(0)_i ) = _l∈I

i0al + ord(ϕ (k)

i (t)). We have observed that ϕ

(j)

i (t) = Fj for

some j. Therefore, _l∈I

i0al+ ord(F  j)

aj

ord(ϕ(j)_i )

≤ c∗_{. Also, since ord(ϕ}(k)

i ) ≤ ord(ϕ

(j)

i ) for all

j < k, we have ord(ϕ(0)_i ) = _l∈Ii0al+ ord(ϕ(k)i ) ≤ c∗ for all i. So, expanding the

divisionseriesuptothedegree(c∗+ 1) is enoughto constructtheArfclosure.

Ifcij = 0 forsomei andj,wedoaninductiononthenumberofj’sforwhichcij = 0

forsomei.

ϕ(0)₁ (t), . . . ϕ(0)_i (t), . . . ϕ(0)s (t), −→ F0 .. . ... ... ... ϕ(j)₁ (t), . . . ϕ(j)_i (t) = ϕ(ji−1)(t) Fj−1 − cij, . . . ϕ (j) s (t), −→ Fj .. . ... ... ... ϕ(k)₁ (t), . . . ϕ(k)_j (t), . . . ϕ(k)s (t), −→ Fk = t + ... Let ϕ(j)_i (t) = b1tα1+ b2tα2+ . . . = ϕ(j_i −1)(t) F_j−1 − cij (α1< α2). (2.3)

It’senoughtoshowthat,wedon’tlosethetermtα1_{inthesteps0,1,...,j}−1 by_expanding

thedivisionseriesuptothedegree(c∗+1).Thereasonisthatafterthej-thstep,thereare nononzerocij’sandfromthepreviouspart,weknowthatα1< aj+ aj+1+ . . . + ak< c∗.

(Notethat,withoutlossofgeneralization,wecanassumethattheithcolumnentersthe algorithmat leastonceafterthej-thstep.)

Hence,byEq. (2.3), ϕ(j_i −1)(t) =cij+ b1tα1+ b2tα2+ . . .  Fj−1 and ϕ(0)_i (t) =cij+ b1tα1+ b2tα2+ . . .   l∈Λi,0 Fl

where Λi,0 = {l : 0 ≤ l ≤ j − 1 and Fl = ϕ(l)i (t)}. As α1 ≤ aj + . . . + ak−1 and

c∗ = a0+ ...+ aj−1+ aj+ ...+ ak−1, we cansay that ϕ(0)i (t) mod tc

∗₊₁

contains the term,whichgivestα1 _{inthej-thstep.Thisshowsthatexpandingthedivisionseriesup}

tothedegreec∗+ 1 insteps0,1,...,j− 1 guaranteesthatthetermtα1 _{isobtainedinthe}

j-thstep.

• Assumetheclaimistrueforabranchhavingcij = 0 forn− 1 differentj’s.Wetake

any branchhaving cij = 0 forn differentj’s.

Letthefirstconstantappearsatthei0-thcolumn,j0-th step.Then,

ϕ(0)₁ (t), . . . ϕ(0)_i (t), . . . ϕ(0)s (t), −→ F0 .. . ... ... ... ϕ(j0) 1 (t), . . . ϕ (j0) i0 (t) = ϕ(j0−1)_i0 (t) Fj0−1 − ci0j0, . . . ϕ (j0) s (t), −→ Fj0 .. . ... ... ... ϕ(k)₁ (t), . . . ϕ(k)_j (t), . . . ϕ(k)s (t), −→ Fk = t + ...

From the inductionassumption,forthe stepsstartingwith j0-thone, itis sufficientto expandthedivisionseriesuptodegreeaj0+ aj0+1+ ...+ ak−1, sincethatistheconductor

of the Arfclosure ofthe ringkJϕ(j0)

1 ,ϕ (j0)

2 ,...,ϕ (j0)

s K. In otherwords, all thesignificant

monomialsthatdeterminetheArfclosurehaveorderslessthanorequaltoaj0+...+ak−1

at j0-thstep.Then,as inthefirstpartof theinductionhypothesis,since

ϕ(j0) i0 (t) = b1t α1_{+ b} 2tα2+ . . . = ϕ(j0−1) i0 (t) Fj0−1 − ci0j0 (α1< α2), (2.4)

we canwriteϕ(0)_i0 as:

ϕ(0)_i0 (t) =cij+ b1tα1+ b2tα2+ . . .   l∈Λi0,0 Fl, where Λi0,0={l : 0≤ l ≤ j0− 1 and Fl = ϕ (l)

i0(t)}.Thenall theimportantmonomials

inthefirststepshaveorderlessthanorequaltoord(_l∈Λ

i0,0Fl)= a0+ a1+ ...+ aj0−1 plusaj0+ aj0+1+ ...+ ak−1,whichisequaltoc∗.Hence,bytruncatingthedivisionseries

inmodtc∗+1,allthesignificanttermstoconstructtheArfclosurearepreserved. 2 Remark2.5.WeshouldnotethatTheorem 2.4 doesnotsaythattheparameterizations

{ϕ(0) 1 (t),ϕ (0) 2 (t),...,ϕ (0) s (t)} and{φ(0)1 (t),φ (0) 2 (t),...,φ (0)

s (t)} (whereφ(0)j isthetruncation

of the series ϕ(0)_j inmod tc∗+1_{)provide thesame curve. Expandingthe divisionseries} up to degree c∗ + 1 in each blow up does not guarantee to obtain the blow up ring

Rj= kJϕ(j)1 (t),ϕ (j)

2 (t),...,ϕ (j)

s (t)K exactly,butitguaranteestoconstructtheArfclosure

correctly.

Example 2.6.LetuscomputetheArfclosure oftheringkJt4_,t6_{+ t}9_,t14_{K withc}∗_{= 10}

R0: t4 t6+ t9 t14 −→ F0= t4

R1: t4 t2+ t5 t10 −→ F1= t2+ t5

R2: t2− t5+ t8− t11 t2+ t5 t8− t11 −→ F2= t2+ t5

R3: −2t3+ 3t6− 4t9 t2+ t5 t6− 2t9 −→ F3= t2+ t5

R4: −2t + 5t4− 9t7+ 14t10 t2+ t5 t4+ ... −→ F4=−2t + 5t4− 9t7+ 14t10 TheArfClosureis,R = k + kt4_{+ k(t}6_{+ t}9_{)+ kt}8_{+ kJtKt}10_{,andthemultiplicitysequence} of thecorresponding branchis:(4,2,2,2,1,1)

WenowhaveaboundforexpandingthedivisionseriestodeterminetheArfclosure.It isobviousthatc∗+ 1 isamuchbetterboundthanc,sincec∗+ 1 ismuchsmallerthanc.

(RecallthatR⊂ R∗andW (R)⊂ W (R∗).Thus,c∗≤ c.)But,itlookslikeasifwearein aviciouscircle:WewanttodeterminetheArfclosureandthemultiplicitysequence,but we needtheconductoroftheArfclosureforthis.Hence,weask thefollowing question:

‘IsthereawaytofindtheconductoroftheArfclosureortogiveaboundforitwithout knowingtheArfclosure?’Toanswerthatquestion,wefirstfocusonplanebranches,but beforethatwegivethefollowing generalremark forallbranches.

Remark 2.7. Let C be a branch given with the parametrization {ϕ1(t),...,ϕn(t)} and

themultiplicitysequencea0,a1,...,ak−1,1,1....Thentheconductorc∗oftheArfclosure oftheringR = kJϕ1(t),...,ϕn(t)K isequaltothesuma0+ a1+ ...+ ak−1.

3. Abound forc∗

Let C be a plane algebroid curve with primitive parametrization {x(t),y(t)} with x(t),y(t) ∈ kJtK, where k is algebraically closed of characteristic 0. With acoordinate changeand interchangingx andy ifnecessary, wecanassumethat

x(t) = tn and y(t) =aiti

withord(y(t))> n andai ∈ k.If β0 := n;β1:= smallest power appearinginy(t),that isnotdivisibleby n;e1 := gcd(β0,β1);continuing inductively,βi := smallest power for

which gcd(β0,β1,...,βi) < gcd(β0,β1,...,βi−1); ei = gcd(β0,β1,...,βi) and eq = 1, the

set{β0,β1,...,βq} iscalledthe characteristic exponents ofC [4].M (n,m) denotingthe

sequenceofdivisorsintheEuclideanalgorithmofn andm,themultiplicitysequenceof

C is

M (β0, β1), M (e1, β2− β1), M (e2, β3− β2), . . . , M (eq−1, βq− βq−1), 1, 1, ... (3.1)

Wefirstgivetheconductorof theArfclosureofaplanebranchintermsofits char-acteristicexponents.

Theorem3.1. LetC be aplane algebroidcurvewith characteristicexponents {β0,β1,...,

βq}.Then

c∗= β0+ βq− 1.

Toprovethis theorem,weneedtwo lemmas.

Lemma3.2.Leta1,a2,. . . ,akbenaturalnumberss.t.gcd(a1,. . . ,ak)= 1.Defineb1= a1,

bi= gcd(bi−1,ai)= gcd(a1,. . . ,ai) (1< i≤ k) inductively.Then,

gcd(b_i−1, ai− ai−1) = bi

Proof. Since bi = gcd(bi−1,ai) and bi−1 = gcd(bi−2,ai−1), we have bi−1 = bih, ai =

bir, ai−1 = bi−1p and bi−2 = bi−1q, where gcd(h,r) = 1 and gcd(p,q) = 1. Then

Thesecond lemmaisthefollowing,and anequivalent isgivenin[7].

Lemma 3.3. (See[7].)Let n and m betwonatural numbers, e= gcd(n,m) and also let di bethedivisorsobtainedby applyingtheEuclideanalgorithm ton andm.In this case



idi= n+ m− e.

Proof of Theorem 3.1. Recalling Remark 2.7 and Eq. (3.1), it is enough to show that thesumM (β0,β1)+ M (e1,β2− β1)+ M (e2,β3− β2)+ . . . + M (eq−1,βq− βq−1) isequal

to β0+ β1− 1. ByLemma 3.3,the sumof themultiplicitiesisβ0+ β1− gcd(β0,β1)+ gcd(β0,β1)+ β2− β1+ gcd(gcd(β0,β1),β2− β1)+ ...+ gcd(β1,...,βq−1)+ βq− βq−1− 1

and byLemma 3.2,this isequalto β0+ βq− 1,whichcompletestheproof. 2

Having determinedc∗ intheplanecase,wecangiveabound forc∗ inthespacecase byusing Theorem 3.1.

Let C be an algebroid curve with embdim(C) > 2 and with local ring R = kJϕ1(t),ϕ2(t),...,ϕs(t)K,where ϕ1(t) = tm11 ϕ2(t) = a21tm21+ a22tm22+ ... + a2r2t m_2r2 .. . ϕs(t) = as1tms1+ as2tms2+ ... + asrst msrs_. m11 ≤ m21 ≤ .... ≤ ms1 and gcd(m11,m21,m22,...,m2r2,...,ms1,ms2,...,msrs) = 1 Then,wecanalwaysdetermineconstantsb2,...,bssuchthatinthesumϕ(t)= b2ϕ2(t)+

b3ϕ3(t)+ ...+ bsϕs(t),noneofthemij’svanish andthegreatestcommondivisorofthe

powersof thetermsofϕ(t) andϕ1(t) isequalto gcd(m11,...,ms1,ms2,...,msrs)= 1.If we considertheplanecurvebranchC with˜ thelocal ringR = k˜ Jϕ1(t),ϕ(t)K,

˜

R⊂ R ⇒ ˜R∗⊂ R∗⇒ WR˜∗⊂ WR∗⇒ ˜c∗≥ c∗.

That is, c∗ is always greater than or equal to the smallest characteristic exponent of ˜

C plus greatest characteristic exponent of C minus˜ 1. As the smallest characteristic exponent is equalto m11 and thegreatest characteristic exponent ofC is˜ lessthan or equaltomsrs,c∗≤ m11+ msrs− 1,andwecanstatethefollowingtheorem,whichgives thebound todeterminetheArfclosurecorrectly.

Theorem 3.4.LetR = kJϕ1(t),ϕ2(t),...,ϕs(t)K be abranch,where

ϕ1(t) = tm11 ϕ2(t) = a21tm21+ a22tm22+ ... + a2r2t m2r2 .. . ϕs(t) = as1tms1+ as2tms2+ ... + asrst msrs

m11 ≤ m21≤ .... ≤ ms1≤ ... ≤ msrs and gcd(m11,m21,...,m2r2,...,ms1,...,msrs)= 1.

Using thebound m11+ msrs for thetruncation ofthe divisionseriesin each blow upis

sufficienttoconstructtheArf Closurecorrectly.

Example3.5.RecallExample 2.6.TocomputetheArfclosureoftheringkJt4_,t6_+t9_,t14_K, wecannow usethebound 4+ 14= 18 whileexpandingthedivisionseriesandwe get:

R0: t4 t6+ t9 t14 R1: t4 t2+ t5 t10 R2: t2− t5+ t8− t11+ t14− t17 t2+ t5 t8− t11+ t14− t17 R3: −2t3+ 3t6− 4t9+ 5t12− 6t15+ 6t18 t2+ t5 t6− 2t9+ 8t12− 4t15+ 4t18 R4: −2t + 5t4− 9t7+ 14t10− 20t13+ 26t16 t2+ t5 t4+ ... AsF0= t4,F1= F2= F3= t2+ t5andF4=−2t+ 5t4− 9t7+ 14t10− 20t13+ 26t16,the ArfClosureisR = k + kt4+ k(t6+ t9)+ kt8+ kJtKt10,andthemultiplicitysequence of thecorresponding branchis: (4,2,2,2,1,1).Hence,theresults arethe samewithwhat wehavefoundinExample 2.6.

4. Hilbertfunctionsoflocalringshaving thesameArf closure

In this section, we present a conjecture of Arslan and Sertöz and give examples supportingthis conjectureobtainedbyusingthealgorithmgiven above.First,we char-acterizetheHilbertfunctionofanArfring.

RecallthattheHilbertfunctionHR(n) ofthelocalringR withthemaximalidealm is

definedtobetheHilbertfunctionoftheassociatedgradedringgrm(R)= ∞ i=0mi/mi+1. Inotherwords, HR(n) = Hgrm(R)(n) = dimR/m  mn/mn+1, n≥ 0.

TheHilbertseries ofR isdefinedtobe

HSR(t) =



n∈N

HR(n)tn.

IthasbeenprovedbyHilbertandSerrethat,HSR(t)= ₍₁h(t)_−t)d,whereh(t) isapolynomial withcoefficientsfrom Z,h(1) isthemultiplicity ofR andd istheKrulldimensionofR.

ItisalsoknownthatthereisapolynomialPR(n)∈ Q[n] calledtheHilbertpolynomialof

R suchthatHR(n)= PR(n) foralln≥ n0,forsomen0∈ N.Thesmallestn0 satisfying thisconditionistheregularityindexoftheHilbertfunctionofR.Wefirstshowthatthe regularityindex oftheHilbertfunctionofanArfringis 1.

Theorem 4.1. (See [21, Theorems 1 and 2].)Let R be a local Cohen–Macaulay ring of dimensiond andmultiplicitye.Then

embdim(R)≤ e − d + 1,

andifthereisequality(R hasmaximalembeddingdimension),thentheassociatedgraded ring of R isCohen–Macaulay.

Theorem 4.2.(See[19,Theorem 2.2].)AnArf ring has maximalembeddingdimension.

Corollary 4.3.The associatedgradedrings of Arfrings are CohenMacaulay.

Proof. ThisisadirectconsequenceofTheorems 4.1 and4.2,as d= 1 forArfrings. 2 We havedim(R)= dim gr(R) andembdim(R)= embdim(gr(R)). Foragraded ring

G and anonzerodivisor x∈ G ofdegree1,we haveembdim(G/x) = embdim(G)− 1,

e(G/x)= e(G),anddim G/x= dim G− 1.Hence,G hasmaximalembeddingdimension ifandonlyifG/x hasmaximalembedding dimension.Furthermore,HSG(t)=HS₁G/x(t)_−t .

(Also,notethattoguaranteetheexistenceofanonzerodivisorofdegree 1,thefieldhas tobeinfinite,andthereisstandardtrickforextendingthefield.)A0-dimensionalgraded ringofmaximalembeddingdimensionandmultiplicity e hasHilbertseries1+(e−1)t,so a1-dimensionalgradedringofmaximalembeddingdimensionhasHilbertseries1+(e_(1−t)−1)t.

As aconsequence,wecanstatethenexttheorem.

Theorem4.4.LetR bealocal ringandR∗itsArfClosure.ThentheHilbertseriesofR∗ is:

PR∗(t) = 1 + (e− 1)t 1− t .

We haveshownthattheregularity index ofthe Hilbertfunctionof anArfringis 1. This shows that,althoughan Arfringisnotgenerally regular,itis verycloseto being regular,sowehavethefollowingquestion:CanweinterprettheArfclosureas aspecific way of taming the singularity? Withthis question inmind and recalling that the Arf closureofaringisobtainedbyenlargingtheringwiththeadditionofnewelementsina certainmanner,wecantrytounderstandtheeffectofaddinganelementontheregularity index of the Hilbert function.The following conjecture due to Arslan and Sertöz says that, while constructing theArf closure, the addition of a new element results with a ringhaving aHilbertfunctionwithasmalleror anequalregularityindex:

Conjecture4.5.If R1 andR2 are twolocal ringshaving thesame Arfclosurewith R1⊂

R2 andPR1(t)= h1(t) 1−t,PR2(t)= h2(t) 1−t ,then wehave degree(h1)≥ degree(h2).

NotethattheregularityindicesofR1andR2aredegree(h1) anddegree(h2).Moreover, theclaimoftheconjectureisnottruefortwoarbitrarylocalrings,oneofwhichcontains theother:

Example 4.6. Consider the rings R1 = kJt10,t15, t17,t18K and R2 = kJt10,t11,t15,

t17,t18K, which do not have the same Arf closure. Although R1 ⊂ R2, we have

PR1(t)=

1+3t+4t2_+2t3

1−t andPR2(t)=

1+4t+4t2_+t4

1−t .

Lastly, we give some examples supporting the conjecture. The next table presents ringshavingtheArfclosure

kJt12, t18, t25, t26, t27, t28, t29, t31, t32, t33, t34, t35K

andtheirHilbertseries.ObservethatwhilegettingclosertotheArfclosure,thedegrees oftheh(t)’s,andsotheregularity indicesneverincrease.

Rings Hilbert Series

kJt12_{, t}18_{, t}25_{, t}26_{K 1 + 3t + 4t}2_{+ 3t}3_{+ t}4 kJt12_{, t}18_{, t}25_{, t}26_{, t}27_{K 1 + 4t + 5t}2_{+ 2t}3 kJt12, t18, t25, t26, t27, t28K 1 + 5t + 5t2+ t3 kJt12, t18, t25, t26, t27, t28, t29K 1 + 6t + 5t2 kJt12_{, t}18_{, t}25_{, t}26_{, t}27_{, t}28_{, t}29_{, t}31_{K 1 + 7t + 4t}2 kJt12_{, t}18_{, t}25_{, t}26_{, t}27_{, t}28_{, t}29_{, t}31_{, t}32_{K 1 + 8t + 3t}2 kJt12, t18, t25, t26, t27, t28, t29, t31, t32, t33K 1 + 9t + 2t2 kJt12_{, t}18_{, t}25_{, t}26_{, t}27_{, t}28_{, t}29_{, t}31_{, t}32_{, t}33_{, t}34_{K 1 + 10t + t}2 kJt12_{, t}18_{, t}25_{, t}26_{, t}27_{, t}28_{, t}29_{, t}31_{, t}32_{, t}33_{, t}34_{, t}35_{K 1 + 11t}

ThenexttablepresentsringshavingtheArfclosure

kJt12, t16+ t30, t20, t31, t33, t34, t35, t37, t38, t39, t41, t42K.

Rings Hilbert Series

kJt12_{, t}16_{+ t}30_{, t}31_{K 1 + 2t + 2t}2_{+ 2t}3_{+ 2t}4_{+ 2t}5_{+ t}6

kJt12, t16+ t30, t20, t31K 1 + 3t + 5t2+ 3t3

kJt12_{, t}16_{+ t}30_{, t}20_{, t}31_{, t}33_{K 1 + 4t + 7t}2

kJt12_{, t}16_{+ t}30_{, t}20_{, t}31_{, t}33_{, t}34_{, t}35_{K 1 + 6t + 5t}2

kJt12_{, t}16_{+ t}30_{, t}20_{, t}31_{, t}33_{, t}34_{, t}35_{, t}37_{, t}38_{, t}39_{, t}41_{, t}42_{K 1 + 11t}

(Here, the Arfclosure computations are done byusing the SINGULAR [17] library “ArfClosure.lib”,whichyoucanfindin[23].ThelibraryusestheArfconstruction algo-rithmgivenabove.)

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