Stratified reduction of singularities of generalized analytic functions

Re . Real Acad. Cienc. Exac as Fis. Na . Se . A-Ma . (2024) 118:4 h ps://doi.o g/10.1007/s13398-023-01486-8 ORIGINAL PAPER S a i ied educ ion o singula i ies o gene alized analy ic unc ions B. Molina-Sampe 1·J. Palma-Má quez2·F. Sanz Sánchez1 Recei ed: 5 Sep embe 2022 / Accep ed: 14 July 2023 © The Au ho (s) 2023 Abs ac Gene alized analy ic unc ions a e na u ally de ined in mani olds wi h bounda y and a e buil om sums o con e gen eal powe se ies wi h non-nega i e eal exponen s. In his pape we deal wi h he p oblem o educ ion o singula i ies o hese unc ions. Namely, we p o e ha a ge m o gene alized analy ic unc ion can be ans o med by a ini e sequence o blowing-ups in o a unc ion which is locally o monomial ype wi h espec o he coo dina es de ining he bounda y o he mani old whe e i is de ined. Keywo ds Blowing-up mo phism ·Reduc ion o singula i ies ·Gene alized powe se ies · P incipializa ion o ideals Ma hema ics Subjec Classi ica ion 14E15 ·14P15 ·16W60 ·32C05 ·32S45 Fi s and hi d au ho s a e pa ially suppo ed by he P ojec “Mé odos asin ó icos, algeb aicos y geomé icos en oliaciones singula es y sis emas dinámicos” (Re .: PID2019-105621GB-100) o he Minis e io de Ciencia in Spain. Fi s au ho is pa ially suppo ed by he “P og ama de becas pos doc o ales DGAPA” o he UNAM in Mexico. The second au ho is pa ially suppo ed by Papii Dgapa UNAM IN110520, by he Is ael Science Funda ion (g an No. 1167/17) and by unding ecei ed om he MINERVA S i ung wi h he unds om he BMBF o he Fede al Republic o Ge many. This p ojec has ecei ed unding om he Eu opean Resea ch Council (ERC) unde he Eu opean Union’s Ho izon 2020 esea ch and inno a ion p og amme (g an ag eemen No. 802107). BB. Molina-Sampe bea iz.molina@u a.es J. Palma-Má quez [email p o ec ed] F. Sanz Sánchez [email p o ec ed]a.es 1Depa amen o de Álgeb a, Análisis Ma emá ico, Geome ía y Topología, Uni e sidad de Valladolid, Valladolid, Spain 2Weizmann Ins i u e o Science, Reho o , Is ael 0123456789().: V,- ol 123 4 Page 2 o 28 B. Molina-Sampe e al. 1 In oduc ion In his pape , a gene alized powe se ies (in n a iables and wi h coe icien s in some ing A) is a powe se ies wi h n- uples o non-nega i e eal numbe s as exponen s and whose suppo is con ained in a ca esian p oduc o nwell-o de ed subse s o R+={ ≥0}.I iswo h o men ion ha his condi ion on he suppo is mo e es ic i e (excep o n=1) han he one used o de ine he Hahn ing A(()),whe eis he g oup Rnwi h he lexicog aphic o de (and whose elemen s a e also called gene alized powe se ies). In oduced and s udied by an den D ies and Speissegge in [6], gene alized powe se ies appea in se e al con ex s. To men ion a ew: as solu ions o di e en ial/ unc ional equa ions; as exp essions o he Riemann ze a- unc ion (o , mo e gene ally, he Di ichle se ies) in a loga i hmic cha ; as asymp o ic expansions o Dulac ansi ion maps o ec o ields (see o ins ance [12,13]); in model heo y and o-minimal geome y ( he pape [6] i sel , o also [17]); as pa ame iza ions o algeb aic cu es in posi i e cha ac e is ic (see o ins ance [18,p.19]). Conside ing eal coe icien s, we ha e a na u al no ion o con e gence o gene alized powe se ies, whose sums p o ide con inuous unc ions on open subse s o he o han Rn +, called gene alized analy ic unc ions. They a e he local pieces o build abs ac ( eal) gene - alized analy ic mani olds, in oduced and de eloped by Ma ín, Rolin and Sanz in [14]. Mo e p ecisely, a gene alized analy ic mani old is a locally inged space M=(M,GM),whe eM is a opological mani old wi h bounda y and GMis a shea o con inuous unc ions locally isomo phic o he shea o gene alized analy ic unc ions on open subse s o Rn +. Sec ions o he shea GMa e called hemsel es gene alized analy ic unc ions on M. The main esul in [14] es ablishes he local educ ion o singula i ies o gene alized ana- ly ic unc ions, in he spi i o Za iski’s local uni o miza ion heo em o algeb aic a ie ies [19] o Hi onaka’s e sion o analy ic a ie ies [10]. The s a emen , o mula ed in analo- gous e ms o hose used in Bie s one–Milman’s pape [4] o eal analy ic unc ions, is he ollowing: Local Monomializa ion Theo em [14]. Le be a gene alized analy ic unc ion on Mand le p∈M. Then he e exis s a neighbou hoodU0o pin M, ini ely many sequences o local blowing-ups {πi:Mi→U0} i=1and compac se s Li⊂Misa is ying ha iπi(Li)is a neighbou hood o pandsuch ha , o e e yi, he o al ans o m i= ◦πiis o monomial ype a e e y q∈Li(i.e., o some coo dina es x=(x1,x2,...,xn)cen e ed a q,weha e i=xαU(x)whe e U(0)= 0). The cen e s o blowing-ups in each sequence πiha e no mal c ossings wi h he bounda y, bu hey a e de ined only in some open se s o he co esponding mani old. In he s anda d eal analy ic case, we ha e s onge global monomializa ion esul s ( ypically called Reduc ion o Singula i ies,see[1,5,7]). They consis , essen ially, in ha in he abo e s a emen , we can ake jus a single sequence ( =1) and he cen e s o blowing-ups a e globally de ined closed analy ic submani olds, ha ing no mal c ossings wi h he bounda y. Such a global esul is no known so a o gene alized analy ic unc ions. The e a e wo main di icul ies ela ed o he e y no ion o a blowing-up mo phism in he ca ego y o gene alized analy ic mani olds. On he one hand, a blowing-up depends on he local coo dina es ha we use o de ine i . Mo e in insically, a blowing-up is no uniquely de ined and depends on he choice o a s anda diza ion o he mani old (o a leas o an open neighbou hood o he cen e o blowing-up). Roughly, a s anda diza ion is a subshea OMo GMsuch ha (M,OM)is a eal analy ic s anda d mani old and om which he shea GMcan be eco e ed by a na u al comple ion adding gene alized se ies (see [14], we ecall his no ion below). Secondly, al hough e e y gene alized analy ic mani old is locally s anda dizable, 123 S a i ied educ ion o singula i ies... Page 3 o 28 4 he e may exis closed submani olds which do no admi s anda dizable neighbou hoods; i.e., such submani olds canno be “geome ic” cen e s o a blowing-up (c . [14,Example 3.20]). Mo ally, a p ocedu e o educ ion o singula i ies o gene alized analy ic unc ions would need o gua an ee ha , in he p ocess, all closed cen e s suscep ible o be blown-up ha e s anda dizable neighbou hoods. I his is al eady p o ed and Yis such a cen e , one needs o show u he mo e ha , among he di e en s anda diza ions a ound Y, he e exis s o which he co esponding blowing-up π: M→M educes he “complexi y” o he unc ion. In his pape , we o e come hese di icul ies o ob ain an in e media e s ep owa ds a global esul , he so-called s a i ied educ ion o singula i ies. Le us explain i . Fi s , we ecall ha , byi s e yde ini ion, hebounda y ∂Mo a gene alizedanaly ic mani oldis ano malc ossing di iso ; i.e., ∂Mis locally gi en by a ini e union o coo dina e hype planes. Mo eo e , he numbe o such hype planes a each poin p o ides a na u al s a i ica ion o Mby (s anda d) analy ic mani olds. A gene alized analy ic unc ion :M→Ris said o be o s a i ied monomial ype i o any gi en p∈M,i Sis he s a um whe e pbelongs, he e exis s a local cha (x=(x1,x2,...,xe), y)cen e ed a psa is ying S={x1=x2= ···= xe=0} and o which (x,y)=xαU(x,y), whe e α∈Re +and U(0,y)≡ 0. Thus, equi ing a unc ion o be o s a i ied monomial ype means o equi e ha i is o monomial ype only wi h espec o he gene alized coo dina es de e mining equa ions o he componen s o he bounda y. In pa icula , he condi ion is emp y i p/∈∂M. Also, i is au oma ic i Shas codimension e=1, aking in he abo e de ini ion α o be he minimum o he suppo o he se ies de ining wi h espec o he single a iable x=x1. Ou main esul may be s a ed now as ollows. Theo em 1.1 (S a i ied Reduc ion o Singula i ies) Le M=(M,GM)be a gene alized analy ic mani old and le :M→Rbe a gene alized analy ic unc ion. Le p ∈M and assume ha he ge m o a p is no iden ically ze o. Then, he e exis a neighbou hood Vp o p in M and a sequence o blowing-ups (M ,GM )π −1 →(M −1,GM −1)π −2 → ··· π1 →(M1,GM1)π0 →(Vp,GM|Vp) such ha he pull-back := ◦π0◦···◦π −1∈GM (M )is o s a i ied monomial ype. Mo eo e , he cen e o each blowing-up πj, wi h j =0,1,..., −1, can be chosen o be he closu e o a codimension wo s a um in Mj,whe eM 0:= Vp. Ou p oo o Theo em 1.1 is cons uc i e in he sense ha each cen e , as well as he s an- da diza ion used o de ine he espec i e blowing-up a each s ep, can be gi en explici ly in e ms o he exp ession o in some ini ial coo dina es o Ma p. Mo eo e , each blowing- up mo phism is locally exp essed as a pu ely monomial map be ween wo domains o Rn + in sui able cha s. Consequen ly, all he p ocess o s a i ied educ ion o singula i ies can be desc ibed using only combina o ics om he s a ing da a gi en simply by he minimal suppo (see Sec .2below) o a gene alized powe se ies ep esen ing a p. The da um o minimal suppo is closely ela ed o ha o he New on polyhed on o a unc ion in he s anda d analy ic case and he e o e, ou esul should be compa ed wi h he combina o ial educ ion o singula i ies s a ed in Molina’s pape [15]. Al hough i has been a sou ce o inspi a ion o us, we canno apply di ec ly he esul s in [15], mos ly because he e is no good no ion o “mul iplici y” in he gene alized non-s anda d si ua ion (any powe unc ion wi h posi i e eal exponen in a gene alized a iable is a genuine change o a iables). 123 4 Page 4 o 28 B. Molina-Sampe e al. We wan o obse e ha Theo em 1.1 is al eady p o ed o dim M=3 in Palma’s pape [16], bu wi h a di e en s a egy o he choice o he sequence o blowing-ups ( o ins ance, he cen e s o blowing-ups may be ei he co ne poin s o closu es o one-dimensional s a a). The pape is s uc u ed as ollows. In Sec . 2we summa ize he basic no ions and p ope ies o gene alized powe se ies and o he ca ego y o gene alized analy ic mani olds, using he men ioned e e ences [6]and [14]. We emphasize he no ion o s anda diza ion, which is c ucial o de ine blowing-ups. In Sec . 3we in oduce he ca ego y o monomial (gene alized o s anda d) analy ic mani- olds. The objec s o his subca ego y a e mani olds ha ing a leas one co ne and equipped wi h an a las o local cha s cen e ed a each co ne poin o which he change o coo dina es is exp essed as a monomial map be ween domains o he local model Rn +. We ep esen hese changes o coo dina es by means o a amily o ma ices o exponen s ( o a simila ea men see o ins ance [2,3,16]), a combina o ial da a which codi ies uniquely he s uc u al shea o he mani old. We de ine also he class o monomial mo phisms and he class o monomial s anda diza ions o monomial mani olds. A e a blowing-up using such a s anda diza ion wi h a cen e which is he closu e o a s a um (a so-called combina o ial cen e ), we ob ain again a monomial mani old and he blowing-up mo phism is a monomial mo phism. The main esul in his sec ion is he abundance o monomial s anda diza ions (P oposi ion 3.16 below). Fu he mo e, we can choose such a monomial s anda diza ion wi h a p esc ibed local exp ession a a gi en co ne poin . Mo ally, local s a egies o educ ion o singula i ies a e suscep ible o be “globalized”. We end his sec ion by in oducing a special class o monomial mani olds, hose ob ained om a gi en one by a sequence o blowing-ups wi h combina o ial cen e s, and using only monomial s anda diza ions. Such a sequence is called a monomial s a and he amily o such s a s is called he monomial “ oû e é oilée”, a e minology ha e okes he one in oduced by Hi onaka in [10,11] o sequences o local blowing-ups in complex analy ic geome y. In Sec . 4we p o ide a p oo o he main Theo em 1.1. Fi s ly, we p o e a esul abou p incipaliza ion o ini ely gene a ed monomial ideal shea es in a gi en monomial mani old. This esul (see Theo em 4.5 below) can be seen as a e sion o ou ca ego y o a well known esul on p incipaliza ion o ideals in he algeb aic o s anda d analy ic si ua ion (see o ins ance Gowa d’s pape [9] o a simple p oo , o see also Fe nández-Duque’s pape [8] o a simila s a emen conce ning he esonances elimina ion o singula i ies o codimension- one analy ic olia ions). Taking in o accoun ha i su ices o ob ain he p incipaliza ion only a he co ne poin s, such a esul can also be ega ded as a globaliza ion o he algo i hm desc ibed in an den D ies and Speissegge ’s pape (see [6, Lemma 4.10]) ha educes he numbe o elemen s in he minimal suppo o a gene alized powe se ies by monomial ans o ma ions o he a iables. Al hough we use ce ain elemen s and a gumen s o ha esul , and despi e o wha we ha e said abo e conce ning he possibili y o globalize a “local s a egy”, ou p oo he e equi es a di e en con ol in a ian . Once we ha e he p incipaliza ion o monomial ideal shea es, he main heo em is con- cluded easily in he case we s a wi h a co ne poin p∈M. In his case, he sequence π0◦π1◦···π −1 o Theo em 1.1 is ac ually a s a in he oû e é oilée o e he ge m o M a p. Finally, he gene al case p∈∂Mis educed o he case o a co ne poin , using ha a ound p he e is a p oduc s uc u e o a neighbou hood o a co ne poin imes a s anda d analy ic mani old wi hou bounda y. 123 S a i ied educ ion o singula i ies... Page 5 o 28 4 2 P elimina ies Wesumma izehe e hebasicno ionsabou heca ego yo gene alizedanaly ic mani oldsand blowing-up mo phisms in i , in oduced by Ma ín, Rolin and Sanz in [14]. These mani olds a e buil om con e gen gene alized powe se ies, ex ensi ely s udied in a pape by an den D ies and Speissegge [6]. 2.1 Fo mal and con e gen gene alized powe se ies Deno e by R+=[0,∞). Tuples o a iables a e deno ed by X,Y,Z, e c., and we implici ly assume ha uples wi h di e en name ha e no common a iables. I Xhas ncomponen s, we say ha Xis an n- uple and so on. Le X=(X1,X2,...,Xn)be an n- uple o a iables and le Abe an in eg al domain. A o mal gene alized powe se ies wi h coe icien s in A in he a iables Xisamaps:Rn +→A, w i en as s= λ∈Rn + sλXλ,whe e Xλ=Xλ1 1Xλ2 2···Xλn n o λ=(λ1,λ 2,...,λ n) and sλ:= s(λ) ∈A, such ha i s suppo Supp(s):= {λ∈Rn +:sλ= 0}is con ained in a ca esian p oduc o nwell-o de ed subse s o R. The se o all such o mal gene alized powe se ies, deno ed by A[[X∗]], wi h he usual addi ion and p oduc ope a ions o powe se ies has an s uc u e o an A-algeb a which is also an in eg al domain. Mo eo e , i Ais a ield, hen A[[X∗]] is a local algeb a (see [6, Co olla y 5.6]), wi h maximal ideal gi en by m={s∈A[[X∗]] : s0=0}. No e ha A[[X∗]] is no noe he ian, in ac , he ideal mis ne e ini ely gene a ed. The minimal suppo o a powe se ies s∈A[[X∗]] is he subse Suppmin(s)⊂Supp(s) composed o he minimal uples o Rn +wi h espec o he (pa ial) di ision o de ≤d, ha is (λ1,λ 2,...,λ n)≤d(μ1,μ 2,...,μ n)i and only i λi≤μi, o alli∈{1,2,...,n}. The condi ion imposed on he suppo o a powe se ies sallows o show ha he mini- mal suppo Suppmin(s)is ini e (see [6, Lemma 4.2]). As a consequence, sadmi s a ini e monomial p esen a ion: s= λ∈Suppmin(s) XλUλ(X), whe e Uλ∈A[[X∗]] sa is ies Uλ(0)= 0, o any λ∈Suppmin(s). Deno e by m(s)= #Suppmin(s). When m(s)=1 o , equi alen ly, he monomial ep esen a ion o shas a single e m, we say ha sis o monomial ype. In his pape , we a e in e es ed in eal gene alized powe se ies, ha is A=R, bu we use di e en ings when we wan o dis inguish some a iables and pu he o he s in o he coe i- cien s. To be p ecise, i Yand Za e uples o kand n−k a iables, espec i ely, we conside R[[(Y,Z)∗]] as a p ope R-subalgeb a o R[[Y∗]][[Z∗]] by he na u al monomo phism s= (λ,μ)∈Rn + aλμYλZμ→ sZ= μ∈Rn−k + AμZμ,whe e Aμ= λ∈Rk + aλμYλ.(1) I p :Rn→Rn−kdeno es he na u al p ojec ion on o he las n−kcoo dina es, o any powe se ies s∈R[[(Y,Z)∗]] we ha e he inclusion Suppmin(sZ)⊂p (Suppmin(s)),andas 123 4 Page 6 o 28 B. Molina-Sampe e al. a consequence we ge he inequali y m(sZ)≤m(s). (2) Le us w i e R[[Y,Z∗]] o deno e he subalgeb a o R[[(Y,Z)∗]] composed by he so- called eal mixed powe se ies: hose o mal eal gene alized powe se ies sin he a iables (Y,Z), such ha he inclusion Supp(s)⊂Nk×Rn−k +holds, o equi alen ly, such ha sZ∈R[[Y]][[Z∗]]. Gi en an n- uple o a iables Xand a poly adius ρ=(ρ1,ρ 2,...,ρ n)∈Rn >0,deno eby R{X∗}ρ he subalgeb a o R[[X∗]] consis ing on hose powe se ies s o which sρ:=  λ∈Supp(s) |sλ|ρλ<∞. The union o he R{X∗}ρalong all he possible poly adius ρ∈Rn >0is again a subalgeb a R{X∗}⊂R[[X∗]], and i s elemen s a e called ( eal) con e gen gene alized powe se ies. We ha e ha R{X∗}is also a local algeb a, whose maximal ideal is gi en by m∩R{X∗}. I Y,Za e uples o kand n−k a iables, espec i ely, and ρ∈Rn >0is a poly adius, an elemen s∈R[[Y,Z∗]] ∩ R{(Y,Z)∗}ρgi es ise o a con inuous unc ion s:Pρ k,n−k→R x=(x1,x2,...,xn)→ λsλxλ,(3) whe e Pρ k,n−k=(−ρ1,ρ 1)×(−ρ2,ρ 2)×···×(−ρk,ρ k)×[0,ρ k+1)×···×[0,ρ n)⊂ Rk×Rn−k +,called hesum o he powe se ies s. Mo eo e , sis ealanaly ica anypoin in he in e io o Pρ k,n−kand i s ge m a 0∈Rnis uniquely de e mined by he se ies s.Wede ine he con e gen mixed powe se ies o be he elemen s o R{Y,Z∗}:=R[[Y,Z∗]] ∩ R{(Y,Z)∗}. 2.2 S anda d and gene alized analy ic mani olds Le Vbe an open subse o Rn +and le g:V→Rbe a con inuous unc ion. Gi en a poin p=(p1,p2,...,pn)∈V, conside Ip:= {i:pi=0}⊂{1,2,...,n}, and pu =#Ip and k=n−.Wesay ha gis gene alized analy ic (o jus G-analy ic)a pi he e exis s s∈R{Y,Z∗},whe eYis a k- uple and Zis an - uple, such ha o any x=(x1,x2,...,xn) in a su icien ly small neighbou hood o 0in Rk×R +,weha e g(p1+xσ(1),p2+xσ(2),...,pn+xσ(n))= s(x1,x2,...,xn), whe e σis a pe mu a ion o he se {1,2,...,n}sa is ying he ela ion j∈Ipi and only i σ(j)∈{k+1,k+2,...,n}. We say ha gis gene alized analy ic in V i i so a e e y poin pin V. In he de ini ion abo e, he se ies sis uniquely de e mined by he ge m o ga p, up o pe mu a ion o he a iables Yand Z, sepa a ely. Thus, he se o ge ms o gene alized analy ic unc ions a pde ines an R-algeb a isomo phic o R{Y,Z∗}. On he o he hand, i g is a gene alized analy ic unc ion a some poin p∈Rn +, hen i is so in a neighbou hood o p in Rn +. Summa izing, he assignmen Gn:V→ Gn(V),whe eVis an open subse o Rn +and Gn(V)is he se o gene alized analy ic unc ions in V, is a shea o R-algeb as o con inuous unc ions o e Rn +, whe e he s alks Gn,pa e local algeb as. Mo eo e Gncon ains he shea Ono analy ic unc ions, whe e On(V)is he R-algeb a o eal unc ions in Vwhich ex end o eal analy ic unc ions on some open neighbou hood o Vin Rn. Wi h his o malism, and aking as local models he locally inged spaces On:= (Rn +,On) and Gn:= (Rn +,Gn), we de ine bo h he ca ego ies o s anda d and gene alized ( eal) 123 S a i ied educ ion o singula i ies... Page 7 o 28 4 analy ic mani olds (wi h bounda y and co ne s). The objec s in hese ca ego ies a e called O-mani olds and G-mani olds, espec i ely. In o de o ea bo h oge he we w i e A o make e e ence ei he o Oo o G,andA o e e ei he o Oo G.AnA-mani old o dimension nis a locally inged space M=(M,AM),whe eMis a second coun able Hausdo opological space ( he unde lying space)andAMis a subshea o he shea C0 Mo ge ms o con inuous eal unc ions on M( he s uc u al shea ), which is locally isomo phic o he local model An. Tha is, gi en p∈M he e is an open neighbou hood Vo pin M, an open subse U o Rn +and a homeomo phism ϕ:V→Uinducing an isomo phism o he locally inged spaces (ϕ, ϕ#):(V,AM|V)∼ −→ (U,An|U), whe e ϕ# p:An,ϕ(p)→AM,pis gi en by he composi ion g→ g◦ϕ(as ge ms). A mo phism be ween wo A-mani olds is jus a mo phism as locally inged spaces, induced by composi ion wi h con inuous maps on he unde lying spaces (wi h an abuse o language, we equen ly iden i y mo phisms wi h he co esponding con inuous maps). A couple (V,ϕ)in he abo e condi ions is called a local cha o Ma p, he componen s x=(x1,x2,...,xn) o he isomo phism ϕ:V→Ua e local coo dina es a p, and a amily o local cha s {(Vj,ϕj)}j∈Jsuch ha M=∪ j∈JVjis an a las o M. Le M=(M,AM)be an A-mani old. No e ha he unde lying space Mis a opological mani old wi h bounda y, deno ed by ∂M, and ha he es ic ion (M ∂M,AM|M ∂M)is a s anda d analy ic mani old wi hou bounda y (consequen ly, gene alized analy ic mani olds wi hou bounda y a e also s anda d). Also he e is a na u al s a i ica ion SMo Mdesc ibed as ollows. I p∈M,and(V,ϕ)is a local cha a p, he numbe epo anishing coo dina es in ϕ(p)(equal o #Iϕ(p)) does no depend on he local cha (V,ϕ)chosen (see [14]). In ha way, he e is a well-de ined map e:M→{0,1,...,n},p→ ep, which is uppe semi-con inuous. The elemen s o SMa e he connec ed componen s o he ibe s o e.Gi enS∈SM, le us w i e eS:= ep,whe epis any poin in S. Obse e ha (S,AM|S)is a s anda d analy ic mani old o dimension n−eS. In pa icula , he bounda y ∂Mco esponds exac ly wi h he poin s p∈Mwi h ep>0, ha is, ∂Mis equal o he union o s a a o dimension s ic ly smalle han n.Weha ealso ha ,∂Mis a no mal c ossings di iso wi h espec o he s uc u al shea . Tha is, o each p∈∂M, he e exis s a local cha (V,ϕ)o Ma psuch ha ∂M∩V={q∈V:x1(q)·x2(q)···· ·xep(q)=0}, whe e (x1,x2,...,xn)a e he coo dina es associa ed o ϕ. Example 2.1 Le ¯ Okbe he shea o eal (s anda d) analy ic unc ions in Rk. The locally inged space (Rk,¯ Ok)is a gene alized and s anda d analy ic mani old, wi h a single cha ψk:Rk→(0,∞)kde ined by (a1,a2,...,ak)→ (ea1,ea2,··· ,eak). We obse e a his poin ha he p oduc is de ined in he ca ego y o A-mani olds. Tha is, gi en wo gene alized o s anda d analy ic mani olds M1=(M1,AM1)and M2= (M2,AM2)o dimensions nand m, espec i ely, he e is a na u al A-mani old o dimension n+m, ha we deno e by M1×M2=(M1×M2,AM1×M2), unique up o isomo phism, sol ing he “p oduc uni e sal p ope y”. Wi hou oo much de ail, he shea AM1×M2is cons uc ed as ollows. Gi en a poin (p,q)∈M1×M2and wo coo dina e cha s ϕ1: 123 4 Page 8 o 28 B. Molina-Sampe e al. V1→U1and ϕ2:V2→U2a pand q espec i ely, we ha e ha AM1×M2,(p,q)={ ◦(ϕ1×ϕ2)(p,q): ∈An+m,(p,q)}, whe e (p,q)=(ϕ1(p), ϕ2(q)). Example 2.2 The p oduc (Rk,¯ Ok)×(Rn−k +,An−k),whe eA∈{O,G}, has a na u al s uc- u e o A-mani old by means o he homeomo phism ψk×id, whe e ψkhas been in oduced in Example 2.1. We e e o his p oduc by w i ing (Rk×Rn−k +,Ak,n−k). Rema k 2.3 Le us conside a poin p∈Mwi h ep=kand le (V,ϕ) be a local cha o Ma p. Up o pe mu a ion, we can assume ha ϕ(p)=(a1,a2,...,ak,0,...,0)wi h ai= 0 o alli∈{1,2,...,k}. We can spli he local coo dina es xde ined by ϕin wo g oups x=(y,z),whe ey=(y1,y2,...,yk)a e s anda d analy ic unc ions a pand z=(zk+1,zk+2,...,zn)a e gene alized unc ions. By means o ansla ions y i=yi−ai in he analy ic coo dina es we ob ain a new isomo phism ϕ:V→ (ψk×id)−1(ϕ(V)) ⊂Rk×Rn−k +. We conside also ϕas a coo dina e cha cen e ed a p in he sense ha ϕ(p)=0∈ Rk×Rn−k +, and we usually assume ha ou cha s a e cen e ed cha s. Le us ecallnow heexp essionincoo dina eso hecon inuousmapsinducingmo phisms o gene alized unc ions (de ails in [14, P oposi ion 3.16]). Conside wo gene alized analy ic mani oldsM1=(M1,GM1)andM2=(M2,GM2)and acon inuous unc ionφ:M1→M2 inducing a mo phism be ween M1and M2.Gi enp∈M1and q=φ(p)∈M2, ake (Vp,ϕp),(Wq,ψ q)cha scen e eda pandq, espec i ely.Followingno a ioninRema k2.3, deno e by yand z he ks anda d and n−kgene alized coo dina es de ining ϕp, espec i ely. Up o pe mu a ion, we can assume also ha he i s kcoo dina es de ining ψqa e s anda d and he o he n−ka e gene alized. Then, he j- h componen ˜ φjo ˜ φ=ψq◦φ◦ϕ−1 pis a gene alized analy ic unc ion and o j=k+1,k+2,...,n,weha e ha ˜ φj=zλjUj(y,z), Uj(0,0)= 0,λ j∈Rn−k + {0}.(4) Mo eo e , i φinduces an isomo phism, we ha e ha φis a homeomo phism, n=n, k=k, hemap ∈Rk→ (˜ φ1( ,0), ˜ φ2( ,0),..., ˜ φk( ,0)) is an analy ic isomo phism, and, i we w i e λj=(λj,1,λj,2,...,λj,n−k)in Eq. (4), up o a pe mu a ion o coo dina es zwe ha e λj,j−k>0,λ j, =0,∈{1,2,...,n−k} { j−k},(5) o all j=k+1,k+2,...,n. We end his sec ion in oducing some no a ion and de ini ions conce ning he s a a o he na u al s a i ica ion SM.Gi enas a umSin SM,deno ebyS he closu e o Sin M,and de ine dim(¯ S):= dim(S). We w i e ZM:= {S⊂M:S∈SM}.Fo j=0,1,...,n, deno e by Zj M he se o elemen s in ZMwi h dimension j, ha is Zj M={¯ S∈ZM:eS=n−j}. The elemen s o Z0 Ma e he s a a o dimension 0, and a e called co ne poin s, he elemen s o Z1 Ma e called edges and he elemen s o Zn−1 Ma e called componen s o ∂M.No e ha ∂Mis he union o i s componen s. Fo each Z∈ZM, we deno e by ZM(Z) he subse o ZMwhose elemen s a e con ained in Z, and o each j=0,1,...,n, we w i e Zj M(Z)=ZM(Z)∩Zj M. We w i e o sho 123 S a i ied educ ion o singula i ies... Page 9 o 28 4 p∈Z0 Mins ead o {p}∈Z0 M, and when no con usion a ises, we will pu Zins ead o ZM, Zjins ead o Zj M,e c. 2.3 Monomial complexi y along s a a We in oduce in his sec ion he concep o monomial complexi y along a s a um and he de ini ion o s a i ied monomial ype unc ion. Le us conside a gene alized analy ic mani old M=(M,GM)and a s a um So i s na u al s a i ica ion S. Take a local cha (V,ϕ)o Mcen e ed a some p∈S, w i e e=eS and k=dim S=n−e. We can spli he coo dina es de ining ϕ, up o eo de hem, as (y,z), whe ey=(y1,y2,...,yk)a es anda danaly iccoo dina esin S∩Vandz=(z1,z2,...,ze) a e gene alized unc ions such ha S∩V={q∈V:z1(q)=z2(q)=··· = ze(q)=0}. Sh inking Vi necessa y, he cha ϕp o ides an isomo phism p ϕ:R{Y,Z∗}→GM,p,s→ s◦ϕ, whe e Yand Za e kand e uples, espec i ely, and sis he sum o he powe se ies s in oduced in Eq. (3). Gi en ∈GM,pand s∈R{Y,Z∗} he mixed powe se ies such ha p ϕ(s)= , we deno e SuppS( ;ϕ) =Supp(sZ)⊂Re +,Suppmin,S( ;ϕ) =Suppmin(sZ)⊂Re +,(6) whe e sZ∈R{Y}{Z∗}has been in oduced in Eq. (1). Lemma 2.4 Le S be a s a um in Swi h e =eS. Take an open subse U o M such ha U∩S=∅, and a unc ion ∈GM(U). Conside wo local cha s (V1,ϕ 1)and (V2,ϕ 2), cen e ed a p and q espec i ely, wi h p,q∈S∩U. The e exis s a uple (γ1,γ 2,...,γ e)∈ Re >0such ha (λ1,λ 2,...,λ e)∈Suppmin,S( q;ϕ2)i and only i (γ1λ1,γ 2λ2,...,γ eλe)∈ Suppmin,S( p;ϕ1). P oo Using ha Sis pa h connec ed and by compac ness o a gi en pa h om p o q,we can educe he p oblem o he case whe e bo h poin s pand qbelong o he same connec ed componen Wo U∩V1∩V2. W i e y=(y1,y2,...,yn−e),z=(z1,z2,...,ze),and also ¯ y=(¯y1,¯y2,..., ¯yn−e),¯ z=(¯z1,¯z2,...,¯ze), whe e, up o eo de ing, (y,z)a e he coo dina e unc ions associa ed o ϕ1and (¯ y,¯ z)a e he ones associa ed o ϕ2,insuchaway ha y|S∩V1,¯ y|S∩V2a e analy ic coo dina es in W∩S. Tha is, we ha e W∩S={z1=z2=···= ze=0}={¯z1=¯z2=···= ¯ze=0}. In iew o Eqs. (4)and(5), up o eo de ing he a iables z, he change o coo dina es ϕ2◦ϕ−1 1sa is ies, o any j=1,2,...,n−eand =1,2,...,e, ha ¯yj=gj(y,z),and ¯z=zγ h(y,z),whe egj,ha e gene alized analy ic unc ions such ha y→ gj(y,0)is a s anda d analy ic non-cons an unc ion, γ>0andh(0,0)= 0. We summa ize hese exp essions by w i ing ¯ y=gand ¯ z=zγh.I 2:= Suppmin,S( q;ϕ2)={μ1,μ 2,...,μ }, he exp ession o in coo dina es (¯ y,¯ z)is |W=¯ zμ11(¯ y,¯ z)+¯ zμ22(¯ y,¯ z)+···+¯ zμ  (¯ y,¯ z), whe e j(¯ y,0)≡ 0, o any j=1,2,..., . Applying he change o coo dina es in o de o ge he exp ession o in (y,z),weob ain |W=zγμ 11(y,z)+zγμ 22(y,z)+···+zγμ  (y,z), k(y,z)=hμkk(g,zγh), 123 4 Page 16 o 28 B. Molina-Sampe e al. 3.3 Abundance o s anda diza ions o monomial mani olds In his sec ion, we de ine m-s anda diza ions, we gi e a cha ac e iza ion o hei combina- o ial da a and we p o e a esul o abundance o m-s anda diza ions o a ixed monomial G-mani old. Le us ix a monomial gene alized analy ic mani old (M,a).Alocal m-s anda diza ion o (M,a)a a co ne poin pis jus an m-cha upde ined in he whole open se V p,such ha i xp∈a, henup◦x−1 pis gi en by monomial ela ions o he o m up,i=xαp,i p,i,whe e αp,i∈R>0, o all i∈Ip.(12) We ep esen his change o coo dina es by means o he map αp:Ip→R>0de ined by i→ αp,i. In ha way, he change o coo dina es up◦x−1 pis codi ied by he ma ix o exponen s Dαp:Ip×Ip→R>0, whe e we ecall ha (once an o de in Ipis ixed) Dαpis a diagonal ma ix wi h he elemen s αp,iin he diagonal. De ini ion 3.12 Anm-s anda diza ion o (M,a)is a pai (O,b),whe eOis a s anda diza ion o Mand b={up}p∈Z0is a monomial a las o N=(M,O)such ha upis a local m- s anda diza ion o (M,a) o e e y co ne poin p∈Z0.Thecombina o ial da a o an m-s anda diza ion (O,b)is he collec ion o maps (O,b)={αp}p∈Z0. Rema k 3.13 I (O,b)and (O,b)a e m-s anda diza ions o (M,a), henbnecessa ily ha b=bas we ha e al eady no ed in Rema k 3.3. No e also ha he m-s anda diza ion (O,b) is comple ely de e mined by he combina o ial da a (O,b). Lemma 3.14 A collec ion o maps ={αp:Ip→R>0}p∈Z0is he combina o ial da a o an m-s anda diza ion o (M,a)i and only i o any pai o co ne poin s p,q∈Z0 he ollowing ela ions hold: αp, =γpq αq,, o all ∈Ip∩Iq,(13) whe e γpq is he weigh connexion unc ion om p o q. P oo Le us assume i s ha =(O,b),whe e(O,b)is an m-s anda diza ion o (M,a). Le us deno e N=(M,O)and le C(N,b)be he combina o ial da a o he monomial s anda d analy ic mani old (N,b).In iewo Eq.(11), i is enough o p o e Eq. (13) o wo co ne poin s pand qconnec ed h ough a compac edge Y. Le us conside he m-cha s up,uq∈b a pand q, espec i ely. The change o coo dina es up◦u−1 qis codi ied by a ma ix o exponen s A=Apq Y∈C(N,b). This change mus be s anda d analy ic in i s domain o de ini ion uq(V p∩V q)=R×Rn−1 +, and his implies Ai∈Z+,(A−1)j∈Z+, o all i∈Iq,j∈Ip,∈IY.(14) Le C=Cpq Y∈C(M,a)and αp,α q∈(O,b).No e ha Ais ob ained as he p oduc A=DαqCD−1 αp:Iq×Ip→R. When ∈Ip∩Iq=IY, in iew o Lemmas 3.7 and 3.9,weha e A =γpq αq, αp, ∈Z+,(A−1) =(Aqp Y) =γqp αp, αq, =αp, γpq αq, =1/A ∈Z+, 123 S a i ied educ ion o singula i ies... Page 17 o 28 4 which shows A =(A−1) =1. F om he e we ge αp, =γpq αq,, and hence sa is ies Eq. (13)aswewan ed. Assumenow ha sa is iesEq.(13) o anypai o co ne poin s p,q∈Z0.A eachco ne poin p∈Z0, conside he m-cha upde ined on V psuch ha he change o coo dina es up◦x−1 psa is ies up,j=xαp,j p,j, o all j∈Ip,whe eαp∈and xp∈a.In ha way,we ge a new monomial a las b={up}p∈Z0o M. Le us see ha he changes o coo dina es uq◦u−1 pa e s anda d analy ic o any pai o co ne poin s pand q.In iewo Eq.(9)i is enough o suppose ha pand qa e connec ed h ough an edge Y. De ining he ma ix A=DαqCD−1 αp, he change o coo dina es uq◦u−1 pis gi en by uq,i= j∈Ip uAij p,j, o any i∈Iq. I su ices o show ha Asa is ies he condi ions in Eq. (14). Indeed, i Ai∈Z+ o i∈Iq and o all ∈IY, henuq,iin he abo e equa ion is s anda d analy ic in e ms o he a iables upin he domain V p∩V q={up,ip= 0}∩{uq,iq= 0}( he same in e changing pand qi (A−1)j∈Z+ o j∈Iqand any ∈IY). Applying Lemma 3.7 we ge ha A =0and (A−1) =0, o all ,∈IYwi h = . Mo eo e , he same lemma assu es ha Ciq=0 and ha (Cip)−1=Cqp ip=0, o all ∈IY; hence, o any such index ∈IYwe ob ain Aiq=Ciqαq,iq/αp, =0,(A−1)ip=(Cip)−1αp,ip/αq, =0. Again by Lemma 3.7 we ge A =Cαq, αp, =γpq αq, αp, ,(A−1) =(C−1)αp, αq, =γqp αp, αq, , o all ∈IY. Using Lemma 3.9 and Eq. (13) we conclude A =(A−1) =1. As a conclusion, he a las bde ines a s anda d analy ic s uc u e N=(M,O)o e M,whe e M=(M,GM); hus O⊂GMis a s anda diza ion o M. Mo eo e , by de ini ion o b,we ha e ha (O,b)is an m-s anda diza ion o (M,a)wi h (O,b)=. In he sequel, a collec ion o maps ={αp:Ip→R>0}p∈Z0is called ealizable o (M,a) i Eq. (13) holds o any pai o co ne poin s p,q∈Z0. De ini ion 3.15 Le upbe a local m-s anda diza ion o (M,a)a a gi en co ne poin p∈Z0. An ex ension o upis a (global) m-s anda diza ion (O,b)o (M,a)such ha up∈b;we sayalso ha (O,b)ex endsup.Wedeno ebyE(up) he se o ex ensions o up. P oposi ion 3.16 Le (M,a)be a monomial gene alized analy ic mani old. Then: (a) The e is a bijec ion be ween he se o m-s anda diza ions o (M,a)and RN >0,whe eN is he numbe o bounda y componen s o ∂M. (b) Gi en a co ne poin p ∈Z0and a local m-s anda diza ion upa p, he e is a bijec i e map RN−n >0→E(up), whe e n is he dimension o M. P oo We s a wi h he p oo o he i s asse ion (a). Le Ibe he se o indices labelling he componen s o ∂M, ha is∂M=i∈IEi,whe eN=#I, and le us ix a collec ion o co ne poin s q={qi}i∈Iin such a way ha qi∈Ei o each i∈I. Gi en a map β∈RI >0, we ake β={αp:Ip→R>0}p∈Z0 o be he amily o maps de ined by αp, =γpq β, o all p∈Z0and ∈Ip. 123 4 Page 18 o 28 B. Molina-Sampe e al. Le us see ha βis a ealizable amily o maps. Fix wo co ne poin s pand q,andle ∈Ip∩Iq.ByEq.(11)weha e ha γpq γqq =γpq . Mo eo e , by he de ini ion o αq, and as a consequence o Lemma 3.9,weha e ha β=γqq αq,. Then we ob ain αp, =γpq β=γpq γqq αq, =γpq αq, which is he equi ed condi ion o β o be ealizable. Now, in iew o Lemma 3.14, he e exis s a unique m-s anda diza ion (Oβ,bβ)wi h (Oβ,bβ)=β.Finally,weshow ha he map q:RI >0→m-s anda diza ions o (M,a),β→ (Oβ,bβ) is a bijec ion. Indeed, i β= β,weha e ha β= βand hence (Oβ,bβ)= (Oβ,bβ) aking in o accoun Rema k 3.13. On he o he hand, gi en an m-s anda diza ion (O,b)wi h combina o ial da a ={αp}p∈Z0,weha e ha (O,b)=q(β),whe eβis de ined by βi=αqi,i, o alli∈I. The p oo o (a) is inished. Le us p o e now hesecond asse ion (b).Deno eby αp:Ip→R>0 hemapo exponen s de ining up, ha is up,i=xαp,i p,i, o all i∈Ip,whe e xp∈a. Conside he injec i e map iαp:RI Ip >0→RI >0,de inedby δ→ iαp(δ) := βδ,whe e βδ i=δii i∈I Ip, αp,ii i∈Ip. Take a collec ion o co ne poin s qp={qi}i∈Isuch ha qi=p, o eachi∈Ip,andqi∈Ei, o each i∈I Ip. Using he no a ions in i em a) abo e, we ha e ha qp(β) ∈E(up)i and only i β|Ip=αp, o equi alen ly β=iαp(β|I Ip). In o he wo ds, we ha e he equali y E(up)=Im(qp◦iαp), and hence we ha e he bijec ion RI Ip >0→E(up)mapping δin o qp(iαp(δ)). We inish jus by no ing ha #Ip=n. 3.4 The monomial Voû e E oilée In his sec ion we gi e he de ini ion o m-combina o ial blowing-up and we in oduce he concep o “monomial oû e é oilée” o e an m-mani old, whose elemen s, called m-s a s, a e sequences o monomial blowing-ups s a ing om ha m-mani old. The e minology is inspi ed by Hi onaka [10,11]. Le (M,a)be a monomial gene alized analy ic mani old. An m-combina o ial cen e o blowing-up o (M,a)is a ipe (Z,O,b),whe eZis a combina o ial geome ic cen e o Mand (O,b)is an m-s anda diza ion o (M,a). Gi en such an m-combina o ial cen e (Z,O,b), we conside he blowing-up πξ:Mξ→Mwi h cen e ξ=(Z,O).Le Ibe an index se labelling he componen s o ∂M. We w i e ∞/∈I o label he excep ional di iso E∞:= π−1 ξ(Z), and we pu Iξ=I∪{∞}as an index se o he componen s o ∂Mξ.Mo e p ecisely, gi en i∈I, i ep esen s bo h he bounda y componen Eio ∂Mand i s s ic ans o m E i=π−1 ξ(Ei Z)⊂∂Mξ, 123 S a i ied educ ion o singula i ies... Page 19 o 28 4 belonging o Zn−1 Mand Zn−1 Mξ, espec i ely. The index ∞∈Iξ ep esen s E∞∈Zn−1 Mξ. P oposi ion 3.17 The e is a monomial a las aξo Mξin such a way ha πξde ines a mo phism o monomial G-mani olds om (Mξ,aξ) o (M,a). P oo Take a co ne poin pin Mξand le p=πξ(p). No e ha pis a co ne poin in M. Le xp∈abe he m-cha o he a las aa p. We dis inguish wo si ua ions: Casep/∈E∞.Weha e ha Ip=Ipand he blowing-up πξinduces an isomo phism be ween V∗ pand V∗ p. We ake a ine coo dina es x po e V pde ined by ˜x p,i=xp,i◦πξ|V∗ p, o all i∈Ip. Thus, he exp ession o πξin coo dina es x pand xpis pu ely monomial. This exp ession can be codi ied wi h he ma ix o exponen s Bp:Ip×Ip→R≥0gi en by Bp(i,j)=δij,i,j∈Ip,(15) whe e δij is he K onecke del a symbol. In o he wo ds, Bp=D1Ip. Casep∈E∞.Weha eIp=Ip { j}∪{∞}, o somej∈IZ(see o ins ance [15] o de ails in he combina o ial ea men o blowing-ups). By hypo hesis, he pai (O,b)is an m-s anda diza ion o (M,a); in pa icula , bis a monomial a las o he s anda d analy ic mani old N=(M,O). Using his in o ma ion, oge he wi h he de ini ion o blowing- up cen e ed a ξ, we ge ha he e exis s an m-cha x pde ined in V psuch ha he map x p◦πξ◦x−1 pis pu ely monomial wi h associa ed ma ix o exponen s Bp:Ip×Ip→R+ gi en by ( ,s)→ ⎧ ⎨ ⎩ 1i =sand ∈Ip { j}, αp,j/αp, i s=∞ and ∈IZ, 0 o he wise. (16) whe e αp∈(O,b). Wi h an app op ia e o de o ows and columns, Bpcan be seen as he uppe iangula ma ix ⎛ ⎝ Idn−s0 0 0Ids−1a 0 0 1 ⎞ ⎠∈Rn×n +, whe e s=#IZand a∈Rs−1 >0is a column ec o whose en ies a e de ined by he quo ien s αp,j/αp, , wi h ∈IZ { j}. The collec ion aξ={x p}p∈Z0 ξwi h Z0 ξ=Z0 Mξis hus a monomial a las in Mξ.Mo e- o e , he blowing-up πξinduces a mo phism om (Mξ,aξ) o (M,a)and he associa ed combina o ial da a is Bπξ={Bp}p∈Z0 ξ. F om now on, gi en an m-combina o ial cen e o blowing-up (Z,O,b) o a monomial gene alized analy ic mani old (M,a), and he blowing-up mo phism πξ:Mξ→M, wi h cen e a ξ=(Z,O),wealwaysconside Mξendowed wi h he monomial a las aξ cons uc ed in P oposi ion 3.17. Mo eo e , we also w i e πξ:(Mξ,aξ)→(M,a), oemphasize ha hemo phismπξis conside ed also as a mo phism o monomial gene alized analy ic mani olds, and we call i an m-combina o ial blowing-up o (M,a). The associa ed 123 4 Page 20 o 28 B. Molina-Sampe e al. combina o ial da a Bπξ={Bp}p∈Z0 ξo his mo phism has been made explici in Eq. (15), o poin s p∈Z0 ξwi h p/∈E∞andinEq.(16), o poin s p∈Z0 ξ(E∞). De ini ion 3.18 Le (M,a)be a monomial gene alized analy ic mani old. An m-s a o e (M,a)is he composi ion σ=π0◦π1◦···◦π −1o a ini e sequence o m-combina o ial blowing-ups. Tha is σ:(M ,a )π −1 −−→ (M −1,a −1)π −2 −−→··· π0 −→ (M0,a0)=(M,a), whe e o each k=0,1,2,..., −1, he mo phism πkis an m-combina o ial blowing-up o (Mk,ak). The in ege and he monomial gene alized analy ic mani old (M ,a )a e called, espec i ely, he age and he end o he m-s a σ. The collec ion Vm (M,a)o all he m-s a s o e (M,a)is called he monomial oû e é oilée o (M,a). 4 S a i ied educ ion o singula i ies ia p incipaliza ion o m-ideals We de o e his sec ion o in oducing he concep o m-ideal, in o de o p o e a heo em o p incipaliza ion. Wi h his esul we p o e he s a i ied educ ion o singula i ies o a global unc ion de ined in gene alized analy ic mani olds admi ing a monomial s uc u e. Finally, we apply his esul o p o e he main esul o his pape s a ed in Theo em 1.1. 4.1 P incipaliza ion o m-ideals Le us ix a monomial gene alized analy ic mani old (M,a),whe eM=(M,GM). Take a global gene alized analy ic unc ion ∈GM(M)and wo co ne poin s p,q∈Z0. Le xq,xq∈abe he m-cha s a pand q, espec i ely, and le Cpq ∈C(M,a)be he ma ix o exponen s codi ying he change o coo dina es xq◦x−1 p. The ela ion be ween he suppo s o a pand qwi h espec o hese coo dina es is gi en by: Suppp( ;xp)={λqCpq :λq∈Suppq( ;xq)}⊂RIp +.(17) De ini ion 4.1 A gene alized analy ic global unc ion m∈GM(M)is said o be an m- unc ion in (M,a)i o each p∈Z0, he eisamapλp:Ip→R+, such ha m|V p=xλp p,whe e xp∈a. The combina o ial da a o mis he lis Lm={λp}p∈Z0. Le us conside an m- unc ion min (M,a)wi h combina o ial da a Lm={λp}p∈Z0.By Eq. (17), o any pai o co ne poin s p,q∈Z0we ha e he ela ion λp=λqCpq,whe e Cpq ∈C(M,a). In pa icula , we ge ha λp, =λq,γpq , o any ∈Ip∩Iq,(18) whe e γpq is he weigh ed connexion unc ion om p o q. Indeed, in iew o Eq. (11), i is enough o check Eq. (18) o he case whe e pand qa e connec ed h ough an edge Y.Fo his case, i holds as a consequence o Lemma 3.7. Rema k 4.2 Gi en a lis o maps L={λp:Ip→R+}p∈Z0sa is ying λp=λqCpq, o any pai o co ne poin s p,q∈Z0, he e exis s an m- unc ion msuch ha Lm=L. 123 S a i ied educ ion o singula i ies... Page 21 o 28 4 De ini ion 4.3 A ini ely gene a ed m-ideal in (M,a)is a shea o ideals J⊂GMgene a ed by ini ely many m- unc ions. Tha is, J=m1GM+m2GM+···+mkGM=: (m1,m2,...,mk), whe e m1,m2,...,mka e m- unc ions called m-gene a o s o J. No a ion 4.4 Le Ibe a ini e index se and le Abe a ini e subse o RI. We deno e by Amin he se o elemen s in A ha a e minimal wi h espec o he di ision o de ≤din RI. Le Jbe an m-ideal in (M,a)wi h se o m-gene a o s G={m1,m2,...,mk}. Fo each i=1,2,...,k, le us w i e Lmi={λi p}p∈Z0. Gi en a co ne poin p∈Z0and xp∈a he m-cha a p, he es ic ion J|V pis an m-ideal in he m-co ne (M|V p,xp)wi h se o m-gene a o s equal o G|V p:= {m1|V p,m2|V p,...,mk|V p}. Conside he se G,p:= {λ1 p,λ 2 p,...,λ k p}⊂RIp +. No e ha i (G,p)min ={μ1 p,μ 2 p,...,μ kp p}, hen J|V p=(xμ1 p p,xμ2 p p,··· ,xμkp p p). (19) The shea o ideals Jis called locally p incipal i a each poin a∈M, hes alkJa⊂GM,a is a p incipal ideal. Using he de ini ion o m-ideal, i is enough o ask his p ope y o he co ne poin s. In e ms o he se in oduced abo e, we ha e ha Jis locally p incipal i and only i (G,p)min is a single on o any p∈Z0. Le mbe an m- unc ion in (M,a)and ake an m-s a σ:(M,a)→(M,a).The o al ans o m σ∗m=m◦σis a again an m- unc ion in (M,a).Mo ep ecisely,i p∈Z0 M and p=σ(p), henλ p∈Lσ∗mis gi en by λ p=λpBσ p,(20) whe e λp∈Lmand Bσ p∈Bσis he ma ix o exponen s codi ying σa p.I Jis an m-ideal gene a ed by G={m1,m2,...,mk}, hen he o al ans o m σ∗Jis also an m-ideal in (M,a)gene a ed by σ∗G:= {σ∗m1,σ∗m2,...,σ∗mk}. The main esul in his sec ion is he ollowing one abou p incipaliza ion o m-ideals. Theo em 4.5 Le Jbe a ini ely gene a ed m-ideal in a monomial gene alized analy ic man- i old (M,a). The e exis s an m-s a σ∈Vm (M,a)such ha σ∗Jis locally p incipal. To p o e his heo em, we can educe ou sel es o he case whe e Jis gene a ed by wo m- unc ions by conside ing a clea ini e ecu ence and he ollowing lemma. Lemma 4.6 Le J=(m1,m2,...,mk)be an m-ideal in (M,a). Assume ha J s := (m ,ms)is locally p incipal o any pai o indices ,s∈{1,2,...,k}.ThenJis locally p incipal. P oo Assume ha he e is a poin p∈Z0such ha Jpis no p incipal. The e exis indices ,s∈{1,2,...,k}such ha λ p∈Lm and λs p∈Lmsa e no compa able o he di ision o de ≤din RIp. No e ha G s,p=(G s,p)min ={λ p,λ s p},whe eG s ={m ,ms},and hence J s is no locally p incipal, which is a con adic ion.  123 4 Page 22 o 28 B. Molina-Sampe e al. Thecaseo wogene a o s.Le us assume ha J=(m,n)is an m-ideal in (M,a)gene a ed by wo m- unc ions and w i e Lm={λp}p∈Z0and Ln={μp}p∈Z0. We in oduce i s se e al de ini ions, mainly inspi ed by he “b-in a ian ” in oduced in [6] by an den D ies and Speissegge . Le Z∈Zn−2be a codimension wo combina o ial geome ic cen e o M.Weknow ha he index se IZhas jus wo elemen s, say IZ={i,j}.Le p∈Z0(Z)be a co ne poin in Z. We say ha Zis uncoupled o Ja p i (λp,i−μp,i)(λp,j−μp,j)<0. We say ha Zis uncoupled o Ji i is so a each p∈Z0(Z). Lemma 4.7 A combina o ial geome ic cen e Z ∈Zn−2is uncoupled o Ji and only i he e is a co ne poin q ∈Z0(Z)such ha Z is uncoupled o Ja q. P oo Assume ha Zis uncoupled o Ja a co ne poin q∈Z0(Z)and ake any o he poin p∈Z0(Z).ByEq.(18), we ha e λp, =λq,γpq , o all∈Ip∩Iq,whe eγpq is he weigh ed connexion unc ion om p o q.Sincep,q∈Z, we know ha IZ={i,j}⊂ Ip∩Iq.Then (λp,i−μp,i)(λp,j−μp,j)=γpq iγpq j(λq,i−μq,i)(λq,j−μq,j)<0, since γpq i>0andγpq j>0. Hence Zis uncoupled o Jalso a p, and we conclude ha Z is uncoupled o J. Obse e ha i p∈Z0and he e a e no uncoupled cen e s o Jpassing h ough p, hen we necessa ily ha e ha λp≤dμpo μp≤dλp, ha isJpis a p incipal ideal. De ini ion 4.8 Le Jbe he amily o codimension wo combina o ial geome ic cen e s in M ha a e uncoupled o J, and de ine he in a ian o J o be In J:= #J. We ha e ha In J=0 i and only i Jis locally p incipal. Thus, he objec i e now is o ind an m-s a σ∈Vm (M,a)such ha In σ∗J=0. Suppose ha In J>0and ixZ∈J. Take a co ne poin p∈Zand pick a local m-s anda diza ion upo (M,a)a pde ined by he map αp:Ip→R>0. We say ha upis adap ed oJwi h espec oZ i αp,j(λp,i−μp,i)+αp,i(λp,j−μp,j)=0. A global m-s anda diza ion (O,b={up}p∈Z0)o (M,a)is said o be adap ed o Jwi h espec oZ i upis adpa ed o Jwi h espec o Z o e e y p∈Z0(Z). Lemma 4.9 An m-s anda diza ion (O,b)is adap ed o Jwi h espec o Z i and only i he e is a co ne poin q ∈Z such ha uq∈bis adap ed o Jwi h espec o Z. P oo Deno e =(O,b). Assume ha he e is a co ne poin q∈Zsuch ha uq∈bis adap ed o Jwi h espec o Z. Take any o he co ne poin p∈Z. In iew o he ealizabili y o es ablished in Lemma 3.14 we know ha αq, =γqp αp,, o all∈Ip∩Iq.Since p,q∈Zwe ha e ha IZ={i,j}⊂Ip∩Iq. Then, by Eq. (18), we ge αp,j(λp,i−μp,i)+αp,i(λp,j−μp,j)=γpq jαq,jγpq i(λq,i−μq,i) +γpq iαq,iγpq j(λq,j−μq,j) =γpq iγpq j[αq,j(λq,i−μq,i) +αq,i(λq,j−μq,j)]=0. 123 S a i ied educ ion o singula i ies... Page 23 o 28 4 As a consequence, he local m-s anda diza ion up∈bis adap ed o Jwi h espec o Z a p. We conclude ha (O,b)is adap ed o Jwi h espec o Z. A codimension wo combina o ial cen e o blowing-up ξ=(Z,O,b)is adap ed o Ji Z∈Jand (O,b)is an m-s anda diza ion adap ed o Jwi h espec o Z. The nex esul assu es he exis ence o such a cen e . Lemma 4.10 Assume ha In J>0. Then, he e exis codimension wo combina o ial cen e s o blowing-up adap ed o J. P oo By de ini ion In J>0 i and only i J=∅. Fix an elemen Z∈Jand le us see ha he e a e m-s anda diza ions adap ed o Jwi h espec o Z. In iew o Lemma 4.9, i is enough o p o e he exis ence o an m-s anda diza ion (O,b)adap ed o Ja a co ne poin p∈Z. Fix any co ne poin p∈Z.SinceZis uncoupled o J,wecanassume,up o exchanging he indices iand j, ha i=λp,i−μp,i>0,and j=μp,j−λp,j>0. Take αp:Ip→R>0 o be a map such ha αp,i=iand αp,j=j, and ake he m-cha up=xαp pde ined in V p. Any m-s anda diza ion ex ending upis adap ed o Jwi h espec o Za he poin pbecause o he de ini ion o αp. Mo eo e , such an ex ension exis s as a consequence o P oposi ion 3.16. We conclude by applying ini ely many imes he ollowing esul . P oposi ion 4.11 Le J=(m,n)be an m-ideal wi h In J>0. Gi en an m-combina o ial cen e o blowing-up ξ=(Z,O,b)adap ed o J, he blowing-up πξ:Mξ→Mcen e ed a ξsa is ies In π∗ ξJ=In J−1. P oo Le us w i e Zξ=ZMξ,π=πξ,E∞=π−1(Z)and IZ={i,j}. Deno e also Lm={λp}p∈Z0,Ln={μp}p∈Z0,Lπ∗m={λ p}p∈Z0 ξ,Lπ∗n={μ p}p∈Z0 ξ. Le Tbeacodimension wocombina o ialgeome iccen e inMdi e en om Z.Deno e by ST he s a um in SMsuch ha ST=T. The closu e To π−1(ST)is a codimension wo geome ic cen e in Mξha ing index se IT=IT={ ,s}. Gi en a co ne poin p∈T, le p=πξ(p).Weha e λp, =λp, ,μ p, =μp, ,λ p,s=λp,s,μ p,s=μp,s, in iew o he ela ion be ween λp,λpand μp,μpes ablishedin Eq. (20), and he exp ession o Bπ p∈Bπgi eninEqs.(15)and(16).Then,weha e ha T∈π∗Ji andonly i T∈J, ha is Tis uncoupled o π∗Ji and only i Tis uncoupled o J. Le us see now ha any elemen in π∗Jis among he ones conside ed be o e. Tha is, le us show ha he e is no codimension wo combina o ial geome ic cen e Zuncoupled o π∗Jcon ained in E∞. Take a codimension wo combina o ial geome ic cen e Z⊂E∞and a poin p∈ Z0 ξ(Z). In iew o Lemma 4.7, i is enough o p o e ha Zis no uncoupled o π∗Ja p. Mo e p ecisely, i we w i e IZ={k,∞},wewan oshow ha (λp,k−μp,k)(λp,∞−μp,∞)≥0. Le us conside p=π(p)and he local da a αp∈(O,b). The co ne poin pbelongs o Z, and he a ine coo dina es up∈bde ine a local m-s anda diza ion adap ed o Jwi h 123 4 Page 24 o 28 B. Molina-Sampe e al. espec o Za p, ha is, we ha e he ela ion αp,j(λp,i−μp,i)+αp,i(λp,j−μp,j)=0. We know ha Ip=Ip { j}∪{∞}, up o exchanging he indices iand j. Hence, he ma ix B:= Bp:Ip×Ip→R+sa is ies, using Eq. (16): B  =1, o ∈Ip { j},B j∞=1,B i∞=αp,j αp,i =μp,j−λp,j λp,i−μp,i ,B s =0 o he wise. By Eq. (20) we ge he ela ions λp,k=λp,k,μp,k=μp,k,and λp,∞=λp,j+B i,∞λp,i=λp,iμp,j−λp,jμp,i λp,i−μp,i =μp,j+B i,∞μp,i=μp,∞. Thus (λp,k−μp,k)(λp,∞−μp,∞)=0, and we a e done.  4.2 S a i ied educ ion o singula i ies in monomial mani olds We use he esul o p incipaliza ion o m-ideals in o de o p o e he ollowing s a emen : P oposi ion 4.12 Le (M,a)be a monomial gene alized analy ic mani old wi h M= (M,GM). Gi en a gene alized analy ic unc ion ∈GM(M), he e is an m-s a σ∈Vm (M,a) such ha he pull-back = ◦σis o s a i ied monomial ype. Fo he p oo o P oposi ion 4.12 we associa e o a ini ely gene a ed m-ideal J ,and we p o e ha he p incipaliza ion o J gi es ise o he s a i ied educ ion o singula i ies o . Gi en q∈Z0and λq∈Suppq( ;xq), i makes sense o de ine he m- unc ion mλqas he one ha ing he collec ion o maps Lmλq={λqCpq}p∈Z0as a combina o ial da a, by Rema k 4.2 and Eq. (17). The m-ideal J associa ed o is he ideal shea gene a ed by he ini e se o m- unc ions G = q∈Z0mλq:λq∈Suppmin,q( ;xq). By de ini ion, no ice ha o any co ne poin q∈Z0,weha e (G ,q)min =Suppmin,q( ;xq). (21) whe e he he no a ion G ,qwas in oduced in Sec . 4.1 abo e. Lemma 4.13 Gi en an m-s a τ:(M,a)→(M,a), we ha e τ∗J =J ,whe e = ◦τ. P oo In iewo Eq. (19),i isenough o p o e ha o anyco ne poin p∈Z0 M heequali y (τ∗G ,p)min =(G ,p)min holds. Deno e o sho 1=τ∗G ,pand 2=G ,p. Fix a poin p∈Z0 Mand conside p=τ(p). W i e := Suppmin,p( ;xp),whe exp∈ a,andle Bτ p∈Bτbe he ma ix o exponen scodi ying τa p.Le := {λpBτ p;λp∈}. We p o e ha bo h min 1and min 2a e equal o min. S ep 1: min 2=min.Recall by Eq. (21) ha min 2=:= Suppmin,p( ;x p),whe e x p∈a.Le ={λ1,λ 2,...,λ k}⊂RIp, ha is, he unc ion a ound he co ne poin phas he ini e p esen a ion |V p=xλ1 pU1+xλ2 pU2+···xλk pUk,whe eUi(p)= 0, o all 123 S a i ied educ ion o singula i ies... Page 25 o 28 4 i=1,2,...,k. Taking in o accoun ha τ(V p)⊂V p, he unc ion = ◦τis w i en in he cha xp∈aas |V p=x˜ λ1 p(U1◦τ|V p)+x˜ λ2 p(U2◦τ|V p)+···x˜ λk p(Uk◦τ|V p), whe e ˜ λi=λiBτ p.SinceBτ pis an in e ible ma ix, we can assu e ha ˜ λ = ˜ λs o any pai o di e en indices ,s∈{1,2,...,k}. This implies ha ={ ˜ λ1,˜ λ2,...,˜ λk}min by de ini ion o minimal suppo o a p. We a e done, since ={ ˜ λ1,˜ λ2,...,˜ λk}. S ep 2: min 1=min.Recall ha ⊂G ,pand deno e ˜ =G ,p .ByEq.(21) we know ha o any μ∈˜  he e exis s λ∈such ha λ≤dμ. No e ha 1=∪˜ , whe e ˜ := {μBτ p;μ∈˜ }. The e o e, we need only o p o e he ollowing claim: I λ,μ :Ip→R+sa is y λ≤dμ, hen λBτ p≤dμBτ p. Fo ha , i is enough o conside he case whe e τ=πξis a single m-combina o ial blowing-up wi h cen e ξ=(Z,O,b). Deno e, as usual, E∞=π−1 ξ(Z). I p/∈E∞o equi alen ly p/∈Z,weha e ha Ip=Ipand Bτ p=D1Ip, and we a e done. Assume ha p∈E∞,andle jbe he index in IZsuch ha Ip {∞} = Ip { j}. Deno e λ=λBτ pand μ=μBτ p.ByEq.(16), we ha e ha λ =λ,μ =μ, and hus λ ≤μ , o all ∈Ip {∞}; whe eas λ ∞= ∈IZ αp,j αp, λ≤ ∈IZ αp,j αp, μ=μ ∞, whe e αp∈(O,b),aswewan ed.  P oo o P opos ion 4.12 In iew o Theo em 4.5, we can ake an m-s a σ:(M,a)→ (M,a)such ha σ∗J islocallyp incipal.ByLemma4.13weknowalso ha σ∗J =J ◦σ. Hence, since G ◦σis a se o gene a o s o J ◦σ,weha e ha (G ◦σ,p)min is a single on o all p∈Z0 M. Finally, by Eq. (21) we ob ain mp( )=#Suppmin,p( ;x p)=#(G ◦σ,p)min =1 o all p∈Z0 M.Sinceais a monomial a las, we know ha M=p∈Z0 MV p. Thus, gi en a s a um S∈SM, he e is a co ne poin p∈Z0 Msuch ha p∈¯ S.In iewo he ho izon al s abili y p ope y o he monomial complexi y es ablished in Lemma 2.6,wege mS( )≤mp( )=1. We conclude ha is o s a i ied monomial ype.  4.3 P oo o he main s a emen We end end he e he p oo o he s a i ied educ ion o singula i ies o gene alized analy ic unc ions, as s a ed in Theo em 1.1. Recall ha we ha e a gene alized analy ic mani old M=(M,GM), and a gene alized analy ic unc ion ∈GM(M)in M. Gi en a poin p∈M, we wan o p o e ha he e exis an open neighbou hood V⊂Mo pand a ini e sequence o blowing-ups σ:M π −1 −→ M −1 π −2 −→··· π0 −→ M0=(V,GM|V), 123