aa r X i v : . [ m a t h . AG ] J un Poincar´e series of embedded filtrations
Ann Lemahieu ∗ Abstract.—
In this article we define a Poincar´e series on a subspace of a complex analytic germ,induced by a multi-index filtration on the ambient space. We compute this Poincar´e series for subspacesdefined by principal ideals. For plane curve singularities and nondegenerate singularities this Poincar´eseries yields topological and geometric information. We compare this Poincar´e series with the one in-troduced in [E,G-Z]. In few cases they are equal and we show that the Poincar´e series we consider inthis paper in general yields more information.
0. Introduction
In [C,D,K] one introduced a Poincar´e series induced by a filtration on the ring ofgerms of a complex variety. This Poincar´e series has been studied for several kindsof singularities, see for example [C,D,G-Z1], [C,D,G-Z2], [C,D,G-Z3], [Eb,G-Z], [CHR],[L], [GP-H] and [N]. In some cases this Poincar´e series determines the topology of thesingularity and is related to the zeta function of monodromy of the singularity.In these works one considers multi-index filtrations defined by valuations on thelocal ring at the singularity and in [Eb,G-Z] and very recently in [E,G-Z] one considersvaluations on an ambient smooth space of the singularity that correspond to facetsof the Newton polyhedron. Here we study Poincar´e series induced by multi-indexfiltrations coming from arbitrary valuations on the ambient space where at least oneof them is centred at the maximal ideal of the local ring of the singularity consideredin the ambient space. In an upcoming paper, we study this Poincar´e series also forvaluations where none of them is centred at the maximal ideal.The Poincar´e series we introduce here is defined in an algebraic way and differsfrom Poincar´e series studied before in the sense that there is no notion of fibre thatcorresponds to our Poincar´e series. We go into this in Section 1. We compute thisPoincar´e series for a subspace corresponding to a principal ideal. A nice A’Campo typeformula shows up. In Section 2 we compare our Poincar´e series with the one defined in[E,G-Z]. The Poincar´e series in [E,G-Z] is consistent with a notion of geometric fibrebut the price to pay is that this filtration is less richer than the one we consider here.This follows also from the formulas they computed for their Poincar´e series. Whenthe Newton polytope is bi-stellar, we show that both Poincar´e series coincide. Theproperty of bi-stellar generalises the notion of stellar introduced in [E,G-Z]. In Section3 we study this Poincar´e series for a plane curve singularity that is a general element ∗ Ann Lemahieu; K.U.Leuven, Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Bel-gium, email: [email protected]. The research was partially supported by the Fund ofScientific Research - Flanders and MEC PN I+D+I MTM2007-64704.
1n an ideal in C { x, y } where the embedded filtration comes from the Rees valuations ofthe ideal. Using the result in [C,D,G-Z2], we show that the Poincar´e series determinesand is determined by the embedded topology of the plane curve singularity. In Section4 we compare different Poincar´e series for affine toric varieties. In Section 5 we studythe Poincar´e series for nondegenerate singularities where the embedded filtration isnow induced from the Newton polyhedron. We show that one can recover the Newtonpolyhedron from the Poincar´e series and thus, in particular, that the zeta function ofmonodromy can be deduced from the Poincar´e series.
1. Poincar´e series associated to embedded filtrations
Let (
X, o ) be a germ of a complex analytic space and let O X,o be the local ring ofgerms of functions on (
X, o ). Let ν = { ν , · · · , ν r } be a set of order functions from O X,o to Z ∪ {∞} , i.e. functions ν j that satisfy ν j ( f + g ) ≥ min { ν j ( f ) , ν j ( g ) } and ν j ( f g ) ≥ ν j ( f ), for all f, g ∈ O X,o , 1 ≤ j ≤ r . An order function ν j is called a valuationif moreover it satisfies ν j ( f g ) = ν j ( f ) + ν j ( g ), for all f, g ∈ O X,o . In particular, when(
X, o ) is irreducible, ν can be a set of discrete valuations of the function field C ( X )whose valuation rings contain O X,o . The set ν defines a multi-index filtration on O X,o by the ideals M ( v ) := { g ∈ O X,o | ν j ( g ) ≥ v j , ≤ j ≤ r } , v ∈ Z r . If the dimensions of the complex vector spaces M ( v ) /M ( v + 1) are finite for all v ∈ Z r ,then originally (see [C,D,K] and [C,D,G-Z2]) the Poincar´e series associated to thismulti-index filtration was defined as P νX ( t , · · · , t r ) := Q rj =1 ( t j − t · · · t r − X v ∈ Z r dim( M ( v ) /M ( v + 1)) t v . Let us now consider an ideal I in O X,o . We will define a Poincar´e series associatedto an embedded filtration on O X,o / I . Let V be the analytic subspace of X determinedby the ideal I and let the map ϕ : O X,o → O
X,o / I define the embedding of V in X . Weset I ( v ) := ϕ ( M ( v ) + I ). In general the ideals I ( v ) define a multi-index filtration on O V,o which is not induced by a set of order functions on V , i.e. there do not have to existorder functions µ , . . . , µ r on V such that I ( v ) = { g ∈ O V,o | µ j ( g ) ≥ v j , ≤ j ≤ r } .The existence of such functions for { I ( v ) } is equivalent with the condition that I ( v ) ∩ I ( v ) = I ( v ) , (1)where v and v are arbitrary tuples in Z r and v is the tuple of the componentwisemaxima of v and v . Indeed, if Condition (1) holds, then one has order functions µ j on V with µ j ( g ) = v j if and only if g ∈ I (0 , . . . , , v j , , . . . , \ I (0 , . . . , , v j +1 , , . . . , Definition 1.
The
Poincar´e series associated to the embedded multi-index filtration given by the ideals I ( v ) is the series P νV ( t , · · · , t r ) := Q rj =1 ( t j − t · · · t r − X v ∈ Z r dim( I ( v ) /I ( v + 1)) t v .
2t is useful to notice that dim( I ( v ) /I ( v + 1)) = dim ( J ( v ) /J ( v + 1)), where J ( v ) := M ( v ) + I . If { I ( v ) } is defined by order functions µ , . . . , µ r , then one has a notionof fibre because to every g ∈ O V,o can can attach a value µ ( g ) = ( µ ( g ) , . . . , µ r ( g )).Concretely, consider the projection map j v : I ( v ) −→ D ( v ) × . . . × D r ( v ) g ( a ( g ) , . . . , a r ( g )) , where for 1 ≤ j ≤ r the space D j ( v ) = I ( v ) /I ( v + e j ) where e j is the r -tuple with j -thcomponent equal to 1 and the other components equal to 0. As Condition (1) thenholds, the following expressions F v := Im j v ∩ ( D ∗ ( v ) × · · · × D ∗ r ( v )) , (2) F v := ( I ( v ) /I ( v + 1)) \ r [ i =1 ( I ( v + e i ) /I ( v + 1)) (3)coincide and one has that µ ( g ) = v if and only if g + I ( v + 1) ∈ F v . The space F v isinvariant with respect to multiplication by nonzero constants; let P F v = F v / C ∗ be theprojectivisation of F v . We then have P νV ( t ) = X v ∈ Z r χ ( P F v ) t v , where χ ( · ) denotes the Euler characteristic. If such functions µ , . . . , µ r do not existfor { I ( v ) } , then (2) and (3) are possible generalisations of the notion of fibre but wethen lose some geometric meaning.In this paper we will investigate the Poincar´e series as in Definition 1 and we will seein Example 1 that this Poincar´e series is different from the two possible ones inducedby the ‘fibres’ (2) and (3).We now compute this Poincar´e series for I = ( h ) a principal ideal. Theorem 1.
Let ( X, o ) be irreducible, I = ( h ) a principal ideal in O X,o and V theanalytic subspace of ( X, o ) determined by the ideal I . Let ν = { ν , . . . , ν r } be a set ofdiscrete valuations of C ( X ) centred at the maximal ideal of O X,o . We write q = ν ( h ) .Then P νV ( t ) = (1 − t q ) P νX ( t ) . (4) Proof.
For a set A ⊂ { , · · · , r } , let 1 A be the r-tuple with j-th component equal to1 if j ∈ A and else equal to 0, 1 ≤ j ≤ r . The coefficient of t v in the left hand side of(4) is ( − r +1 X A ⊂{ , ··· ,r } ( − A dim O X,o J ( v − A + 1) . The coefficient of t v in the right hand side of (4) is equal to( − r +1 X A ⊂{ , ··· ,r } ( − A dim O X,o M ( v − A + 1) − ( − r +1 X A ⊂{ , ··· ,r } ( − A dim O X,o M ( v − A − q + 1) . A ⊂ { , · · · , r } , it is enough to prove forsome j ∈ A that dim J ( v − A + 1) J ( v − A \{ j } + 1)= dim M ( v − A + 1) M ( v − A \{ j } + 1) − dim M ( v − A − q + 1) M ( v − A \{ j } − q + 1) . This follows immediately from the fact that the kernel of the projection map M ( v − A + 1) M ( v − A \{ j } + 1) −→ J ( v − A + 1) J ( v − A \{ j } + 1)is equal to ( h ) M ( v − A − q + 1)( h ) M ( v − A \{ j } − q + 1) . (cid:4) Example 1.
Let X = C and h ( x, y ) = x y + y . We choose monomial valuations ν and ν on O X,o given by ν ( x a y b ) = 2 a + 3 b and ν ( x a y b ) = 4 a + 3 b . One can computethat the coefficient of t t in (1 − t q ) P νX ( t ) is 0 although both fibres F (20 , as definedin (2) and (3) contain the monomial x y . (cid:3)
2. Geometric embedded filtrations versus algebraic embeddedfiltrations.
Let h : ( C d , o ) → ( C , o ) be a germ of a holomorphic function. Recently in [E,G-Z] oneconsidered a Poincar´e series on O V,o = O C d ,o / ( h ), induced by a Newton filtration . Wewrite h ( x ) = P k ∈ Z d ≥ a k x k , where k = ( k , . . . , k d ) and x k = x k · . . . · x k d d . The supportof h is supp h := { k ∈ Z d ≥ | a k = 0 } . The
Newton polyhedron of h at the origin is theconvex hull in R d ≥ of S k ∈ supp h k + R d ≥ and the Newton polytope of h at the origin isthe compact boundary of the Newton polyhedron of h at the origin.Let ν , . . . , ν r be the monomial valuations on O C d ,o corresponding to the facets ofthe Newton polytope of h , i.e. for a compact facet τ with the affine space through τ given by the equation a x + · · · + a d x d = N τ , the corresponding valuation ν acts asfollows: ν ( x m . . . x m d d ) = a m + · · · + a d m d . W. Ebeling and S. M. Gusein-Zade studythe Poincar´e series P { ω i } ( t ) on V induced by the order functions ω i ( g ) := max { ν i ( g ′ ) | g ′ − g ∈ ( h ) } . On the other hand we can consider the Poincar´e series as defined in Definition 1, with I = ( h ). The Poincar´e series in [E,G-Z] has a more geometric meaning because onehas fibres F v and P { ω i } ( t ) = P v ∈ Z r χ ( P F v ) t v . However, our Poincar´e series - which israther algebraic - contains more information about the singularity ( V, o ). In general, theembedded filtration we introduce in this article is richer because I ( v ) is not necessarily4etermined by I ( v , , . . . , , I (0 , v , , . . . , , . . . , I (0 , . . . , , v d ), whereas the idealsthat appear in the Poincar´e series P { ω i } ( t ) are. We comment further on this in thissection.In [E,G-Z] a Newton polytope is called stellar if all its facets have a common vertex.If the Newton polytope of h is stellar, then they show ([E,G-Z, Theorem 2]) that P { ω i } ( t ) = (1 − t ν ( h ) ) P νX ( t ) . However, if the Newton polytope is not stellar, the information on (
V, o ) can be lost inthe Poincar´e series P { ω i } ( t ). Indeed, for h a germ of a holomorphic function on ( C , o )with o an isolated critical point of h , they prove ([E,G-Z, Theorem 1]) that P { ω i } ( t ) = P νX ( t ) . Proposition 2.
If the ideals I ( v ) satisfy Condition (1), then P { ω i } ( t ) = P νV ( t ) .Proof. It is sufficient to verify that then I (0 , . . . , , v i , , . . . ,
0) = { g ∈ O V,o | ω i ( g ) ≥ v i } . (cid:4) We will now characterise the germs h of holomorphic functions on C d for which Con-dition (1) is satisfied, and thus for which these geometric and algebraic Poincar´e seriescoincide. Obviously this will depend on the Newton polytope of h . We will use thefollowing lemma to give the characterisation. Lemma 3.
Let { M ( v ) } be ideals in a local ring R that satisfy Condition (1) andconsider an ideal I in R . Then for v , v ∈ Z r , the following conditions are equivalent:1. ( M ( v ) + I ) ∩ ( M ( v ) + I ) = M ( v ) + I where v is the tuple of the componentwisemaxima of v and v ;2. ( M ( v ) + M ( v )) ∩ I = ( M ( v ) ∩ I ) + ( M ( v ) ∩ I ) .Proof. Suppose that the first equation of sets holds. Take f ∈ ( M ( v ) + M ( v )) ∩ I .Then f = f + f with f ∈ I , f ∈ M ( v ) and f ∈ M ( v ). Thus f ∈ ( M ( v ) + I ) ∩ ( M ( v ) + I ). By the hypothesis it then follows that f ∈ M ( v ) + I and so wecan write f = g + k with g ∈ M ( v ) and k ∈ I . We have f = ( f − g ) + ( f + g ) and f − g ∈ M ( v ) ∩ I and f + g ∈ M ( v ) ∩ I . Hence f ∈ ( M ( v ) ∩ I ) + ( M ( v ) ∩ I ).We now suppose that the second equation of sets holds. We consider the followingexact sequences: → ( M ( v ) ∩ I ) / ( M ( v ) ∩ I ) → M ( v ) /M ( v ) → ( M ( v ) + I ) / ( M ( v ) + I ) → → ( M ( v ) ∩ I ) / ( M ( v ) ∩ I ) → M ( v ) /M ( v ) → ( M ( v ) + I ) / ( M ( v ) + I ) → We take a set B ′ i of elements in M ( v i ) ∩ I that give rise to a basis of the vector space( M ( v i ) ∩ I ) / ( M ( v ) ∩ I ) , i ∈ { , } . For i ∈ { , } , we use the exact sequences (5) and(6) to add to B ′ i a set of elements B ′′ i of M ( v i ) whose images are a basis of the quotient( M ( v i ) + I ) / ( M ( v ) + I ). So the classes of the elements of B i := B ′ i ∪ B ′′ i form a basisof M ( v i ) /M ( v ), i = 1 ,
2. We also have that the classes of the elements in B ∪ B M ( v ) + M ( v )) /M ( v ). Analogously,the classes of the elements in B ′ ∪ B ′ are a system of generators for the vector space[( M ( v ) ∩ I ) + ( M ( v ) ∩ I )] / ( M ( v ) ∩ I ).Condition (1) implies that the set B ∪ B consists of elements of M ( v ) + M ( v )whose classes module M ( v ) are linearly independent and hence the classes of the el-ements in B ∪ B are a basis of ( M ( v ) + M ( v )) /M ( v ). Indeed, if x is a linearcombination of elements in B and y is a linear combination of elements of B and if x + y = z with z ∈ M ( v ), then x = z − y ∈ M ( v ) ∩ M ( v ). Condition (1) then impliesthat x, y ∈ M ( v ).Let us now take an element f ∈ ( M ( v ) + I ) ∩ ( M ( v ) + I ), so f = f + k = f + k , f i ∈ M ( v i ) and k i ∈ I , i = 1 ,
2. For i = 1 ,
2, we can write f i = f ′ i + f ′′ i + m i with f ′ i a linear combination of elements in B ′ i , f ′′ i a linear combination of elementsin B ′′ i and m i ∈ M ( v ). Then g := k − k = f − f ∈ ( M ( v ) + M ( v )) ∩ I =( M ( v ) ∩ I ) + ( M ( v ) ∩ I ) by the hypothesis. Hence g = g ′ + g ′ + m , with g ′ i a linearcombination of elements of B ′ i , i = 1 ,
2, and m ∈ M ( v ) ∩ I . We obtain( g ′ − f ′ ) − f ′′ + ( g ′ + f ′ ) + f ′′ = m − m − m. The right hand side of this equality is contained in M ( v ). The left hand side is a sumof four terms that are linear combinations of elements respectively from B ′ , B ′′ , B ′ and B ′′ . As the classes of the elements in B ∪ B are a basis of ( M ( v ) + M ( v )) /M ( v ),we get in particular f ′′ = 0. Hence f = m + ( f ′ + k ) ∈ M ( v ) + I . (cid:4) Definition 2.
A Newton polytope is called bi-stellar if every two facets of the Newtonpolytope have a non-empty intersection.
Proposition 4.
The Newton polytope of h is bi-stellar if and only if the ideals M ( v ) +( h ) satisfy Condition (1).Proof. Say the Newton polytope of h has r compact facets inducing the monomialvaluations ν , . . . , ν r on C d . Suppose that the Newton polytope of h is bi-stellar. ByLemma 3, it suffices to show that for all v , v ∈ Z r one has that( I ( v ) ∩ ( h )) + ( I ( v ) ∩ ( h )) = ( I ( v ) + I ( v )) ∩ ( h ) . Let gh ∈ I ( v ) + I ( v )) and q := ν ( h ). We write g = g + g with g = P λ a x a and x a / ∈ I ( v − q )+ I ( v − q ), for all x a in supp g , and g = P λ b x b with x b ∈ I ( v − q )+ I ( v − q ),for all x b in supp g . Suppose that g = 0. We take a monomial x a in supp g . Thenthere exist i, j ∈ { , . . . , r } such that ν i ( x a ) < v ,i − q i and ν j ( x a ) < v ,j − q j .We first consider the case where i = j . Let N be the set of the monomials in supp g that are minimal for the pair ( ν i , ν j ), i.e. x c ∈ N if and only if there does not exista monomial x d in supp g for which ν i ( x d ) < ν i ( x c ) and ν j ( x d ) < ν j ( x c ). Let M be theset of monomials x m in supp h for which ν i ( x m ) = q i and ν j ( x m ) = q j . As the Newtonpolytope is bi-stellar, M is not empty. For the monomials x m x c with x m ∈ M and x c ∈ N , we thus have that ν i ( x m x c ) < v ,i and ν j ( x m x c ) < v ,j . As such monomials donot appear in the support of gh they should be canceled. It follows that at least onesuch monomial x m x c has to be equal to a monomial x h x a ′ with x h ∈ supp h \ M and6 a ′ in supp g . Say ν i ( x h ) > q i . We then find that ν i ( x a ′ ) < ν i ( x c ) and ν j ( x a ′ ) ≤ ν j ( x c )which contradicts the fact that x c ∈ N and thus g = 0.Suppose now that i = j . Let N be the set of monomials with support in g that areminimal for the valuation ν i , i.e. x c ∈ N if and only if there does not exist a monomial x d in supp g for which ν i ( x d ) < ν i ( x c ). Let M be the set of monomials x m in thesupport of h for which ν i ( x m ) = q i . Analogously there then has to be a monomial x m x c with x m ∈ M and x c ∈ N that is equal to a monomial x h x a ′ , with x h ∈ supp h \ M and x a ′ ∈ supp g . Again we get a contradiction because x c would not be minimal for ν i .We now suppose that the ideals M ( v ) + ( h ) satisfy Condition (1). If the Newtonpolytope of h would not be bi-stellar, then there would exist two valuations ν i , ν j with i, j ∈ { , . . . , r } for which the sets of monomials M i = { x m ∈ supp( h ) | ν i ( x m ) = q i } and M j = { x m ∈ supp( h ) | ν j ( x m ) = q j } would be disjoint. Let h i be the part of h with support in M i , so ν i ( h i ) = q i and ν i ( h − h i ) > q i . Then h = h i + ( h − h i ) with ν i ( h i ) < ν i ( h − h i ), ν j ( h − h i ) < ν j ( h i ) and ν ( h ) ≤ ν ( h i ) and ν ( h ) ≤ ν ( h − h i ). ByLemma 3 it follows that h would be contained in ( M ( ν ( h i )) + M ( ν ( h − h i ))) ∩ ( h ) butnot in ( M ( ν ( h i )) ∩ ( h )) + ( M ( ν ( h − h i ))) ∩ ( h )), contradicting Condition (1). (cid:4)
3. Embedded filtrations for plane curve singularities
Let X = C and let I be a primary ideal in O X,o . If φ : Z → C is a principilisationof I , then φ is realised by blowing up a constellation of points { Q σ } σ ∈ G . The map φ factorises through the normalised blowing up of I which we will denote by Bl I ( C ).Let σ be the morphism Z → Bl I ( C ) in this factorisation. For σ ∈ G , we denote theexceptional divisor of the blowing-up in Q σ by E σ , as well as its strict transform underfollowing blowing-ups, and D := ∪ σ ∈ G E σ . Blowing up a point Q σ induces a discretevaluation ν σ on C ( X ) \ { } : for g ∈ C ( X ) \ { } , the value ν σ ( g ) is the order of thepullback of g along E σ . The valuation ν σ is called Rees for I if its centre in Bl I ( X ) isa divisor. We have that ν σ is Rees for I if and only if the strict transform of a generalelement in I intersects E σ (see for example [L-VP, Lemma 8]). Say that E , . . . , E r arethe exceptional components that give rise to Rees valuations ν , . . . , ν r .Let us now consider a general element h in the ideal I and let V be the hypersur-face given by { h = 0 } . In [C,D,G-Z2] one studied the Poincar´e series P νV ( t ) that isdefined by the filtration on O V,o induced by the essential valuations ν of the minimalresolution of the plane curve V . One showed that that Poincar´e series contained thesame information as the embedded topology of the curve and that P νV ( t, . . . , t ) equalsthe zeta function of monodromy.We will now study the Poincar´e series of the embedded filtration on O V,o induced bythe Rees valuations ν = ( ν , . . . , ν r ). For 1 ≤ j ≤ r , suppose that E j is intersected n j times by the strict transform of { h = 0 } . Let E • σ be E σ without the intersection pointswith the other components of D and let E ◦ σ be E σ without the intersection points of theother components of φ − ( h − { o } ). Let I be the intersection matrix of the { E σ } σ ∈ G andlet M = − I − . Let C σ be a curvette through E σ (i.e. the projection by φ of a smoothcurve transversal to E σ and not intersecting other components of D ). The entry m σ,τ in M is then also equal to ν τ ( C σ ). 7 heorem 5. The Poincar´e series P νV ( t ) determines and is determined by the embeddedtopology of { h = 0 } .Proof. By Theorem 1, P νV ( t , · · · , t r ) = (1 − t q · · · t q r r ) P νX ( t , · · · , t r ), with q = ν ( h ).The Poincar´e series P νX ( t ) induced by plane divisorial valuations is computed in [D,G-Z].For σ ∈ G , let m σ = ( m σ, , . . . , m σ,r ). Then one has P νX ( t , . . . , t r ) = Y σ ∈ G (1 − t m σ ) − χ ( E • σ ) . As E j is intersected n j times by the strict transform of { h = 0 } , 1 ≤ j ≤ r , we get q i = P rj =1 n j m i,j , for 1 ≤ i ≤ r . If the curve is irreducible (i.e. r = 1) and if E isintersected by the strict transform then n = 1 and q = m , . We then have P νV ( t ) = (1 − t m , ) Y σ ∈ G (1 − t m σ, ) − χ ( E • σ ) = Y σ ∈ G (1 − t q σ ) − χ ( E ◦ σ ) . Hence it follows that P νV ( t ) is then equal to the zeta function of monodromy ζ V ( t )([A’C]).Suppose now that the curve is reducible. We will show that the factor (1 − t q ) cannot be canceled by a factor (1 − t m σ ) − χ ( E • σ ) of P νX ( t ). As q i = P rj =1 n j m i,j , we havethat q i > m i,j for all j ∈ { , · · · , r } . If ν σ is not a Rees valuation, then there existsalways a valuation ν j which is Rees and such that Q j lies above Q σ . Then m σ,j < m j,j and thus q i > m σ,i for all σ ∈ G . Thus q is the biggest exponent in the cyclotomicfactors in P νV ( t ). This makes that we can extract the value q and the Poincar´e series P νX ( t ) from the Poincar´e series P νV ( t ). In [C,D,G-Z2] it has been shown that P νX ( t )determines the dual graph of the divisors { E σ } σ ∈ G and thus the matrix M . As M isinvertible, it follows that we can now compute the numbers n j , 1 ≤ j ≤ r . Hence thedual resolution graph of { h = 0 } is known and the Poincar´e series P νV ( t ) determinesthe embedded topology of { h = 0 } . (cid:4) For the Poincar´e series we study here we get that the zeta function of monodromy ζ V ( t ) = P νV ( t n , . . . , t n r ) Q rj =1 (1 − t q j ) n j (1 − t P rj =1 n j q j ) . Remark 1. If V = { h = 0 } is a reduced plane curve singularity and Z → C is a con-crete embedded resolution of singularities for V , then the function h becomes a generalelement for some convenient primary ideal I such that IO Z is locally principal. HenceTheorem 5 can be applied to any reduced curve singularity and a chosen embeddedresolution for it and, in particular, for the minimal one.8 . Poincar´e series for toric varieties We now consider the particular case where V is an affine toric variety that is a completeintersection. Let S be a semigroup in M ∼ = Z d such that S +( − S ) = M and S ∩ ( − S ) = 0and let V = Spec C [ S ] be the associated affine toric variety. Let ˇ σ be the cone generatedby S and let { s , · · · , s d + p } be a system of generators of S = ˇ σ ∩ M . Suppose that theembedding of V in C d + p is given by the map ε : C [ x , · · · , x d + p ] → C [ S ] x k χ s k . We set deg( x k ) = s k , 1 ≤ k ≤ d + p . The toric complete intersection V is givenby an ideal generated by binomials h i = x α i − x β i ∈ C [ x , · · · , x d + p ], 1 ≤ i ≤ p .Moreover deg( x α i ) = deg( x β i ) and supp( x α i ) ∩ supp( x β i ) = ∅ . The Newton polytopeof each toric hypersurface h i is a segment τ i , connecting α i and β i . Let τ ∗ i be thedual space in R ∗ d + p ≥ to this segment and let H ∗ i be the hyperplane passing through τ ∗ i . The equation of H ∗ i is P d + pk =1 ( α ik − β ik ) x k = 0. Set τ ∗ := τ ∗ ∩ · · · ∩ τ ∗ p and H ∗ := H ∗ ∩ · · · ∩ H ∗ p . Let N be the dual space to M and let σ be the dual cone toˇ σ . A primitive element n in σ ∩ N defines a discrete valuation ν of C ( V ) by setting ν ( P m ∈ F a m χ m ) = min {h m, n i | m ∈ F, a m = 0 } .A finite set of valuations ν in σ induces a Poincar´e series P νV for V . On the otherhand valuations µ in τ ∗ give rise to ambient ideals M ( v ) ⊂ X = C d + p and hence to aPoincar´e series P µV for V . We now show how both Poincar´e series are related. Theorem 6.
The cones σ and τ ∗ are isomorphic and under this isomorphism one has P νV ( t ) = P µV ( t ) .Proof. We show that there is an isomorphism θ : R d → H ∗ that maps σ to τ ∗ . Let ν ∈ R d and let µ = ( µ , · · · , µ d + p ) be the vector such that µ k = h s k , ν i , 1 ≤ k ≤ d + p .Then obviously P d + pk =1 ( α ik − β ik ) µ k = 0 for 1 ≤ i ≤ p and this implies that µ ∈ H ∗ . Forthe opposite direction, take µ ∈ H ∗ . The equality P d + pk =1 ( α ik − β ik ) µ k = 0 implies thatthere exists a ν ∈ R d such that µ k = h s k , ν i , 1 ≤ k ≤ d + p . This ν is then unique. As σ is the dual cone to ˇ σ , it follows that σ maps to τ ∗ .The Poincar´e series P µV with respect to the valuations µ , · · · , µ r ∈ τ ∗ is inducedby the ideals J ( v ) = ( x λ | h λ, µ j i ≥ v j , ≤ j ≤ r ) + ( h , · · · , h p ) . As h λ, µ j i = h s, ν j i , where s = P d + pk =1 λ k s k , it follows that x λ ∈ J ( v ) if and only if ε ( x λ ) = χ s , with h s, ν j i ≥ v j , 1 ≤ j ≤ r .The Poincar´e series P νV with respect to the corresponding valuations ν j is induced bythe ideals ( χ s | h s, ν j i ≥ v j , ≤ j ≤ r ) . It now follows that both Poincar´e series coincide. (cid:4) . Embedded filtrations for nondegenerate singularities Let h be the germ of a holomorphic function on C d defining a hypersurface singularity( V, o ). Let ν = { ν , . . . , ν r } be the monomial valuations corresponding to the facetsof the Newton polyhedron of h , including the non-compact facets. The centre of thevaluations ν i could then be a prime ideal different from the maximal ideal of O C d ,o .We suppose that there is at least one compact facet such that there is at least onevaluation with centre in the maximal ideal. The definition of Poincar´e series for affinetoric varieties can be extended for a set of valuations which contains at least onevaluation with centre in the maximal ideal. Indeed, notice that the χ ( P F v ) are thenfinite numbers (see also [GP-H] for an equivalent definition using graded rings) suchthat the Poincar´e series P v ∈ Z r χ ( P F v ) t v is well-defined. Theorem 7.
Suppose that h is nondegenerate with respect to its Newton polyhedron N in the origin and that N has at least one compact facet. Let ν = { ν , · · · , ν r } bethe monomial valuations on C d induced by the facets of N . Then the Poincar´e series P νV ( t ) contains the same information as the Newton polyhedron of h and in particulardetermines the zeta function of monodromy of h .Proof. Let ν be a valuation in ν with centre in the maximal ideal. The coefficientof t v in the series (1 − t q · · · t q r r )(1 − t ν , · · · t ν r, r ) · · · (1 − t ν ,d · · · t ν r,d r ) (7)with q = ν ( h ), can also be written as X A ⊂{ , ··· ,r } ( − A dim M ( v − A + 1) M ( v − A + e + 1) − X A ⊂{ , ··· ,r } ( − A dim M ( v − q − A + 1) M ( v − q − A + e + 1) . Notice that these dimensions are finite because ν has centre in the maximal ideal. Onecan now argue in the same way as in the proof of Theorem 1 to obtain that P νV ( t ) = (1 − t q ) P ν C d ( t ) . No factors cancel in (7). Indeed, from the fact that ν contains the valuations (1 , , . . . , . . . , (0 , . . . , ,
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