Combinatorial duality for Poincar\'e series, polytopes and invariants of plumbed 3-manifolds

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincar\'e series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of the coefficients). We show that the (a Gorenstein type) symmetry of the zeta function combined with Ehrhart-Macdonald-Stanley reciprocity (of Ehrhart theory of polytopes) provide a simple expression for the periodic constant. Using these dualities we also find a multivariable polynomial generalization of the Seiberg-Witten invariant, and we compute it in terms of lattice points of certain polytopes. All these invariants are also determined via lattice point counting, in this way we establish a completely general topological analogue of formulae of Khovanskii and Morales valid for singularities with non-degenerate Newton principal part.

1. Introduction 1 2. Preliminaries 4 3. Ehrhart-Macdonald-Stanley duality for rational functions 6 4. Duality for multivariable Poincaré series 10 5. The 'polynomial part' of the series Z(t) 11 6.Topological polytopes, counting lattice points and Seiberg-Witten invariants 14 References 20 1. Introduction 1.1.In order to study the topological invariants of links of complex normal surface singularities one usually uses the dual resolution graph of the singularity.Indeed, such links appear as plumbed 3-manifolds associated with these graphs, and the combinatorics of the graph is an ideal source to codify the topological invariants, even if they were originally defined by completely different methods of algebraic or differential topology.E.g., in order to determine the Seiberg-Witten invariants of a rational homology sphere link, a possible procedure is the following.Firstly, one defines from the graph a multivariable rational function, the so-called 'zeta function' Z(t) (it is also called the 'multivariable topological Poincraré series', as the topological analogue of a 'multivariable analytical Poincaré series' introduced by Campillo, Delgado and Gusein-Zade [CDGZ04,CDGZ08]).This has a natural decomposition Z = h∈H Z h , where H is the first homology of the link M (and it indexes also Spin c (M )).Then one shows that a summation associated with special truncations of the coefficients of Z h behaves like a quasipolynomial.And finally it turns out that the evaluation of the quasipolynomial at the origin (this value is called the 'periodic constant pc(Z h ) of Z h ') provides the Seiberg-Witten invariant associated with the corresponding spin c -structure indexed by h.(For details see section 2.) This is a rather complex regularization procedure.Usually it is hard to find the quasipolynomial, one needs to know all the coefficients of Z h and to understand their asymptotic behaviour.
In fact, there exists an expected reformulation/generalization as follows.Particular examples and families suggest that there must exists an object which is even more general and guides the above periodic constant computation as well: one predicts a unique canonical decomposition of Z h into a sum Z neg h + P + h such that Z neg h is a rational function 'with negative degree' (or, with zero periodic constant) and P + h a finite polynomial, such that P + h (1) = pc(Z h ).In this way, one finds a multivariable polynomial generalization of the Seiberg-Witten invariants, namely P + h .The first goal of the present article is to provide from Z(t) a simple expression for pc(Z h ), or for the Seiberg-Witten invariants of the link.This is done using combination of two dualities: one of them is the topological trace of a Gorenstein type duality, the second is the Ehrhart-Macdonald-Stanley duality of Ehrhart theory.It turns out that pc(Z h ) is an easy precise sum of coefficient of the 'dual' series Z [ZK ]−h , where [Z K ] − h ∈ H is the Gorenstein type dual of h.The second goal is to make the connection with Ehrhart theory deeper: in this way we show that the decomposition Z neg h + P + h exists indeed, and we determine each P + h in language of lattice points of some polytopes.In this way we get such an expressions for the Seiberg-Witten invariants as well.
1.2.Let us provide a more detailed discussion regarding the connection with lattice polytopes.
The interactions between polyhedral combinatorics and algebraic geometry is a classical and very intensive research theme.The most classical example is the toric geometry which compiles questions of algebraic geometry in terms of combinatorics of convex cones and lattice polytopes.This leads to an infusion of combinatorial methods and formulas into the geometry and topology of complex analytic/algebraic varieties.
In the theory of complex normal surface singularities there are several results which establish the above interaction in the case of hypersurface, or, more generally, of isolated complete intersection singularities, with Newton non-degenerate principal part.For such germs one defines Newton polyhedrons using the nontrivial monomials of the defining equations, from which several invariants are expressed, see e.g.[AGZV12, K76, MT80, V76, BN07, S15, Kh78, Mor84, Ok90, Ok97, BA07].
From a purely topological point of view, a formula by [FS90] provides the Casson invariant of a Brieskorn homology sphere (as the Euler characteristic of instanton Floer homology) in terms of counting lattice points inside a tetrahedron in R 3 (see also [NW90]).In fact, this lattice point counting formula can also be generalized to Seifert homology spheres with at most 4 singular fibers [Nic01].Note that for rational homology sphere 3-manifolds the Seiberg-Witten invariant is the generalization of the Casson invariant.Furthermore, for links of Newton non-degenerate hypersurface germs the Seiberg-Witten invariant can be equated with the geometric genus of the singularity proven by [S15], for which a lattice point counting formula in terms of the Newton polyhedron is given by [MT80].However, this family of germs is rather restrictive and does not give (even the idea) how to find topologically the polytopes in general.Thus, the following goal is very natural: In this direction a new channel has been opened in the work of the first and the last author [LN14] by developing a connection between the Seiberg-Witten invariant of rational homology sphere links and equivariant multivariable Ehrhart theory of dilated polytopes: it has been shown that equivariant multivariable Ehrhart quasipolynomials (endowed with their Ehrhart-Macdonald-Stanley reciprocity) appear naturally in the study of multivariable rational 'zeta functions' using the coefficient function and the periodic constant of their Taylor expansion at the origin.However, in that work the strength of the reciprocity and its geometric reinterpretations were not exploited totally.Though this connection suggested the existence of a 'polynomial -negative degree part' decomposition of multivariable topological Poincaré series, hence a polynomial generalization of the Seiberg-Witten invariant, and it generated an intense activity (see e.g.[LN14,LSz16,LSz17]), the complete explanation waited till the present manuscript.
1.3.The final output of the present paper is a complete answer to the Problem formulated above.The main result transforms the equivariant Ehrhart-Macdonald-Stanley reciprocity to the level of series, giving a duality between two objects: the periodic constant of the Taylor expansion at the origin and a finite sum of coefficients of the Taylor expansion at infinity.This, applied to the multivariable topological Poincaré series of normal surface singularities, gives rise to an identification of the Seiberg-Witten invariants indexed by h with a finite sum of coefficients of the 'dual' series indexed by ([Z k ] − h).(Here −Z K is the cycle of the dualizing sheaf.)This can be regarded as an extension of the Z K -symmetry given by the Riemann-Roch formula for the Euler characteristic of line bundles on the resolution.One can also be viewed as a combinatorial analog of the Laufer's duality for equivariant geometric genera of the singularity.
Moreover, it turns out that this duality at the level of series provides the wished multivariable 'polynomial -negative degree part' decomposition of the Poincaré series as well.It also motivates the definition of the topological polytopes associated with a plumbing graph of any rational homology sphere link, which provides an inclusion-exclusion lattice point counting formula for the Seiberg-Witten invariants, similar to the one of Khovanskii [Kh78] and Morales [Mor84] for the geometric genus of Newton non-degenerate isolated complete intersection singularities.
1.4.The organization of the paper is the following.Section 2 contains preliminaries about plumbing graphs, manifolds, their Seiberg-Witten invariants, and also Poincaré series and their counting functions and periodic constants.In section 3 we shortly review the equivariant multivariable Ehrhart theory from [LN14,LSz16], then we formulate the Ehrhart-Macdonald-Stanley duality for rational functions.Section 4 contains the study of the duality for multivariable topological Poincaré and the identification of the Seiberg-Witten invariants with finite sum of certain coefficients of the dual series.Section 5 describes the 'polynomial -negative degree part' decomposition.It contains a brief motivation and history of the problem, and some examples as well.Lastly, in section 6 we define the topological polytopes and we prove the lattice point counting formula for the polynomial part of the Poincaré series, in particular, for the Seiberg-Witten invariants.The results of this section are illustrated on a concrete example in Example 6.2.14.We also discuss the similarities with the Khovanskii-Morales formulas using Minkowski sums of associated polyhedrons.AN was partially supported by ERC Adv.Grant LDTBud of A. Stipsicz at Rényi Institute of Math., Budapest.JN was partially supported by NKFIH Grant K119670, JN and AN were partially supported by NKFIH Grant K112735.
2.1.Plumbing graphs.Plumbed 3-manifolds.We fix a connected plumbing graph Γ whose associated intersection matrix is negative definite.We denote the corresponding oriented plumbed 3-manifold by M = M (Γ).In this article we always assume that M is a rational homology sphere, equivalently, Γ is a tree with all genus decorations zero.
We use the notation V for the set of vertices, δ v for the valency of a vertex v, N for the set of nodes (vertices with δ v ≥ 3), and E for the set of end-vertices (vertices with δ v = 1).
Let X be the plumbed 4-manifold with boundary associated with Γ, hence ∂ X = M .Its second homology L := H 2 ( X, Z) is a lattice, freely generated by the classes of 2-spheres {E v } v∈V , with a negative definite intersection form ( , ).Furthermore, H 2 ( X, Z) can be identified with the dual lattice The 4-manifold X has a complex structure.In fact, any such M (Γ) is the link of a complex normal surface singularity (X, o), which has a resolution X → X with resolution graph Γ (see e.g.[N99]), and the complex analytic smooth surface X as a C ∞ -manifold is the plumbed 4-manifold associated with Γ.Let K ∈ L ′ be its canonical cycle.Though the complex structure is not unique, K is determined topologically by L via the adjunction formulae (K + E v , E v ) + 2 = 0 for all v.In some cases, it is more convenient to use Z K := −K ∈ L ′ , hence, by adjunction formulae, (2.1.1) In the sequel |A| denotes the cardinality of the set A.

The series Z(t).
The multivariable topological Poincaré series is the Taylor expansion ] at the origin of the 'zeta-function' (2.2.1) where Fix h ∈ H.We define a 'counting function' of the coefficients of Z h by (2.2.3) The appearance of this type of truncation {l ′ x, [l ′ ] = h} for the counting function in the above sum is motivated by the results below, see e.g.2.3.2 (or [N12]).
2.3.Seiberg-Witten invariants of M .Let σ can be the canonical spin c -structure on X identified by c 1 ( σ can ) = Z K , and let σ can ∈ Spin c (M ) be its restriction to M , called the canonical spin cstructure on M .Spin c (M ) is an H-torsor with action denoted by * .
We denote by sw σ (M ) ∈ Q the Seiberg-Witten invariant of M indexed by the spin c -structures σ ∈ Spin c (M ) (cf. [Lim00,Nic04]).(We will use the sign convention of [BN10,N11].) In the last years, several combinatorial expressions were established for the Seiberg-Witten invariants.For rational homology spheres, Nicolaescu [Nic04] showed that sw(M ) is equal to the Reidemeister-Turaev torsion normalized by the Casson-Walker invariant.In the case when M is a negative definite plumbed rational homology sphere, a combinatorial formula for Casson-Walker invariant in terms of the plumbing graph can be found in Lescop [Les96], and the Reidemeister-Turaev torsion is determined by Némethi and Nicolaescu [NN02] using Dedekind-Fourier sums.
A different combinatorial formula of {sw σ (M )} σ was proved in [N11] using qualitative properties of the coefficients of the series Z(t).
If we fix h ∈ H and we write l ′ = l + r h with l ∈ L, then the right-hand side of (2.3.2) is a multivariable quadratic polynomial on L, a fact which will be exploited conceptually next.
2.4.Periodic constants.One of the key tools of the present article is an invariant associated with series motivated by properties of Hilbert-Samuel functions used in algebraic geometry and singularity theory.It creates a bridge with Ehrhart theory and the properties of its quasipolynomials.It is called the periodic constant of the series.For one-variable series they were introduced in [NO09, O08], see also [BN10], the multivariable generalization is treated in [LN14].Let S(t) = l≥0 c l t l ∈ Z[[t]] be a formal power series with one variable.Assume that for some p ∈ Z >0 the counting function Then the constant term Q (p) (0) is independent of p and it is called the periodic constant pc(S) of the series S. E.g., if S(t) is a finite polynomial, then pc(S) exists and it equals S(1).If the coefficients of S(t) are given by a Hilbert function l → c(l), which admits a Hilbert polynomial H(l) with c(l) = H(l) for l ≫ 0, then one shows that S reg (t) = l≥0 H(l)t l has zero periodic constant and pc(S) = pc(S − S reg ) + pc(S reg ) = (S − S reg )(1), measuring the difference between the Hilbert function and Hilbert polynomial.
For pc(S), valid for the Taylor expansion of special rational functions, see 5.1.
For the multivariable case we consider a (negative) definite lattice L = Z E v v , its dual lattice L ′ , a series S(t) ∈ Z[[L ′ ]] (e.g.Z(t)), and its well-defined counting function Q h = Q h (S(t)) as in (2.2.3) for fixed h ∈ L ′ /L.(In fact, the definition extends to the case of any two free Z-modules L ⊂ L ′ of the same rank with H = L ′ /L, the setup of Section 3.) Assume that there exist a real cone K ⊂ L ′ ⊗ R whose affine closure is top-dimensional, l ′ * ∈ K, a sublattice L ⊂ L of finite index, and a quasipolynomial Q h (l) (l ∈ L) (written also as Then we say that the counting function Q h (or just S h (t)) admits a quasipolynomial in K, namely Q h (l), and also an (equivariant, multivariable) periodic constant associated with K, which is defined by The definition does not depend on the choice of the sublattice L, which corresponds to the choice of p in the one-variable case.This is responsible for the name 'periodic' in the definition.The definition is independent of the choice of l ′ * as well.By the general theory of multivariable Ehrhart-type quasipolynomials (counting lattice points with special coefficients in polytopes attached to Z(t)) one can construct a conical chamber decomposition of the space L ′ ⊗ R, such that each cone satisfies the above definition (hence provides a periodic constant), for details see [LN14] or [SzV03].This decomposition, in principle, divides S ′ R := R ≥0 E * v v∈V into several sub-cones (hence, providing different quasipolynomials and periodic constants associated with these sub-cones of S ′ R ).However, Theorem 2.3.1 guarantees that this is not the case, the whole S ′ R admits a unique quasipolynomial, in particular, a periodic constant (cf.also with [LSz16]).Hence, Theorem 2.3.1 reads as follows.
Theorem 2.4.2.[N11] The counting function of Z h (t) in the cone S ′ R admits the (quasi)polynomial The right-hand side of (2.4.4) with opposite sign is called the r h -normalized Seiberg-Witten invariant of M , and it is denoted by sw norm h (M ).

Ehrhart-Macdonald-Stanley duality for rational functions
3.1.We wish to apply equivariant Ehrhart theory for the computation of the periodic constant of the topological Poincaré series.For this we review some results from the Ehrhart theory.We will use a similar setup as in section 2 to make the comparison easier: we will fix a free Z-module L, another one L ′ ⊃ L of the same rank (but in this Ehrhart context L ′ is not necessarily the dual of L, in fact L carries no intersection form at all).We write H for L ′ /L, d for the order of H, and we fix a bases {E v } v∈V in L.
We consider multivariable rational functions (in variables t L ′ ) of type , , and fractions of 'polynomials' 'rational functions'.)We also assume that all the coordinates a i,v of a i are strict positive.
We consider the Taylor expansion T z(t) of z(t) at the origin and also define the Taylor expansion of z(t) at infinity T ∞ z is obtained by the substitution s = 1/t into the Taylor expansion at s = 0 of the function

The series z, T z and T ∞ z have equivariant decompositions
They are defined similarly as the decomposition of Z in 2.1.(Note that the operation T z → T ∞ z, defined via the original z, preserves all the h-components.) We also might eliminate some of the variables: for any subset I ⊂ V we substitute t i = 1 in z(t) for all i / ∈ I; in this way we obtain z(t I ).We call this procedure 'reduction'.This procedure at the level of T z is a summation of some of the coefficients.Since the series associated with the denominator of z is supported in the cone Z ≥0 a i , the summation is finite.
Note that the H-decomposition of the restricted functions are not well-defined.That is, from the restricted function z(t ∈I in general one cannot recover anymore the restriction of the H-component z h (t).(That is, from the exponent of t l ′ I one cannot recover [l ′ ] ∈ H. Here, and in the sequel, for any l ′ ∈ L ′ we write t l ′ I for v∈I t ) Hence, in the presence of decomposition and reduction the only well-defined object is (z h )| ti=1,i / ∈I , the reduction of z h , which will be denoted by z h (t I ).Furthermore, if π I is the natural projection associated with the reduction (elimination of the I-components), then π I (L ′ )/π I (L) usually is not isomorphic to H. Hence, we keep H as an index set and we never consider π I (L ′ )/π I (L).
3.3.We fix h ∈ H and I ⊂ V. We define two functions associated with the coefficients of T z h (t I ): the first is called the counting function The definition selects only the coefficients of T z h (t I ), hence if we write T z h (t It depends only on x| I .This truncation and counting function does not (naturally) appear in Ehrhart theory, but this is what is imposed by Theorem 2.3.1.
Our second function is called the modified counting function; it is defined by where for any a, b ∈ R |I| we say that a ≺ b if a v < b v for all v ∈ I. Similarly as above, one also has q h,I (x) = k ′ ≺x|I z (h) (k ′ ).For any h and I by inclusion-exclusion principle This modified counting function will be related to the usual counting functions from the Ehrhart theory.It satisfies several useful properties (e.g.convexity, reciprocity, see below).
3.4.Ehrhart theory of polytopes associated with the denominator of z.In this subsection we follow the equivariant version from [LN14,LSz16].Associated with the vectors {a i } n i=1 (the exponents in the denominator of z(t)) we define two objects.Firstly, let l : R n → L ′ ⊗ R be the map given by l(x) = n i=1 x i a i and consider the representation ρ : Z n → H defined by the composition Then, the vectors {a i } n i=1 and any I ⊂ V, I = ∅ (which might vary, cf.(3.3.3))determines the family of closed dilated convex polytopes indexed (or, dilated by) I .We assume that l ′ I moves in some region of π I (L ′ ) in such a way that P (l ′ ) I stays combinatorially stable.In such a case we can associate to the stable combinatorial type of the dilated polytopes the set (dilated family) of facets F (l ′ ) I .Moreover, any choice of a subset of facets in a fixed stable topological type provides a 'stable' (dilated) subset of facets G I ; we denote this choice by G I ⊂ F I .Furthermore, for any h ∈ H and choice G I ⊂ F I we consider the counting function of specially chosen lattice points identified by According to the equivariant Ehrhart theory, applied to the dilated polytopes P (l ′ ) I , and for any h ∈ H and G I , the counting function Q h,GI (l ′ I ) is a quasipolynomial, whenever l ′ I moves in some region of π I (L ′ ) in such a way that P (l ′ ) I stays combinatorially stable.The inequalities of (3.4.1) can be viewed as a vector partition [SzV03,LSz16], the variance of the combinatorial type of P (l ′ ) I is determined by the following chamber decomposition of π This construction shows that the chamber decomposition is independent of the choices of h ∈ H and G I ⊂ F I .
Therefore, if l ′ I moves in such a chamber c, then the above counting function of lattice points (We warn the reader that usually the parameter 3) in both sides we use the same chamber c, and from the pair l ′ I , −l ′ I only one of them can be in c provided that l ′ I = 0.) 3.5.A distinguished subset of facets G I ⊂ F I is defined as the coordinate facets {P We denote it by G co I .Then the theory above has the following consequences regarding the modified counting function q h,I of T z h (t I ) defined in (3.3.2):Corollary 3.5.1.We fix h and I.
(a) q h,I is a piecewise quasipolynomial, which can be written as (b) For a fixed chamber c of R |I| define the quasipolynomial h,I (l ′ ).(c) For any fixed chamber c the modified counting function q h,I admits a quasipolynomial in the sense of Subsection 2.4 q c h,I , which satisfies for l ′ = l + r h the identity q c h,I (l) = q c h,I (l ′ ), and a periodic constant pc c (q h,I ) = q c h,I (0) = q c h,I (r h ) associated with the chamber c.
This pc c (q h,I ) will be denoted by mpc c (T z h (t I )).We call it the modified periodic constant of T z h (t I ) associated with h, I and the chamber c. (The terminology and the notation emphasize the presence of different cuts in the counting functions.)3.6.In general, the computation of quasipolynomials and their periodic constants (either modified or not) is hard: it measures the asymptotic behaviour of the coefficients in a certain cone.The next result based on the equivariant Ehrhart-Macdonald-Stanley reciprocity (3.4.3) shows that (under some condition) the modified periodic constant equals with a finite sum given by the coefficients of the Taylor expansion at infinity.Theorem 3.6.1.Fix h and I as above.Assume that there exists a chamber c such that b k | I ∈ c for all k.Write the h-component of the Taylor expansion at infinity as I ( l).
Proof.By Corollary 3.5.1 the function q h,I on the set . Hence, by definitions, mpc c (T z h (t I )) = pc c (q h,I ) exists and equals

First we claim that
for all v ∈ I (where the strictness of the inequality is imposed by the boundary condition On the other hand, since b k ∈ c, from (3.4.3) one has and x i > 0 for all i.On the other hand, in the expansion at infinity,

Duality for multivariable Poincaré series
4.1.In this section, we will return to the situation of section 2: we consider the rational function f (t) from (2.2.1) associated with a connected negative definite plumbing graph Γ, its expansion at the origin Z(t), its expansion at infinity T ∞ f (t), and their equivariant versions and reductions.We also use all the notations from section 3 regarding the counting and modified counting functions associated with f and Z, S ′ R is the real Lipman cone, and for any I we consider the projection π I : R E v v∈V → R E v v∈I = R |I| , denoted also as x → x| I .The projected (real) Lipman cone is the cone in R |I| defined as π I (S ′ R ).We also review a reduction analogue of Theorem 2.4.2, which is the prototype of several results showing that for certain choices of I, the reduction Z(t I ) still preserves the 'Seiberg-Witten information' (for more see [LNN17]).
Theorem 4.1.1.[LN14] The counting function of Z h (t N ) in the cone π N (S ′ R ) admits a quasipolynomial and a periodic constant, and Such a result has the following advantages: the number of reduced variables (i.e., in this case, the number of nodes) usually is considerably less than the number of vertices, a fact which reduces the complexity of the calculations.Moreover, the reduced series reflects more conceptually the complexity of the manifold M (using only one variable for each Seifert 3-manifold piece in its JSJdecomposition).Furthermore, the reduced series can be compared/linked with other (geometrically or analytically defined) objects as well (see e.g.[BN10,N08]).
The above theorem has several generalizations, however the number of variables cannot be decreased arbitrarily, it is obstructed by the normalized Seiberg-Witten invariants of the complementary subgraph Γ(V \ I), for details see [LNN17].4.2.As we already mentioned, the modified counting function has some additional nice properties (compared with the original counting function).Regarding it, in the sequel we will use several results from [LSz16] (where q h,I associated with f is called the 'coefficient function', since q h,I (l ′ 0 ) is the coefficient of t 4.3.'Chamber property'.If we wish to apply Theorem 3.6.1 for q h,I , we need to find a chamber associated with the denominator of the rational function f (t I ), which contains all vectors of type b k | I where b k are the exponents appearing in the numerator.The next proposition shows the existence of a chamber which contains the whole projected real Lipman cone.In order to give some intuition for this fact, we also provide an intermediate step of its proof.For a subset I ⊂ V, I = ∅ we define its closure I as the set of vertices of that connected minimal full subgraph Γ I of Γ, which contains I.We denote by δ v,I the valency of a vertex v ∈ I in the graph Γ I .
In [LSz16,Lemma 11] is proved that f (t I ) has a product decomposition of type where R is a polynomial supported on π I (S ′ ), in particular it has no pole.Hence, the possible poles of f via the I-reduction are drastically reduced from the set of poles of {1 − t E * v } v∈E to the set of poles of {1 − t Remark 4.3.3.In [LSz16,LNN17] is also proved the following 'convexity property': 4.4.Duality for counting functions and periodic constants.Now we are ready to prove our main theorem: a duality/pairing between special evaluations of the counting functions associated with f and the periodic constants.The duality is the upshot of two 'symmetries', manifested at two different levels.The first one is the equivariant Ehrhart-Macdonald-Stanley reciprocity of the polytopes, while the second is a topological imprint of the Gorenstein duality present at the level of the topological Poincaré series: a {x ↔ Z K − x} symmetry.
Theorem 4.4.1.Fix any That is, (in principle, the hardly computable) periodic constant of a series Z h can be determined as a precise finite coefficient counting of the dual series Z [ZK ]−h .

Proof. By the inclusion-exclusion principle (3.3.3) (b) is implied by (a). Next we prove (a).
The substitution x → Z K − x in f , together with the identities ).This on the Taylor expansion level transforms into the symmetry Furthermore, (4.3.1) and the sentence after it shows that in the numerator of f (t I ) all the exponents are situated in the projected Lipman cone π I (S ′ R ).In particular, by Lemma 4.3.2,they are contained in a fixed chamber c ⊂ R |I| which contains π I (S ′ R ).Therefore Theorem 3.6.1 gives But the right-hand side is exactly the counting function q ; that is, the Seiberg-Witten invariant can be expressed via the counting function as a finite sum of Z-coefficients.

The 'polynomial part' of the series Z(t)
5.1.The 'Polynomial-negative degree part' decomposition: motivation and history.Consider a one-variable rational function z(t) = B(t)/A(t) with A(t) = n i=1 (1 − t ai ) and a i > 0. In [BN10, 7.0.2] is observed that any such function has a unique decomposition of the form z(t) = P + (t) + z neg (t), where P + (t) is a polynomial (with non-negative exponents) and z neg (t) is a rational function of negative degree.Furthermore, z(t) admits a periodic constant (associated with the Taylor expansion of z and the cone R ≥0 ), which equals P + (1).The decomposition can be established using the Euclidean division algorithm.P + is called the polynomial part while the rational function z neg is the negative degree part of the decomposition.
This one-variable periodic constant computation was useful in theorems of type 4.1.1 with only one node (Seifert 3-manifolds), or surgery formulas along one node, cf.[BN10].Basically, in these cases, the concrete computation of the periodic constant was based on the computation of P + .
Though the multivariable generalization was hardly needed, till now the complete answer was not found.The main questions were: (a) what are the universal properties of the parts P + and z neg , which guarantee that a decomposition z = P + +z neg exists and it is unique; and, (b) what algorithm provides this decomposition.Additionally, the wished decomposition must satisfy (at least) the basic property: (c) pc(z) = P + (1), where pc is associated with the Lipman cone (or, at least, with some subcone of it).(Hence, when all these are satisfied, whenever the Seiberg-Witten invariant is computable via such periodic constant, e.g. when I contains all the nodes, cf.Theorem 4.1.1,or, the 'bad' vertices of Γ, cf.[LNN17], then P + is a polynomial generalization of the Seiberg-Witten invariant.) For functions with two variables in [LN14] a decomposition was established satisfying all the required properties, based on a 'two-variable division procedure'.Furthermore, for more variables, [LSz16] constructed a candidate polynomial P + 1 based on the combinatorics of Γ, Ehrhart theory and reduction to the one-and two-variable divisions.It satisfied (c), but it didn't answer (a) in a natural way.Later, [LSz17] considered another polynomial P + 2 (with P + 2 = P + 1 in general) constructed via an inductive multivariable Euclidean division.It answered (a)-(b), but (c) was not established, so it was not clear if P + 2 is helpful at all in pc (or Seiberg-Witten) computations.The authors of [LSz17, 4.4] not quite conjectured, but asked convincingly whether this P + 2 is the right candidate for the polynomial part.Below we show that this is indeed the case.

The polynomial part by division. [LSz17]
We review in short the needed statement of [LSz17] reorganized from the point of view of the present note.We start, similarly as in section 3, with a pair of free Z-modules L ⊂ L ′ , and a general multivariable rational function (in variables 0 for all k and 0 ≺ a i for all i.(In our application z will be f h (t I ) for some I = ∅.)Proposition 5.2.1.[LSz17] (a) z(t) can be written in the following form by a 'multivariable Euclidean division algorithm': , where ι S,j ∈ Z, b S,j ⊀ 0 for all (S, j), and b S,j ≺ a i for all (S, j) and all i ∈ S whenever S = ∅.(b) z(t) has a decomposition of type P + (t) + z neg (t) with the next properties: (i) P + (t) is a finite sum (polynomial) j n j t cj with c j ⊀ 0 for all j; (ii) z neg (t) is a rational function with negative degree in all variables t i .Furthermore, a decomposition of z(t) with properties (i)-(ii) is unique.(c) In fact, the terms P + and z neg of the decomposition from (b) are given via (5.2.2) as follows: P + (resp.z neg ) is the sum of terms from (5.2.2) over S = ∅ (resp.S = ∅).
Proof.In (a) we use induction.Assume that we have an expression g(t) = t b / i∈S (1 − t ai ) such that b ⊀ 0 and there exists some i 0 such that b ⊀ a i0 .Then we replace g by −t b−ai 0 / i∈S\{i0} (1 − t ai ) + t b−ai 0 / i∈S (1 − t ai ).Note that b − a i0 ⊀ 0, hence the new fractions have similar form.Starting from the original expression of z, this step repeated whenever is applicable, we get (5.2.2).
(b) Define P + and z neg as is indicated in (c).Then properties (i)-(ii) are automatically satisfied.Next, we prove the uniqueness of the decomposition.We need to show that if P + (t) + z neg (t) = 0 ( †) and if P + and z neq satisfy (i)-(ii), then both are zero.But, if P + is non-zero then by ( †) and (ii) P + has negative degrees in all variables, a fact which contradicts (i).
5.3.The polynomial part of f h (t I ) by duality.Assume again that we are in the situation of a plumbing and its rational function f (t) as in Sections 2 and 4.
Theorem 5.3.2.Consider the decomposition f h (t I ) = P + h,I (t I ) + f neg h,I (t I ) from (5.3.1).(a) P + h,I (t I ) and f neg h,I (t I ) satisfy the requirements (i)-(ii) from Proposition 5.2.1(b).In particular, by the uniqueness of the decomposition, this decomposition agrees with the decomposition from 5.2.1(a)-(c) given by Euclidean division. (b) Proof.(a) By its definition, P + h,I (t I ) is a finite sum, hence f neg h,I (t I ) = f h (t I ) − P + h,I (t I ) is a rational function.Since in its expansion all monomials w(l ′ )t l ′ satisfy l ′ ≺ 0, it has negative degree in all the variables.
(b) By the above transformations w(l We illustrate the above duality results on the following example. Example 5.3.4.Let us consider the Brieskorn sphere M = Σ(2, 5, 7).Recall that the entries of E * v 's are the corresponding columns of the matrix {−(E * i , E * j )} i,j , which is in fact −I −1 , the negative of the inverse of the intersection matrix.In this case the graph and −I −1 are the following: Therefore, the zeta-function reduced to the set of nodes N = {v 0 } and its 'polynomial-negative degree part' decomposition (obtained by simple division of polynomials) can be written as The same result follows by duality as well.Indeed, the Taylor expansion is Z(t) = 1 + t 10 + t 14 + . . .and one calculates Z K = (12, 6, 3, 4, 2) where we use short notation (l 1 , l 2 , l 3 , l 4 , l 5 ) for l = 5 i=1 l i E i .Hence, for t ZK −E Z(t −1 ) we get t 11 + t 1 + t −3 + • • • , which gives P + (t) = t 11 + t again.5.4.Example.Plane curve singularities.Let us describe the analogue of (3.6.2) and of the 'polynomial-negative degree part' decomposition for the embedded situation of an irreducible plane curve singularity g : (C 2 , 0) → (C, 0).In this case the series z(t) is the Taylor expansion at the origin of the monodromy zeta function ζ(t) = ∆(t)/(1 − t), cf.[A'C75, AGZV12].A'Campo's formula constructs ζ(t) via (2.2.1), but now applied to the minimal embedded resolution graph of the plane germ, whose unique (−1)-vertex has an extra arrowhead-neighbour (where the arrow represents the strict transform of {g = 0}, thus the (−1)-vertex is considered to be a node of the graph), and this series is reduced to the variable of the (−1)-node.
Let M ⊂ Z ≥0 be the semigroup of g consisting of all intersection multiplicities with all possible analytic germs.It is known [CDG03] that z(t) = s∈M t s .Let µ be the Milnor number of g.
We determine the polynomial part of z(t) and we verify the analogue of (3.6.2).Here the needed Gorenstein duality reads as follows: ( †) s ∈ M if and only if µ − 1 − s ∈ M.This also shows that Z \ M is finite, its cardinality is exactly µ/2, and the largest element of Z \ M is µ − 1.
Since z(t) = s∈M t s = s≥0 t s − s ∈M,s≥0 t s we get that z neg (t) = s≥0 t s = 1/(1 − t), and P + (t) = − s ∈M,s≥0 t s .On the other hand, by duality ( †) one has whose part with positive exponents is exactly P + (t).6. Topological polytopes, counting lattice points and Seiberg-Witten invariants 6.1.Motivation: polytopes and invariants of normal surface singularities.The idea which connects geometric and topological invariants of normal surface singularities with the number of lattice points and the volume of certain polytopes has a long history and it culminates in toric geometry.At the level of singularities, the very first class where this phenomenon appears is the Newton non-degenerate hypersurface singularities (see eg. [AGZV12]).In this case one defines the Newton polytope P associated with the defining equation and one proves that several invariants of the singularity can be recovered from P, see the following non-complete list of articles: [K76, V76, BN07, D78, MT80, S15].In this article we would like to highlight only the geometric genus p g which equals the number of lattice points with strictly positive coordinates in P, cf.[MT80].Moreover, [S15] shows that in the case when the germ has a rational homology sphere link M , one has p g = −sw norm 0 (M ) supporting the Seiberg-Witten invariant conjecture of Némethi and Nicolaescu [NN02] for this case and, in particular, expressing the Seiberg-Witten invariant as a lattice point counting in the Newton polytope.
This analytic situation can be generalized to the case of isolated complete intersection singularities with certain non-degeneracy condition, cf.[Kh78, Mor84, Ok90, Ok97, BA07].For such germs, p g is expressed by [Kh78] and [Mor84] as an alternating sum of lattice points in certain polytopes obtained as Minkowski sums of Newton polyhedra defined by the equations of the germ.
Having in mind the above correspondence p g = −sw norm 0 (M ) between the geometric genus and the Seiberg-Witten invariant (formulated by 'Seiberg-Witten invariant conjecture' and valid for certain analytic types of singularities), it is natural to ask if there is an analogous interpretation of the Seiberg-Witten invariant in terms of an alternating sum of lattice points in certain polytopes associated with the topological type.We will call such expression inclusion-exclusion lattice point counting, or IELP counting.
The main result (and novelty) of this section is a positive answer to this question, valid for any normal surface singularity with rational homology sphere link.Though in the case of p g and Merle-Teissier-Khovanskii-Morales type formulas the polytopes were associated with the equation of the germ, in this topological version they are associated with the resolution graph.
The wished expression starts with the results from [LN14], which identifies the Seiberg-Witten invariant with certain coefficients of the Ehrhart quasipolynomial of a dilated polytope associated with the plumbing graph of the manifold M .However, merely with the techniques of [LN14] a concrete lattice point counting was obstructed.The point is that the duality and construction of P + in the present note unlock the obstructions of [LN14] in the direction of an IELP counting.6.2.Preparation for an IELP expression.In section 3.4 we have defined the linear map l : and we have considered the representation given by the composition ρ : Set also the notation l I for the map defined by the composition of l with the projection π I : R |V| → R |I| .In particular, for I = {v} the maps l v are the components of l.
Recall also that for any I ⊂ V, I = ∅ we have defined the closed dilated convex polytopes P v , where P In this section (similarly as in [LN14]) we consider also the dilated concave polytopes as well: (6.2.1) ¿From the definition is clear that P (l ′ ) I depends only on the restriction l ′ | I .Furthermore, it can happen that in the union the contribution of some P (l ′ ) v is superfluous.This is what we clarify next.If for some I ⊂ V one has l ′ = v∈I a v E * v with all a v = 0, then we say that the E * -support of l ′ is I, and we write sp * (l ′ ) = I.In the sequel we are interested mainly in those cases when the E * -support of l ′ is contained in N .For such cycles we have the following additional 'inclusion property' of the polytopes P (l ′ ) I .Lemma 6.2.2.Assume that sp * (l ′ ) = I ⊂ N , I = ∅.Then P Proof.The I = N case appears in the proof of Theorem 5.4.2(a) of [LN14].The general case is rather similar, for the convenience of the reader we sketch the proof.We assume that v ∈ I. Let Γ 1 be that connected component of the graph Γ \ I which contains v. Then consider those vertices {u k } k∈I ′ of I, which are adjacent to Γ 1 (I ′ ⊂ I, |I ′ | > 0).We claim that there exists positive rational numbers {r k } k∈I ′ such that (6.2.3) where i is the natural inclusion L(Γ 1 ) ֒→ L(Γ).The numbers r k are determined in such a way that the cycle D from the left-hand side of (6.2.3) has vanishing multiplicity along each E u k .This provides a linear system with |I ′ | variables and equations with matrix (E * u k , E * u l ) k,l which is negative definite.E.g., for further references, if ).Once we have determined D one verifies (via intersection with base elements E v ) that it is supported on Γ 1 and in fact equals with the cycle from the right-hand side of (6.2.3).
(6.2.3) shows that ( †) r u E u for certain positive rational numbers {r u } u∈V(Γ1) .This shows that for any i ∈ I one has Furthermore, the intersection of ( †) with E * e for e ∈ E gives The duality results of the present article motivate us to distinguish a special element from L ′ (6.2.4) which serves the role of a special parameter and defines the polytope = N and by Lemma 6.2.2 we obtain P . We call it the topological polytope associated with the graph Γ and we denote it by P top N .A series of sub-polytopes of P we denote by R h (P Proof.We need to compute P + h,N (1).Let us fix some I ⊂ V, and we reformulate P + h,I (t I ).Sometimes is more convenient to calculate P + h,I (t I ) via the truncation of the series Z [ZK ]−h (t I ), the 'dual' polynomial part (6.2.7) P + h,I (t I ) := using the defining identity (5.3.1)(6.2.8) Then (6.2.7) and (2.2.1) gives for the dual polynomial part (6.2.9) and the sum runs over all 0 ≤ k v ≤ δ v − 2 for every v ∈ N and . Now apply (6.2.8), (2.1.1)and replace k v by δ v − 2 − k v in order to get for the polynomial part (6.2.10) , where the sum runs over all 0 ≤ k v ≤ δ v − 2 and x ∈ P If we specify I = N , Lemma 6.2.2 implies that the sum in (6.2.10) runs over all I m ⊂ N m , I m = ∅, and lattice points of with all coordinates being strictly positive.Now, by Corollary 5.3.3 the formula (6.2.6) follows.Remark 6.2.11.In the above proof the expressions with the binomial coefficients coming from the numerator of f (t) have appeared rather naturally.However, we prefer the multiset language of (6.2.6) too, since provides a different interpretation for the appearance of the binomial coefficients, and furthermore, in this way we obtain a perfect similarity with the geometric genus formula of Khovanskii [Kh78] and Morales [Mor84] for Newton non-degenerate isolated complete intersection singularities (supporting the intimate relationship between these two invariants).Corollary 6.2.12.If the conditions Z K | N ≤ E * v | N for every node v ∈ N are satisfied then the polynomial part can be expressed as where Proof.In (6.2.10)only the submultiset I m = N m has non-trivial contribution.
Example 6.2.13.Let M be a negative definite Seifert fibered rational homology sphere, ie.Γ is a star-shaped tree.Assume that the central node v 0 has d ≥ 3 legs, each with determinant α i ≥ 2 (i = 1, . . ., d), and let e be the orbifold Euler number of M .Then by the formula (2.1.1)one calculates , 11.1]).Hence the assumption of Corollary 6.2.12 reads as Notice that this cannot happen if d ≥ 6 and it is always satisfied whenever d = 3.Hence, in particular, Proposition 6.2.12 implies a well-known result in the literature for Brieskorn homology spheres Σ(p, q, r), saying that the normalized Seiberg-Witten invariant, or, equivalently, the normalized Casson invariant, can be expressed as the number of lattice points in the polytope but not on the non-coordinate facet.In fact, in this case P v0 coincides with the Newton polytope associated with the equation of the Brieskorn germ ({x p + y q + z r = 0}, 0).More details and different aspects can be found in Example 6.2.14.Let M be the manifold associated with the following plumbing graph Γ: M is realized e.g. as the link of the Newton non-degenerate hypersurface singularity f : (C 3 , 0) → (C, 0), f (x, y, z) = x 13 + y 13 + z 3 + x 2 y 2 .In the sequel, following the above discussions, we present the computation of the (dual) polynomial part using the lattice points of the topological polytope.
6.3.Minkowski sums of polyhedrons associated with nodes.For all the polytopes P considered in this subsection we denote by P ⊳ := P \ (F \ G co ) the corresponding semi-open polytope and define the unbounded convex polyhedron P + := R |E| ≥0 \ P ⊳ .In the sequel we discuss relations between polyhedrons of type P + I m and Minkowski sums of (P ) + .In particular, we prove for special cases a Khovanskii-Morales type IELP expression for the normalized Seiberg-Witten invariants using Minkowski sums.Lemma 6.3.1.For any v ∈ N let E v be the set of those end-vertices w ∈ E for which the unique minimal connected full subgraph [w, v] does not contain any other nodes than v. Then for any v ′ ∈ N we have  ) + is also a convex polyhedron bounded from below by the convex hull of its vertices and the coordinate hyperplanes.Any vertex v of the Minkowski sum can be written as ) + .Then we can write l(v j v ) = r −1 v E * w for some w ∈ E and r v ∈ Q >0 as in the proof of Lemma 6.2.2.Therefore we have l(v j v ) ≥ E * v which implies the inequality l N (v k ) ≥ v∈N k v E * v | N for all vertices, hence for all v∈N k v (P ) + too.(b) Since the coefficients k v1 , k v2 do not change the combinatorial type of neither of the two polyhedrons (see eg. [GrS93] for Minkowski sums) we assume that k v1 = k v2 = 1.By (a) we have to prove that the compact faces of ( P ) + can be written as Minkowski sums of compact faces of (P (E * v 1 ) v1 ) + and (P ) + .By the definition of P these compact faces are given by the hyperplanes l v1 = (E * v1 + E * v2 )| v1 , l v2 = (E * v1 + E * v2 )| v2 and their intersections.Now, consider the stratification E = E 1 ∐ E 2 of the set of ends so that E i are those end-vertices which are contained in the same connected subgraph as v i if we cut out [v 1 , v 2 ] \ {v 1 , v 2 } from Γ.We denote by v (j) i the (non-zero) vertex of P * vi which are situated on the coordinate axis corresponding to w ∈ E j and let F v (j) i,w be the face generated by the vertices of type v (j) i,w for some fixed i and j.Then our claim is that the compact facets of P are given by the Minkowski sums , and their intersection is given by F v  where Rh (P) is a ρ −1 ([l ′ ] − h)-lattice point counting with all coordinates being strictly positive and not in the interior of P.
. TL was supported by ERCEA Consolidator Grant 615655 -NMST and by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

P
(l ′ ) I depends only on l ′ I := l ′ | I .We denote by F (l ′ ) I the set of facets of P (l ′ ) GI continuously to the closure c of c as a quasipolynomial; all these extensions glue together to a continuous piecewise quasipolynomial of R |I| .This piecewise quasipolynomial is the expression Q h,GI from (3.4.2). (This means that Q h,GI (l ′ I ) in terms of {Q c h,GI } c can be redefined as follows: for any l ′ I ∈ R |I| first find a chamber c such that l ′ I ∈ c and then set Q h,GI (l ′ I ) := Q c h,GI (l ′ I ).)By the equivariant Ehrhart-Macdonald-Stanley reciprocity law, for any fixed h, G I ⊂ F I and chamber c with l ′ I ∈ c one has

I
} v∈E I .Here E I is the set of end-vertices of Γ I ; note that E I ⊂ I. Therefore, by the construction of the chamber decomposition of R |I| (cf.3.4) the chambers associated with f (t I ) are determined by the bases selected from {E * v | I , E u | I : v ∈ E(Γ I ), u ∈ I}.These and a lattice-combinatorial argument provide Proposition 4.3.2.[LSz16] The interior of the projected Lipman cone int (π I (S ′ R )) is contained entirely in a (big) chamber c of f (t I ).
This proves the first identity.For the second one use Theorem 4.4.1 (b).Corollary 4.4.4 and Theorem 5.3.2combined gives Corollary 5.3.3.For any h ∈ H one has P + h,N (1) = −sw norm h (M ).