Stratified reduction of singularities of generalized analytic functions

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2024) 118:4 https://doi.org/10.1007/s13398-023-01486-8 ORIGINAL PAPER Stratified reduction of singularities of generalized analytic functions B. Molina-Samper1·J. Palma-Márquez2·F. Sanz Sánchez1 Received: 5 September 2022 / Accepted: 14 July 2023 © The Author(s) 2023 Abstract Generalized analytic functions are naturally defined in manifolds with boundary and are built from sums of convergent real power series with non-negative real exponents. In this paper we deal with the problem of reduction of singularities of these functions. Namely, we prove that a germ of generalized analytic function can be transformed by a finite sequence of blowing-ups into a function which is locally of monomial type with respect to the coordinates defining the boundary of the manifold where it is defined. Keywords Blowing-up morphism ·Reduction of singularities ·Generalized power series · Principialization of ideals Mathematics Subject Classification 14E15 ·14P15 ·16W60 ·32C05 ·32S45 First and third authors are partially supported by the Project “Métodos asintóticos, algebraicos y geométricos en foliaciones singulares y sistemas dinámicos” (Ref.: PID2019-105621GB-100) of the Ministerio de Ciencia in Spain. First author is partially supported by the “Programa de becas postdoctorales DGAPA” of the UNAM in Mexico. The second author is partially supported by Papiit Dgapa UNAM IN110520, by the Israel Science Fundation (grant No. 1167/17) and by funding received from the MINERVA Stiftung with the funds from the BMBF of the Federal Republic of Germany. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 802107). BB. Molina-Samper [email protected] J. Palma-Márquez [email protected] F. Sanz Sánchez [email protected]a.es 1Departamento de Álgebra, Análisis Matemático, Geometría y Topología, Universidad de Valladolid, Valladolid, Spain 2Weizmann Institute of Science, Rehovot, Israel 0123456789().: V,-vol 123 4 Page 2 of 28 B. Molina-Samper et al. 1 Introduction In this paper, a generalized power series (in nvariables and with coefficients in some ring A) is a power series with n-tuples of non-negative real numbers as exponents and whose support is contained in a cartesian product of nwell-ordered subsets of R+={r≥0}.Itisworth to mention that this condition on the support is more restrictive (except for n=1) than the one used to define the Hahn ring A(()),whereis the group Rnwith the lexicographic order (and whose elements are also called generalized power series). Introduced and studied by van den Dries and Speissegger in [6], generalized power series appear in several contexts. To mention a few: as solutions of differential/functional equations; as expressions of the Riemann zeta-function (or, more generally, the Dirichlet series) in a logarithmic chart; as asymptotic expansions of Dulac transition maps of vector fields (see for instance [12,13]); in model theory and o-minimal geometry (the paper [6] itself, or also [17]); as parametrizations of algebraic curves in positive characteristic (see for instance [18,p.19]). Considering real coefficients, we have a natural notion of convergence for generalized power series, whose sums provide continuous functions on open subsets of the orthant Rn +, called generalized analytic functions. They are the local pieces to build abstract (real) generalized analytic manifolds, introduced and developed by Martín, Rolin and Sanz in [14]. More precisely, a generalized analytic manifold is a locally ringed space M=(M,GM),whereM is a topological manifold with boundary and GMis a sheaf of continuous functions locally isomorphic to the sheaf of generalized analytic functions on open subsets of Rn +. Sections of the sheaf GMare called themselves generalized analytic functions on M. The main result in [14] establishes the local reduction of singularities of generalized analytic functions, in the spirit of Zariski’s local uniformization theorem of algebraic varieties [19] or Hironaka’s version for analytic varieties [10]. The statement, formulated in analogous terms to those used in Bierstone–Milman’s paper [4] for real analytic functions, is the following: Local Monomialization Theorem [14]. Let fbe a generalized analytic function on Mand let p∈M. Then there exists a neighbourhoodU0of pin M, finitely many sequences of local blowing-ups {πi:Mi→U0}r i=1and compact sets Li⊂Misatisfying that iπi(Li)is a neighbourhood of pandsuchthat,foreveryi, the total transform fi=f◦πiis of monomial type at every q∈Li(i.e., for some coordinates x=(x1,x2,...,xn)centered at q,wehave fi=xαU(x)where U(0)= 0). The centers of blowing-ups in each sequence πihave normal crossings with the boundary, but they are defined only in some open sets of the corresponding manifold. In the standard real analytic case, we have stronger global monomialization results (typically called Reduction of Singularities,see[1,5,7]). They consist, essentially, in that in the above statement, we can take just a single sequence (r=1) and the centers of blowing-ups are globally defined closed analytic submanifolds, having normal crossings with the boundary. Such a global result is not known so far for generalized analytic functions. There are two main difficulties related to the very notion of a blowing-up morphism in the category of generalized analytic manifolds. On the one hand, a blowing-up depends on the local coordinates that we use to define it. More intrinsically, a blowing-up is not uniquely defined and depends on the choice of a standardization of the manifold (or at least of an open neighbourhood of the center of blowing-up). Roughly, a standardization is a subsheaf OMof GMsuch that (M,OM)is a real analytic standard manifold and from which the sheaf GMcan be recovered by a natural completion adding generalized series (see [14], we recall this notion below). Secondly, although every generalized analytic manifold is locally standardizable, 123 Stratified reduction of singularities... Page 3 of 28 4 there may exist closed submanifolds which do not admit standardizable neighbourhoods; i.e., such submanifolds cannot be “geometric” centers for a blowing-up (cf. [14,Example 3.20]). Morally, a procedure for reduction of singularities of generalized analytic functions would need to guarantee that, in the process, all closed centers susceptible to be blown-up have standardizable neighbourhoods. If this is already proved and Yis such a center, one needs to show furthermore that, among the different standardizations around Y, there exists for which the corresponding blowing-up π: M→Mreduces the “complexity” of the function. In this paper, we overcome these difficulties to obtain an intermediate step towards a global result, the so-called stratified reduction of singularities. Let us explain it. First, we recall that, byitsverydefinition,theboundary ∂Mofa generalizedanalytic manifoldis anormalcrossing divisor; i.e., ∂Mis locally given by a finite union of coordinate hyperplanes. Moreover, the number of such hyperplanes at each point provides a natural stratification of Mby (standard) analytic manifolds. A generalized analytic function f:M→Ris said to be of stratified monomial type if for any given p∈M,ifSis the stratum where pbelongs, there exists a local chart (x=(x1,x2,...,xe), y)centered at psatisfying S={x1=x2= ···= xe=0} and for which f(x,y)=xαU(x,y), where α∈Re +and U(0,y)≡ 0. Thus, requiring a function to be of stratified monomial type means to require that it is of monomial type only with respect to the generalized coordinates determining equations of the components of the boundary. In particular, the condition is empty if p/∈∂M. Also, it is automatic if Shas codimension e=1, taking in the above definition αto be the minimum of the support of the series defining fwith respect to the single variable x=x1. Our main result may be stated now as follows. Theorem 1.1 (Stratified Reduction of Singularities) Let M=(M,GM)be a generalized analytic manifold and let f :M→Rbe a generalized analytic function. Let p ∈M and assume that the germ of f at p is not identically zero. Then, there exist a neighbourhood Vp of p in M and a sequence of blowing-ups (Mr,GMr)πr−1 →(Mr−1,GMr−1)πr−2 → ··· π1 →(M1,GM1)π0 →(Vp,GM|Vp) such that the pull-back f := f◦π0◦···◦πr−1∈GMr(Mr)is of stratified monomial type. Moreover, the center of each blowing-up πj, with j =0,1,...,r−1, can be chosen to be the closure of a codimension two stratum in Mj,whereM 0:= Vp. Our proof of Theorem 1.1 is constructive in the sense that each center, as well as the standardization used to define the respective blowing-up at each step, can be given explicitly in terms of the expression of fin some initial coordinates of Mat p. Moreover, each blowingup morphism is locally expressed as a purely monomial map between two domains of Rn + in suitable charts. Consequently, all the process of stratified reduction of singularities can be described using only combinatorics from the starting data given simply by the minimal support (see Sect.2below) of a generalized power series representing fat p. The datum of minimal support is closely related to that of the Newton polyhedron of a function in the standard analytic case and therefore, our result should be compared with the combinatorial reduction of singularities stated in Molina’s paper [15]. Although it has been a source of inspiration for us, we cannot apply directly the results in [15], mostly because there is no good notion of “multiplicity” in the generalized non-standard situation (any power function with positive real exponent in a generalized variable is a genuine change of variables). 123 4 Page 4 of 28 B. Molina-Samper et al. We want to observe that Theorem 1.1 is already proved for dim M=3 in Palma’s paper [16], but with a different strategy for the choice of the sequence of blowing-ups (for instance, the centers of blowing-ups may be either corner points or closures of one-dimensional strata). The paper is structured as follows. In Sect. 2we summarize the basic notions and properties of generalized power series and of the category of generalized analytic manifolds, using the mentioned references [6]and [14]. We emphasize the notion of standardization, which is crucial to define blowing-ups. In Sect. 3we introduce the category ofmonomial (generalized or standard) analytic manifolds. The objects of this subcategory are manifolds having at least one corner and equipped with an atlas of local charts centered at each corner point for which the change of coordinates is expressed as a monomial map between domains of the local model Rn +. We represent these changes of coordinates by means of a family of matrices of exponents (for a similar treatment see for instance [2,3,16]), a combinatorial data which codifies uniquely the structural sheaf of the manifold. We define also the class of monomial morphisms and the class of monomial standardizations of monomial manifolds. After a blowing-up using such a standardization with a center which is the closure of a stratum (a so-called combinatorial center), we obtain again a monomial manifold and the blowing-up morphism is a monomial morphism. The main result in this section is the abundance of monomial standardizations (Proposition 3.16 below). Furthermore, we can choose such a monomial standardization with a prescribed local expression at a given corner point. Morally, local strategies of reduction of singularities are susceptible to be “globalized”. We end this section by introducing a special class of monomial manifolds, those obtained from a given one by a sequence of blowing-ups with combinatorial centers, and using only monomial standardizations. Such a sequence is called a monomial star and the family of such stars is called the monomial “voûte étoilée”, a terminology that evokes the one introduced by Hironaka in [10,11] for sequences of local blowing-ups in complex analytic geometry. In Sect. 4we provide a proof of the main Theorem 1.1. Firstly, we prove a result about principalization of finitely generated monomial ideal sheaves in a given monomial manifold. This result (see Theorem 4.5 below) can be seen as a version for our category of a well known result on principalization of ideals in the algebraic or standard analytic situation (see for instance Goward’s paper [9] for a simple proof, or see also Fernández-Duque’s paper [8]for a similar statement concerning the resonances elimination for singularities of codimensionone analytic foliations). Taking into account that it suffices to obtain the principalization only at the corner points, such a result can also be regarded as a globalization of the algorithm described in van den Dries and Speissegger’s paper (see [6, Lemma 4.10]) that reduces the number of elements in the minimal support of a generalized power series by monomial transformations of the variables. Although we use certain elements and arguments of that result, and despite of what we have said above concerning the possibility to globalize a “local strategy”, our proof here requires a different control invariant. Once we have the principalization of monomial ideal sheaves, the main theorem is concluded easily in the case we start with a corner point p∈M. In this case, the sequence π0◦π1◦···πr−1for Theorem 1.1 is actually a star in the voûte étoilée over the germ of M at p. Finally, the general case p∈∂Mis reduced to the case of a corner point, using that around pthere is a product structure of a neighbourhood of a corner point times a standard analytic manifold without boundary. 123 Stratified reduction of singularities... Page 5 of 28 4 2 Preliminaries Wesummarizeherethebasicnotionsaboutthecategoryofgeneralizedanalytic manifoldsand blowing-up morphisms in it, introduced by Martín, Rolin and Sanz in [14]. These manifolds are built from convergent generalized power series, extensively studied in a paper by van den Dries and Speissegger [6]. 2.1 Formal and convergent generalized power series Denote by R+=[0,∞). Tuples of variables are denoted by X,Y,Z, etc., and we implicitly assume that tuples with different name have no common variables. If Xhas ncomponents, we say that Xis an n-tuple and so on. Let X=(X1,X2,...,Xn)be an n-tuple of variables and let Abe an integral domain. A formal generalized power series with coefficients in A inthevariables Xisamaps:Rn +→A, written as s= λ∈Rn + sλXλ,where Xλ=Xλ1 1Xλ2 2···Xλn nfor λ=(λ1,λ 2,...,λ n) and sλ:= s(λ) ∈A, such that its support Supp(s):= {λ∈Rn +:sλ= 0}is contained in a cartesian product of nwell-ordered subsets of R. The set of all such formal generalized power series, denoted by A[[X∗]], with the usual addition and product operations of power series has an structure of an A-algebra which is also an integral domain. Moreover, if Ais a field, then A[[X∗]] is a local algebra (see [6, Corollary 5.6]), with maximal ideal given by m={s∈A[[X∗]] : s0=0}. Note that A[[X∗]] is not noetherian, in fact, the ideal mis never finitely generated. The minimal support of a power series s∈A[[X∗]] is the subset Suppmin(s)⊂Supp(s) composed of the minimal tuples of Rn +with respect to the (partial) division order ≤d,thatis (λ1,λ 2,...,λ n)≤d(μ1,μ 2,...,μ n)if and only if λi≤μi,foralli∈{1,2,...,n}. The condition imposed on the support of a power series sallows to show that the minimal support Suppmin(s)is finite (see [6, Lemma 4.2]). As a consequence, sadmits a finite monomial presentation: s= λ∈Suppmin(s) XλUλ(X), where Uλ∈A[[X∗]] satisfies Uλ(0)= 0, for any λ∈Suppmin(s). Denote by m(s)= #Suppmin(s). When m(s)=1 or, equivalently, the monomial representation of shas a single term, we say that sis of monomial type. In this paper, we are interested in real generalized power series, that is A=R, but we use different rings when we want to distinguish some variables and put the others into the coefficients. To be precise, if Yand Zare tuples of kand n−kvariables, respectively, we consider R[[(Y,Z)∗]] as a proper R-subalgebra of R[[Y∗]][[Z∗]] by the natural monomorphism s= (λ,μ)∈Rn + aλμYλZμ→ sZ= μ∈Rn−k + AμZμ,where Aμ= λ∈Rk + aλμYλ.(1) If pr :Rn→Rn−kdenotes the natural projection onto the last n−kcoordinates, for any power series s∈R[[(Y,Z)∗]] we have the inclusion Suppmin(sZ)⊂pr(Suppmin(s)),andas 123 4 Page 6 of 28 B. Molina-Samper et al. a consequence we get the inequality m(sZ)≤m(s). (2) Let us write R[[Y,Z∗]] to denote the subalgebra of R[[(Y,Z)∗]] composed by the socalled real mixed power series: those formal real generalized power series sin the variables (Y,Z), such that the inclusion Supp(s)⊂Nk×Rn−k +holds, or equivalently, such that sZ∈R[[Y]][[Z∗]]. Given an n-tuple of variables Xand a polyradius ρ=(ρ1,ρ 2,...,ρ n)∈Rn >0,denoteby R{X∗}ρthe subalgebra of R[[X∗]] consisting on those power series sfor which sρ:=  λ∈Supp(s) |sλ|ρλ<∞. The union of the R{X∗}ρalong all the possible polyradius ρ∈Rn >0is again a subalgebra R{X∗}⊂R[[X∗]], and its elements are called (real) convergent generalized power series. We have that R{X∗}is also a local algebra, whose maximal ideal is given by m∩R{X∗}. If Y,Zare tuples of kand n−kvariables, respectively, and ρ∈Rn >0is a polyradius, an element s∈R[[Y,Z∗]] ∩ R{(Y,Z)∗}ρgives rise to a continuous function fs:Pρ k,n−k→R x=(x1,x2,...,xn)→ λsλxλ,(3) where Pρ k,n−k=(−ρ1,ρ 1)×(−ρ2,ρ 2)×···×(−ρk,ρ k)×[0,ρ k+1)×···×[0,ρ n)⊂ Rk×Rn−k +,calledthesum of the power series s. Moreover, fsisrealanalyticat anypointin the interior of Pρ k,n−kand its germ at 0∈Rnis uniquely determined by the series s.Wedefinethe convergent mixed power series to be the elements of R{Y,Z∗}:=R[[Y,Z∗]] ∩ R{(Y,Z)∗}. 2.2 Standard and generalized analytic manifolds Let Vbe an open subset of Rn +and let g:V→Rbe a continuous function. Given a point p=(p1,p2,...,pn)∈V, consider Ip:= {i:pi=0}⊂{1,2,...,n}, and put =#Ip and k=n−.Wesaythatgis generalized analytic (or just G-analytic)atpif there exists s∈R{Y,Z∗},whereYis a k-tuple and Zis an -tuple, such that for any x=(x1,x2,...,xn) in a sufficiently small neighbourhood of 0in Rk×R +,wehave g(p1+xσ(1),p2+xσ(2),...,pn+xσ(n))=fs(x1,x2,...,xn), where σis a permutation of the set {1,2,...,n}satisfying the relation j∈Ipif and only if σ(j)∈{k+1,k+2,...,n}. We say that gis generalized analytic in V if it so at every point pin V. In the definition above, the series sis uniquely determined by the germ of gat p, up to permutation of the variables Yand Z, separately. Thus, the set of germs of generalized analytic functions at pdefines an R-algebra isomorphic to R{Y,Z∗}. On the other hand, if g is a generalized analytic function at some point p∈Rn +, then it is so in a neighbourhood of p in Rn +. Summarizing, the assignment Gn:V→ Gn(V),whereVis an open subset of Rn +and Gn(V)is the set of generalized analytic functions in V, is a sheaf of R-algebras of continuous functions over Rn +, where the stalks Gn,pare local algebras. Moreover Gncontains the sheaf Onof analytic functions, where On(V)is the R-algebra of real functions in Vwhich extend to real analytic functions on some open neighbourhood of Vin Rn. With this formalism, and taking as local models the locally ringed spaces On:= (Rn +,On) and Gn:= (Rn +,Gn), we define both the categories of standard and generalized (real) 123 Stratified reduction of singularities... Page 7 of 28 4 analytic manifolds (with boundary and corners). The objects in these categories are called O-manifolds and G-manifolds, respectively. In order to treat both together we write Ato make reference either to Oor to G,andAto refer either to Oor G.AnA-manifold of dimension nis a locally ringed space M=(M,AM),whereMis a second countable Hausdorff topological space (the underlying space)andAMis a subsheaf of the sheaf C0 Mof germs of continuous real functions on M(the structural sheaf), which is locally isomorphic to the local model An. That is, given p∈Mthere is an open neighbourhood Vof pin M, an open subset U of Rn +and a homeomorphism ϕ:V→Uinducing an isomorphism of the locally ringed spaces (ϕ, ϕ#):(V,AM|V)∼ −→ (U,An|U), where ϕ# p:An,ϕ(p)→AM,pis given by the composition g→ g◦ϕ(as germs). A morphism between two A-manifolds is just a morphism as locally ringed spaces, induced by composition with continuous maps on the underlying spaces (with an abuse of language, we frequently identify morphisms with the corresponding continuous maps). A couple (V,ϕ)in the above conditions is called a local chart of Mat p, the components x=(x1,x2,...,xn) of the isomorphism ϕ:V→Uare local coordinates at p, and a family of local charts {(Vj,ϕj)}j∈Jsuch that M=∪ j∈JVjis an atlas of M. Let M=(M,AM)be an A-manifold. Note that the underlying space Mis a topological manifold with boundary, denoted by ∂M, and that the restriction (M\∂M,AM|M\∂M)is a standard analytic manifold without boundary (consequently, generalized analytic manifolds without boundary are also standard). Also there is a natural stratification SMof Mdescribed as follows. If p∈M,and(V,ϕ)is a local chart at p, the number epof vanishing coordinates in ϕ(p)(equal to #Iϕ(p)) does not depend on the local chart (V,ϕ)chosen (see [14]). In that way, there is a well-defined map e:M→{0,1,...,n},p→ ep, which is upper semi-continuous. The elements of SMare the connected components of the fibers of e.GivenS∈SM, let us write eS:= ep,wherepis any point in S. Observe that (S,AM|S)is a standard analytic manifold of dimension n−eS. In particular, the boundary ∂Mcorresponds exactly with the points p∈Mwith ep>0, that is, ∂Mis equal to the union of strata of dimension strictly smaller than n.Wehavealsothat,∂Mis a normal crossings divisor with respect to the structural sheaf. That is, for each p∈∂M, there exists a local chart (V,ϕ)of Mat psuch that ∂M∩V={q∈V:x1(q)·x2(q)···· ·xep(q)=0}, where (x1,x2,...,xn)are the coordinates associated to ϕ. Example 2.1 Let ¯ Okbe the sheaf of real (standard) analytic functions in Rk. The locally ringed space (Rk,¯ Ok)is a generalized and standard analytic manifold, with a single chart ψk:Rk→(0,∞)kdefined by (a1,a2,...,ak)→ (ea1,ea2,··· ,eak). We observe at this point that the product is defined in the category of A-manifolds. That is, given two generalized or standard analytic manifolds M1=(M1,AM1)and M2= (M2,AM2)of dimensions nand m, respectively, there is a natural A-manifold of dimension n+m, that we denote by M1×M2=(M1×M2,AM1×M2), unique up to isomorphism, solving the “product universal property”. Without too much detail, the sheaf AM1×M2is constructed as follows. Given a point (p,q)∈M1×M2and two coordinate charts ϕ1: 123 4 Page 8 of 28 B. Molina-Samper et al. V1→U1and ϕ2:V2→U2at pand qrespectively, we have that AM1×M2,(p,q)={f◦(ϕ1×ϕ2)(p,q):f∈An+m,(p,q)}, where (p,q)=(ϕ1(p), ϕ2(q)). Example 2.2 The product (Rk,¯ Ok)×(Rn−k +,An−k),whereA∈{O,G}, has a natural structure of A-manifold by means of the homeomorphism ψk×id, where ψkhas been introduced in Example 2.1. We refer to this product by writing (Rk×Rn−k +,Ak,n−k). Remark 2.3 Let us consider a point p∈Mwith ep=kand let (V,ϕ) be a local chart of Mat p. Up to permutation, we can assume that ϕ(p)=(a1,a2,...,ak,0,...,0)with ai= 0foralli∈{1,2,...,k}. We can split the local coordinates xdefined by ϕin two groups x=(y,z),wherey=(y1,y2,...,yk)are standard analytic functions at pand z=(zk+1,zk+2,...,zn)are generalized functions. By means of translations y i=yi−ai in the analytic coordinates we obtain a new isomorphism ϕ:V→ (ψk×id)−1(ϕ(V)) ⊂Rk×Rn−k +. We consider also ϕas a coordinate chart centered at p in the sense that ϕ(p)=0∈ Rk×Rn−k +, and we usually assume that our charts are centered charts. Letusrecallnowtheexpressionincoordinatesofthecontinuousmapsinducingmorphisms of generalized functions (details in [14, Proposition 3.16]). Consider two generalized analytic manifoldsM1=(M1,GM1)andM2=(M2,GM2)and acontinuousfunctionφ:M1→M2 inducing a morphism between M1and M2.Givenp∈M1and q=φ(p)∈M2,take (Vp,ϕp),(Wq,ψ q)chartscenteredat pandq,respectively.FollowingnotationinRemark2.3, denote by yand zthe kstandard and n−kgeneralized coordinates defining ϕp, respectively. Up to permutation, we can assume also that the first kcoordinates defining ψqare standard and the other n−kare generalized. Then, the j-th component ˜ φjof ˜ φ=ψq◦φ◦ϕ−1 pis a generalized analytic function and for j=k+1,k+2,...,n,wehavethat ˜ φj=zλjUj(y,z), Uj(0,0)= 0,λ j∈Rn−k +\{0}.(4) Moreover, if φinduces an isomorphism, we have that φis a homeomorphism, n=n, k=k,themapt∈Rk→ (˜ φ1(t,0), ˜ φ2(t,0),..., ˜ φk(t,0)) is an analytic isomorphism, and, if we write λj=(λj,1,λj,2,...,λj,n−k)in Eq. (4), up to a permutation of coordinates zwe have λj,j−k>0,λ j, =0,∈{1,2,...,n−k}\{ j−k},(5) for all j=k+1,k+2,...,n. We end this section introducing some notation and definitions concerning the strata of the natural stratification SM.GivenastratumSin SM,denotebySthe closure of Sin M,and define dim(¯ S):= dim(S). We write ZM:= {S⊂M:S∈SM}.Forj=0,1,...,n, denote by Zj Mthe set of elements in ZMwith dimension j,thatis Zj M={¯ S∈ZM:eS=n−j}. The elements of Z0 Mare the strata of dimension 0, and are called corner points, the elements of Z1 Mare called edges and the elements of Zn−1 Mare called components of ∂M.Notethat ∂Mis the union of its components. For each Z∈ZM, we denote by ZM(Z)the subset of ZMwhose elements are contained in Z, and for each j=0,1,...,n, we write Zj M(Z)=ZM(Z)∩Zj M. We write for short 123 Stratified reduction of singularities... Page 9 of 28 4 p∈Z0 Minstead of {p}∈Z0 M, and when no confusion arises, we will put Zinstead of ZM, Zjinstead of Zj M,etc. 2.3 Monomial complexity along strata We introduce in this section the concept of monomial complexity along a stratum and the definition of stratified monomial type function. Let us consider a generalized analytic manifold M=(M,GM)and a stratum Sof its natural stratification S. Take a local chart (V,ϕ)of Mcentered at some p∈S, write e=eS and k=dim S=n−e. We can split the coordinates defining ϕ, up to reorder them, as (y,z), wherey=(y1,y2,...,yk)arestandardanalyticcoordinatesin S∩Vandz=(z1,z2,...,ze) are generalized functions such that S∩V={q∈V:z1(q)=z2(q)=··· = ze(q)=0}. Shrinking Vif necessary, the chart ϕprovides an isomorphism p ϕ:R{Y,Z∗}→GM,p,s→ fs◦ϕ, where Yand Zare kand etuples, respectively, and fsis the sum of the power series s introduced in Eq. (3). Given f∈GM,pand s∈R{Y,Z∗}the mixed power series such that p ϕ(s)=f, we denote SuppS(f;ϕ) =Supp(sZ)⊂Re +,Suppmin,S(f;ϕ) =Suppmin(sZ)⊂Re +,(6) where sZ∈R{Y}{Z∗}has been introduced in Eq. (1). Lemma 2.4 Let S be a stratum in Swith e =eS. Take an open subset U of M such that U∩S=∅, and a function f ∈GM(U). Consider two local charts (V1,ϕ 1)and (V2,ϕ 2), centered at p and q respectively, with p,q∈S∩U. There exists a tuple (γ1,γ 2,...,γ e)∈ Re >0such that (λ1,λ 2,...,λ e)∈Suppmin,S(fq;ϕ2)if and only if (γ1λ1,γ 2λ2,...,γ eλe)∈ Suppmin,S(fp;ϕ1). Proof Using that Sis path connected and by compactness of a given path from pto q,we can reduce the problem to the case where both points pand qbelong to the same connected component Wof U∩V1∩V2. Write y=(y1,y2,...,yn−e),z=(z1,z2,...,ze),and also ¯ y=(¯y1,¯y2,..., ¯yn−e),¯ z=(¯z1,¯z2,...,¯ze), where, up to reordering, (y,z)are the coordinate functions associated to ϕ1and (¯ y,¯ z)are the ones associated to ϕ2,insuchaway that y|S∩V1,¯ y|S∩V2are analytic coordinates in W∩S. That is, we have W∩S={z1=z2=···= ze=0}={¯z1=¯z2=···= ¯ze=0}. In view of Eqs. (4)and(5), up to reordering the variables z, the change of coordinates ϕ2◦ϕ−1 1satisfies, for any j=1,2,...,n−eand =1,2,...,e,that¯yj=gj(y,z),and ¯z=zγ h(y,z),wheregj,hare generalized analytic functions such that y→ gj(y,0)is a standard analytic non-constant function, γ>0andh(0,0)= 0. We summarize these expressions by writing ¯ y=gand ¯ z=zγh.If2:= Suppmin,S(fq;ϕ2)={μ1,μ 2,...,μ t}, the expression of fin coordinates (¯ y,¯ z)is f|W=¯ zμ11(¯ y,¯ z)+¯ zμ22(¯ y,¯ z)+···+¯ zμtt(¯ y,¯ z), where j(¯ y,0)≡ 0, for any j=1,2,...,t. Applying the change of coordinates in order to get the expression of fin (y,z),weobtain f|W=zγμ 11(y,z)+zγμ 22(y,z)+···+zγμ tt(y,z), k(y,z)=hμkk(g,zγh), 123 4 Page 16 of 28 B. Molina-Samper et al. 3.3 Abundance of standardizations of monomial manifolds In this section, we define m-standardizations, we give a characterization for their combinatorial data and we prove a result of abundance of m-standardizations of a fixed monomial G-manifold. Let us fix a monomial generalized analytic manifold (M,a).Alocal m-standardization of (M,a)at a corner point pis just an m-chart updefined in the whole open set V p,such that if xp∈a,thenup◦x−1 pis given by monomial relations of the form up,i=xαp,i p,i,where αp,i∈R>0,for all i∈Ip.(12) We represent this change of coordinates by means of the map αp:Ip→R>0defined by i→ αp,i. In that way, the change of coordinates up◦x−1 pis codified by the matrix of exponents Dαp:Ip×Ip→R>0, where we recall that (once an order in Ipis fixed) Dαpis a diagonal matrix with the elements αp,iin the diagonal. Definition 3.12 Anm-standardization of (M,a)is a pair (O,b),whereOis a standardization of Mand b={up}p∈Z0is a monomial atlas of N=(M,O)such that upis a local mstandardization of (M,a)for every corner point p∈Z0.Thecombinatorial data of an m-standardization (O,b)is the collection of maps (O,b)={αp}p∈Z0. Remark 3.13 If (O,b)and (O,b)are m-standardizations of (M,a),thenbnecessarily that b=bas we have already noted in Remark 3.3. Note also that the m-standardization (O,b) is completely determined by the combinatorial data (O,b). Lemma 3.14 A collection of maps ={αp:Ip→R>0}p∈Z0is the combinatorial data of an m-standardization of (M,a)if and only if for any pair of corner points p,q∈Z0the following relations hold: αp, =γpq αq,,for all ∈Ip∩Iq,(13) where γpq is the weight connexion function from p to q. Proof Let us assume first that =(O,b),where(O,b)is an m-standardization of (M,a). Let us denote N=(M,O)and let C(N,b)be the combinatorial data of the monomial standard analytic manifold (N,b).InviewofEq.(11), it is enough to prove Eq. (13) for two corner points pand qconnected through a compact edge Y. Let us consider the m-charts up,uq∈b at pand q, respectively. The change of coordinates up◦u−1 qis codified by a matrix of exponents A=Apq Y∈C(N,b). This change must be standard analytic in its domain of definition uq(V p∩V q)=R×Rn−1 +, and this implies Ai∈Z+,(A−1)j∈Z+,for all i∈Iq,j∈Ip,∈IY.(14) Let C=Cpq Y∈C(M,a)and αp,α q∈(O,b).NotethatAis obtained as the product A=DαqCD−1 αp:Iq×Ip→R. When ∈Ip∩Iq=IY, in view of Lemmas 3.7 and 3.9,wehave A =γpq αq, αp, ∈Z+,(A−1) =(Aqp Y) =γqp αp, αq, =αp, γpq αq, =1/A ∈Z+, 123 Stratified reduction of singularities... Page 17 of 28 4 which shows A =(A−1) =1. From here we get αp, =γpq αq,, and hence satisfies Eq. (13)aswewanted. AssumenowthatsatisfiesEq.(13)foranypairofcornerpoints p,q∈Z0.Ateachcorner point p∈Z0, consider the m-chart updefined on V psuch that the change of coordinates up◦x−1 psatisfies up,j=xαp,j p,j,forall j∈Ip,whereαp∈and xp∈a.Inthatway,we get a new monomial atlas b={up}p∈Z0of M. Let us see that the changes of coordinates uq◦u−1 pare standard analytic for any pair of corner points pand q.InviewofEq.(9)it is enough to suppose that pand qare connected through an edge Y. Defining the matrix A=DαqCD−1 αp, the change of coordinates uq◦u−1 pis given by uq,i= j∈Ip uAij p,j,for any i∈Iq. It suffices to show that Asatisfies the conditions in Eq. (14). Indeed, if Ai∈Z+for i∈Iq and for all ∈IY,thenuq,iin the above equation is standard analytic in terms of the variables upin the domain V p∩V q={up,ip= 0}∩{uq,iq= 0}(the same interchanging pand qif (A−1)j∈Z+for j∈Iqand any ∈IY). Applying Lemma 3.7 we get that Ar=0and (A−1)r=0, for all r,∈IYwith r= . Moreover, the same lemma assures that Ciq=0 and that (Cip)−1=Cqp ip=0, for all ∈IY; hence, for any such index ∈IYwe obtain Aiq=Ciqαq,iq/αp, =0,(A−1)ip=(Cip)−1αp,ip/αq, =0. Again by Lemma 3.7 we get A =Cαq, αp, =γpq αq, αp, ,(A−1) =(C−1)αp, αq, =γqp αp, αq, , for all ∈IY. Using Lemma 3.9 and Eq. (13) we conclude A =(A−1) =1. As a conclusion, the atlas bdefines a standard analytic structure N=(M,O)over M,where M=(M,GM); thus O⊂GMis a standardization of M. Moreover, by definition of b,we have that (O,b)is an m-standardization of (M,a)with (O,b)=. In the sequel, a collection of maps ={αp:Ip→R>0}p∈Z0is called realizable for(M,a) if Eq. (13) holds for any pair of corner points p,q∈Z0. Definition 3.15 Let upbe a local m-standardization of (M,a)at a given corner point p∈Z0. An extension of upis a (global) m-standardization (O,b)of (M,a)such that up∈b;we sayalsothat(O,b)extendsup.WedenotebyE(up)the set of extensions of up. Proposition 3.16 Let (M,a)be a monomial generalized analytic manifold. Then: (a) There is a bijection between the set of m-standardizations of (M,a)and RN >0,whereN is the number of boundary components of ∂M. (b) Given a corner point p ∈Z0and a local m-standardization upat p, there is a bijective map RN−n >0→E(up), where n is the dimension of M. Proof We start with the proof of the first assertion (a). Let Ibe the set of indices labelling the components of ∂M,thatis∂M=i∈IEi,whereN=#I, and let us fix a collection of corner points q={qi}i∈Iin such a way that qi∈Eifor each i∈I. Given a map β∈RI >0, we take β={αp:Ip→R>0}p∈Z0to be the family of maps defined by αp, =γpq β,for all p∈Z0and ∈Ip. 123 4 Page 18 of 28 B. Molina-Samper et al. Let us see that βis a realizable family of maps. Fix two corner points pand q,andlet ∈Ip∩Iq.ByEq.(11)wehavethatγpq γqq =γpq . Moreover, by the definition of αq, and as a consequence of Lemma 3.9,wehavethatβ=γqq αq,. Then we obtain αp, =γpq β=γpq γqq αq, =γpq αq, which is the required condition for βto be realizable. Now, in view of Lemma 3.14,there exists a unique m-standardization (Oβ,bβ)with (Oβ,bβ)=β.Finally,weshowthatthe map q:RI >0→m-standardizations of (M,a),β→ (Oβ,bβ) is a bijection. Indeed, if β= β,wehavethatβ= βand hence (Oβ,bβ)= (Oβ,bβ) taking into account Remark 3.13. On the other hand, given an m-standardization (O,b)with combinatorial data ={αp}p∈Z0,wehavethat(O,b)=q(β),whereβis defined by βi=αqi,i,foralli∈I. The proof of (a) is finished. Letus prove nowthesecond assertion (b).Denoteby αp:Ip→R>0themapof exponents defining up,thatis up,i=xαp,i p,i,for all i∈Ip,where xp∈a. Consider the injective map iαp:RI\Ip >0→RI >0,definedby δ→ iαp(δ) := βδ,where βδ i=δiif i∈I\Ip, αp,iif i∈Ip. Take a collection of corner points qp={qi}i∈Isuch that qi=p, for eachi∈Ip,andqi∈Ei, for each i∈I\Ip. Using the notations in item a) above, we have that qp(β) ∈E(up)if and only if β|Ip=αp, or equivalently β=iαp(β|I\Ip). In other words, we have the equality E(up)=Im(qp◦iαp), and hence we have the bijection RI\Ip >0→E(up)mapping δinto qp(iαp(δ)). We finish just by noting that #Ip=n. 3.4 The monomial Voûte Etoilée In this section we give the definition of m-combinatorial blowing-up and we introduce the concept of “monomial voûte étoilée” over an m-manifold, whose elements, called m-stars, are sequences of monomial blowing-ups starting from that m-manifold. The terminology is inspired by Hironaka [10,11]. Let (M,a)be a monomial generalized analytic manifold. An m-combinatorial center of blowing-up for (M,a)is a tripet (Z,O,b),whereZis a combinatorial geometric center for Mand (O,b)is an m-standardization of (M,a). Given such an m-combinatorial center (Z,O,b), we consider the blowing-up πξ:Mξ→Mwith center ξ=(Z,O).LetIbe an index set labelling the components of ∂M. We write ∞/∈Ito label the exceptional divisor E∞:= π−1 ξ(Z), and we put Iξ=I∪{∞}as an index set for the components of ∂Mξ.More precisely, given i∈I, it represents both the boundary component Eiof ∂Mand its strict transform E i=π−1 ξ(Ei\Z)⊂∂Mξ, 123 Stratified reduction of singularities... Page 19 of 28 4 belonging to Zn−1 Mand Zn−1 Mξ, respectively. The index ∞∈Iξrepresents E∞∈Zn−1 Mξ. Proposition 3.17 There is a monomial atlas aξof Mξin such a way that πξdefines a morphism of monomial G-manifolds from (Mξ,aξ)to (M,a). Proof Take a corner point pin Mξand let p=πξ(p). Note that pis a corner point in M. Let xp∈abe the m-chart of the atlas aat p. We distinguish two situations: Casep/∈E∞.WehavethatIp=Ipand the blowing-up πξinduces an isomorphism between V∗ pand V∗ p. We take affine coordinates x pover V pdefined by ˜x p,i=xp,i◦πξ|V∗ p,for all i∈Ip. Thus, the expression of πξin coordinates x pand xpis purely monomial. This expression can be codified with the matrix of exponents Bp:Ip×Ip→R≥0given by Bp(i,j)=δij,i,j∈Ip,(15) where δij is the Kronecker delta symbol. In other words, Bp=D1Ip. Casep∈E∞.WehaveIp=Ip\{ j}∪{∞},forsomej∈IZ(see for instance [15] for details in the combinatorial treatment of blowing-ups). By hypothesis, the pair (O,b)is an m-standardization of (M,a); in particular, bis a monomial atlas of the standard analytic manifold N=(M,O). Using this information, together with the definition of blowingup centered at ξ, we get that there exists an m-chart x pdefined in V psuch that the map x p◦πξ◦x−1 pis purely monomial with associated matrix of exponents Bp:Ip×Ip→R+ given by (r,s)→ ⎧ ⎨ ⎩ 1ifr=sand r∈Ip\{ j}, αp,j/αp,rif s=∞ and r∈IZ, 0 otherwise. (16) where αp∈(O,b). With an appropriate order of rows and columns, Bpcan be seen as the upper triangular matrix ⎛ ⎝ Idn−s0 0 0Ids−1a 0 0 1 ⎞ ⎠∈Rn×n +, where s=#IZand a∈Rs−1 >0is a column vector whose entries are defined by the quotients αp,j/αp,r, with r∈IZ\{ j}. The collection aξ={x p}p∈Z0 ξwith Z0 ξ=Z0 Mξis thus a monomial atlas in Mξ.Moreover, the blowing-up πξinduces a morphism from (Mξ,aξ)to (M,a)and the associated combinatorial data is Bπξ={Bp}p∈Z0 ξ. From now on, given an m-combinatorial center of blowing-up (Z,O,b)for a monomial generalized analytic manifold (M,a), and the blowing-up morphism πξ:Mξ→M, with center at ξ=(Z,O),wealwaysconsiderMξendowed with the monomial atlas aξ constructed in Proposition 3.17. Moreover, we also write πξ:(Mξ,aξ)→(M,a), toemphasizethatthemorphismπξis considered also as a morphism of monomial generalized analytic manifolds, and we call it an m-combinatorial blowing-up of (M,a). The associated 123 4 Page 20 of 28 B. Molina-Samper et al. combinatorial data Bπξ={Bp}p∈Z0 ξof this morphism has been made explicit in Eq. (15), for points p∈Z0 ξwith p/∈E∞andinEq.(16), for points p∈Z0 ξ(E∞). Definition 3.18 Let (M,a)be a monomial generalized analytic manifold. An m-star over (M,a)is the composition σ=π0◦π1◦···◦πr−1of a finite sequence of m-combinatorial blowing-ups. That is σ:(Mr,ar)πr−1 −−→ (Mr−1,ar−1)πr−2 −−→··· π0 −→ (M0,a0)=(M,a), where for each k=0,1,2,...,r−1, the morphism πkis an m-combinatorial blowing-up of (Mk,ak). The integer rand the monomial generalized analytic manifold (Mr,ar)are called, respectively, the age and the end of the m-star σ. The collection Vm (M,a)of all the m-stars over (M,a)is called the monomial voûte étoilée of (M,a). 4 Stratified reduction of singularities via principalization of m-ideals We devote this section to introducing the concept of m-ideal, in order to prove a theorem of principalization. With this result we prove the stratified reduction of singularities for a global function defined in generalized analytic manifolds admitting a monomial structure. Finally, we apply this result to prove the main result of this paper stated in Theorem 1.1. 4.1 Principalization of m-ideals Let us fix a monomial generalized analytic manifold (M,a),whereM=(M,GM). Take a global generalized analytic function f∈GM(M)and two corner points p,q∈Z0. Let xq,xq∈abe the m-charts at pand q, respectively, and let Cpq ∈C(M,a)be the matrix of exponents codifying the change of coordinates xq◦x−1 p. The relation between the supports of fat pand qwith respect to these coordinates is given by: Suppp(f;xp)={λqCpq :λq∈Suppq(f;xq)}⊂RIp +.(17) Definition 4.1 A generalized analytic global function m∈GM(M)is said to be an m-function in (M,a)if for each p∈Z0,thereisamapλp:Ip→R+, such that m|V p=xλp p,where xp∈a. The combinatorial data of mis the list Lm={λp}p∈Z0. Let us consider an m-function min (M,a)with combinatorial data Lm={λp}p∈Z0.By Eq. (17), for any pair of corner points p,q∈Z0we have the relation λp=λqCpq,where Cpq ∈C(M,a). In particular, we get that λp, =λq,γpq ,for any ∈Ip∩Iq,(18) where γpq is the weighted connexion function from pto q. Indeed, in view of Eq. (11), it is enough to check Eq. (18) for the case where pand qare connected through an edge Y.For this case, it holds as a consequence of Lemma 3.7. Remark 4.2 Given a list of maps L={λp:Ip→R+}p∈Z0satisfying λp=λqCpq,forany pair of corner points p,q∈Z0, there exists an m-function msuch that Lm=L. 123 Stratified reduction of singularities... Page 21 of 28 4 Definition 4.3 Afinitely generated m-ideal in (M,a)is a sheaf of ideals J⊂GMgenerated by finitely many m-functions. That is, J=m1GM+m2GM+···+mkGM=: (m1,m2,...,mk), where m1,m2,...,mkare m-functions called m-generators of J. Notation 4.4 Let Ibe a finite index set and let Abe a finite subset of RI. We denote by Amin the set of elements in Athat are minimal with respect to the division order ≤din RI. Let Jbe an m-ideal in (M,a)with set of m-generators G={m1,m2,...,mk}. For each i=1,2,...,k, let us write Lmi={λi p}p∈Z0. Given a corner point p∈Z0and xp∈a the m-chart at p, the restriction J|V pis an m-ideal in the m-corner (M|V p,xp)with set of m-generators equal to G|V p:= {m1|V p,m2|V p,...,mk|V p}. Consider the set G,p:= {λ1 p,λ 2 p,...,λ k p}⊂RIp +. Note that if (G,p)min ={μ1 p,μ 2 p,...,μ kp p},then J|V p=(xμ1 p p,xμ2 p p,··· ,xμkp p p). (19) The sheaf of ideals Jis called locally principal if at each point a∈M,thestalkJa⊂GM,a is a principal ideal. Using the definition of m-ideal, it is enough to ask this property for the corner points. In terms of the set introduced above, we have that Jis locally principal if and only if (G,p)min is a singleton for any p∈Z0. Let mbe an m-function in (M,a)and take an m-star σ:(M,a)→(M,a).Thetotal transform σ∗m=m◦σis a again an m-function in (M,a).Moreprecisely,ifp∈Z0 M and p=σ(p),thenλ p∈Lσ∗mis given by λ p=λpBσ p,(20) where λp∈Lmand Bσ p∈Bσis the matrix of exponents codifying σat p.IfJis an m-ideal generated by G={m1,m2,...,mk}, then the total transform σ∗Jis also an m-ideal in (M,a)generated by σ∗G:= {σ∗m1,σ∗m2,...,σ∗mk}. The main result in this section is the following one about principalization of m-ideals. Theorem 4.5 Let Jbe a finitely generated m-ideal in a monomial generalized analytic manifold (M,a). There exists an m-star σ∈Vm (M,a)such that σ∗Jis locally principal. To prove this theorem, we can reduce ourselves to the case where Jis generated by two m-functions by considering a clear finite recurrence and the following lemma. Lemma 4.6 Let J=(m1,m2,...,mk)be an m-ideal in (M,a). Assume that Jrs := (mr,ms)is locally principal for any pair of indices r,s∈{1,2,...,k}.ThenJis locally principal. Proof Assume that there is a point p∈Z0such that Jpis not principal. There exist indices r,s∈{1,2,...,k}such that λr p∈Lmrand λs p∈Lmsare not comparable for the division order ≤din RIp. Note that Grs,p=(Grs,p)min ={λr p,λ s p},whereGrs ={mr,ms},and hence Jrs is not locally principal, which is a contradiction.  123 4 Page 22 of 28 B. Molina-Samper et al. Thecaseoftwogenerators.Let us assume that J=(m,n)is an m-ideal in (M,a)generated by two m-functions and write Lm={λp}p∈Z0and Ln={μp}p∈Z0. We introduce first several definitions, mainly inspired by the “b-invariant” introduced in [6] by van den Dries and Speissegger. Let Z∈Zn−2be a codimension two combinatorial geometric center for M.Weknow that the index set IZhas just two elements, say IZ={i,j}.Letp∈Z0(Z)be a corner point in Z. We say that Zis uncoupled fo Jat p if (λp,i−μp,i)(λp,j−μp,j)<0. We say that Zis uncoupled forJif it is so at each p∈Z0(Z). Lemma 4.7 A combinatorial geometric center Z ∈Zn−2is uncoupled for Jif and only if there is a corner point q ∈Z0(Z)such that Z is uncoupled for Jat q. Proof Assume that Zis uncoupled for Jat a corner point q∈Z0(Z)and take any other point p∈Z0(Z).ByEq.(18), we have λp, =λq,γpq ,forall∈Ip∩Iq,whereγpq is the weighted connexion function from pto q.Sincep,q∈Z, we know that 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 for Jalso at p, and we conclude that Z is uncoupled for J. Observe that if p∈Z0and there are no uncoupled centers for Jpassing through p,then we necessarily have that λp≤dμpor μp≤dλp,thatisJpis a principal ideal. Definition 4.8 Let Jbe the family of codimension two combinatorial geometric centers in Mthat are uncoupled for J, and define the invariant of Jto be InvJ:= #J. We have that InvJ=0 if and only if Jis locally principal. Thus, the objective now is to find an m-star σ∈Vm (M,a)such that Invσ∗J=0. Suppose that InvJ>0andfixZ∈J. Take a corner point p∈Zand pick a local m-standardization upof (M,a)at pdefined by the map αp:Ip→R>0. We say that upis adapted toJwith respect toZ if αp,j(λp,i−μp,i)+αp,i(λp,j−μp,j)=0. A global m-standardization (O,b={up}p∈Z0)of (M,a)is said to be adapted to Jwith respect toZ if upis adpated to Jwith respect to Zfor every p∈Z0(Z). Lemma 4.9 An m-standardization (O,b)is adapted to Jwith respect to Z if and only if there is a corner point q ∈Z such that uq∈bis adapted to Jwith respect to Z. Proof Denote =(O,b). Assume that there is a corner point q∈Zsuch that uq∈bis adapted to Jwith respect to Z. Take any other corner point p∈Z. In view of the realizability of established in Lemma 3.14 we know that αq, =γqp αp,,forall∈Ip∩Iq.Since p,q∈Zwe have that IZ={i,j}⊂Ip∩Iq. Then, by Eq. (18), we get 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 Stratified reduction of singularities... Page 23 of 28 4 As a consequence, the local m-standardization up∈bis adapted to Jwith respect to Z at p. We conclude that (O,b)is adapted to Jwith respect to Z. A codimension two combinatorial center of blowing-up ξ=(Z,O,b)is adapted to Jif Z∈Jand (O,b)is an m-standardization adapted to Jwith respect to Z. The next result assures the existence of such a center. Lemma 4.10 Assume that InvJ>0. Then, there exist codimension two combinatorial centers of blowing-up adapted to J. Proof By definition InvJ>0 if and only if J=∅. Fix an element Z∈Jand let us see that there are m-standardizations adapted to Jwith respect to Z. In view of Lemma 4.9, it is enough to prove the existence of an m-standardization (O,b)adapted to Jat a corner point p∈Z. Fix any corner point p∈Z.SinceZis uncoupled for J,wecanassume,upto exchanging the indices iand j,that i=λp,i−μp,i>0,and j=μp,j−λp,j>0. Take αp:Ip→R>0to be a map such that αp,i=iand αp,j=j, and take the m-chart up=xαp pdefined in V p. Any m-standardization extending upis adapted to Jwith respect to Zat the point pbecause of the definition of αp. Moreover, such an extension exists as a consequence of Proposition 3.16. We conclude by applying finitely many times the following result. Proposition 4.11 Let J=(m,n)be an m-ideal with InvJ>0. Given an m-combinatorial center of blowing-up ξ=(Z,O,b)adapted to J, the blowing-up πξ:Mξ→Mcentered at ξsatisfies Invπ∗ ξJ=InvJ−1. Proof Let us write Zξ=ZMξ,π=πξ,E∞=π−1(Z)and IZ={i,j}. Denote also Lm={λp}p∈Z0,Ln={μp}p∈Z0,Lπ∗m={λ p}p∈Z0 ξ,Lπ∗n={μ p}p∈Z0 ξ. Let TbeacodimensiontwocombinatorialgeometriccenterinMdifferentfrom Z.Denote by STthe stratum in SMsuch that ST=T. The closure Tof π−1(ST)is a codimension two geometric center in Mξhaving index set IT=IT={r,s}. Given a corner point p∈T, let p=πξ(p).Wehave λp,r=λp,r,μ p,r=μp,r,λ p,s=λp,s,μ p,s=μp,s, inview ofthe relation between λp,λpand μp,μpestablishedin Eq. (20), and the expression of Bπ p∈BπgiveninEqs.(15)and(16).Then,wehavethatT∈π∗Jifandonly if T∈J, that is Tis uncoupled for π∗Jif and only if Tis uncoupled for J. Let us see now that any element in π∗Jis among the ones considered before. That is, let us show that there is no codimension two combinatorial geometric center Zuncoupled for π∗Jcontained in E∞. Take a codimension two combinatorial geometric center Z⊂E∞and a point p∈ Z0 ξ(Z). In view of Lemma 4.7, it is enough to prove that Zis not uncoupled for π∗Jat p. More precisely, if we write IZ={k,∞},wewanttoshowthat (λp,k−μp,k)(λp,∞−μp,∞)≥0. Let us consider p=π(p)and the local data αp∈(O,b). The corner point pbelongs to Z, and the affine coordinates up∈bdefine a local m-standardization adapted to Jwith 123 4 Page 24 of 28 B. Molina-Samper et al. respect to Zat p, that is, we have the relation αp,j(λp,i−μp,i)+αp,i(λp,j−μp,j)=0. We know that Ip=Ip\{ j}∪{∞}, up to exchanging the indices iand j. Hence, the matrix B:= Bp:Ip×Ip→R+satisfies, using Eq. (16): B  =1,for ∈Ip\{ j},B j∞=1,B i∞=αp,j αp,i =μp,j−λp,j λp,i−μp,i ,B rs =0 otherwise. By Eq. (20) we get the relations 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 are done.  4.2 Stratified reduction of singularities in monomial manifolds We use the result of principalization of m-ideals in order to prove the following statement: Proposition 4.12 Let (M,a)be a monomial generalized analytic manifold with M= (M,GM). Given a generalized analytic function f ∈GM(M), there is an m-star σ∈Vm (M,a) such that the pull-back f =f◦σis of stratified monomial type. For the proof of Proposition 4.12 we associate to fa finitely generated m-ideal Jf,and we prove that the principalization of Jfgives rise to the stratified reduction of singularities of f. Given q∈Z0and λq∈Suppq(f;xq), it makes sense to define the m-function mλqas the one having the collection of maps Lmλq={λqCpq}p∈Z0as a combinatorial data, by Remark 4.2 and Eq. (17). The m-ideal Jfassociated to f is the ideal sheaf generated by the finite set of m-functions Gf= q∈Z0mλq:λq∈Suppmin,q(f;xq). By definition, notice that for any corner point q∈Z0,wehave (Gf,q)min =Suppmin,q(f;xq). (21) where the the notation Gf,qwas introduced in Sect. 4.1 above. Lemma 4.13 Given an m-star τ:(M,a)→(M,a), we have τ∗Jf=Jf,where f= f◦τ. Proof Inviewof Eq. (19),itisenough to provethatfor anycorner point p∈Z0 Mtheequality (τ∗Gf,p)min =(Gf,p)min holds. Denote for short 1=τ∗Gf,pand 2=Gf,p. Fix a point p∈Z0 Mand consider p=τ(p). Write := Suppmin,p(f;xp),wherexp∈ a,andletBτ p∈Bτbethe matrix of exponentscodifying τat p.Let:= {λpBτ p;λp∈}. We prove that both min 1and min 2are equal to min. Step 1: min 2=min.Recall by Eq. (21)thatmin 2=:= Suppmin,p(f;x p),where x p∈a.Let={λ1,λ 2,...,λ k}⊂RIp, that is, the function faround the corner point phas the finite presentation f|V p=xλ1 pU1+xλ2 pU2+···xλk pUk,whereUi(p)= 0, for all 123 Stratified reduction of singularities... Page 25 of 28 4 i=1,2,...,k. Taking into account that τ(V p)⊂V p, the function f=f◦τis written in the chart xp∈aas f|V p=x˜ λ1 p(U1◦τ|V p)+x˜ λ2 p(U2◦τ|V p)+···x˜ λk p(Uk◦τ|V p), where ˜ λi=λiBτ p.SinceBτ pis an invertible matrix, we can assure that ˜ λr= ˜ λsfor any pair of different indices r,s∈{1,2,...,k}. This implies that ={ ˜ λ1,˜ λ2,...,˜ λk}min by definition of minimal support of fat p. We are done, since ={ ˜ λ1,˜ λ2,...,˜ λk}. Step 2: min 1=min.Recall that ⊂Gf,pand denote ˜ =Gf,p\.ByEq.(21) we know that for any μ∈˜ there exists λ∈such that λ≤dμ. Note that 1=∪˜ , where ˜ := {μBτ p;μ∈˜ }. Therefore, we need only to prove the following claim: If λ,μ :Ip→R+satisfy λ≤dμ, then λBτ p≤dμBτ p. For that, it is enough to consider the case where τ=πξis a single m-combinatorial blowing-up with center ξ=(Z,O,b). Denote, as usual, E∞=π−1 ξ(Z). If p/∈E∞or equivalently p/∈Z,wehavethatIp=Ipand Bτ p=D1Ip, and we are done. Assume that p∈E∞,andlet jbe the index in IZsuch that Ip\{∞} = Ip\{ j}. Denote λ=λBτ pand μ=μBτ p.ByEq.(16), we have that λ =λ,μ =μ, and thus λ ≤μ , for all ∈Ip\{∞}; whereas λ ∞= ∈IZ αp,j αp, λ≤ ∈IZ αp,j αp, μ=μ ∞, where αp∈(O,b),aswewanted.  Proof of Propostion 4.12 In view of Theorem 4.5, we can take an m-star σ:(M,a)→ (M,a)suchthatσ∗Jfislocallyprincipal.ByLemma4.13weknowalso thatσ∗Jf=Jf◦σ. Hence, since Gf◦σis a set of generators of Jf◦σ,wehavethat(Gf◦σ,p)min is a singleton for all p∈Z0 M. Finally, by Eq. (21) we obtain mp(f)=#Suppmin,p(f;x p)=#(Gf◦σ,p)min =1 for all p∈Z0 M.Sinceais a monomial atlas, we know that M=p∈Z0 MV p. Thus, given a stratum S∈SM, there is a corner point p∈Z0 Msuch that p∈¯ S.Inviewofthe horizontal stability property for the monomial complexity established in Lemma 2.6,weget mS(f)≤mp(f)=1. We conclude that fis of stratified monomial type.  4.3 Proof of the main statement We end end here the proof of the stratified reduction of singularities for generalized analytic functions, as stated in Theorem 1.1. Recall that we have a generalized analytic manifold M=(M,GM), and a generalized analytic function f∈GM(M)in M. Given a point p∈M, we want to prove that there exist an open neighbourhood V⊂Mof pand a finite sequence of blowing-ups σ:Mr πr−1 −→ Mr−1 πr−2 −→··· π0 −→ M0=(V,GM|V), 123