Multisummability for generalized power series

Abstract We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both 
$\mathbb {R}_{\mathcal {G}}$
 and the reduct of 
$\mathbb {R}_{\text {an}^*}$
 generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the gamma function on 
$(0,\infty )$
 and the zeta function on 
$(1,\infty )$
 .


Introduction
We generalize the theory of multisummability in the positive real direction, as discussed in [2,9,12], to certain non-convergent power series with real non-negative exponents (introduced in [11, p. 4377]).Examples of such series are Dirichlet series (after the change of variables s = − log x), and asymptotic expansions of certain solutions of differential equations [13] and of certain functions appearing in Dulac's problem [3].
Our main motivation here comes from o-minimality: summation processes induce a quasianalyticity property which is usually needed to prove that a given structure is o-minimal.In their paper [10], Van den Dries, Macintyre and Marker show that neither Euler's Gamma function Γ restricted to (0, +∞), nor the Riemann Zeta function ζ restricted to (1, +∞), are definable in the o-minimal structure R an,exp [10,Theorem 5.11 and Corollary 5.14].Subsequently, Van den Dries and Speissegger constructed the o-minimal expansions (R an * , exp) [11,12] and (R G , exp) [12], and they proved that ζ↾ (0,+∞) is definable in the former, but not in the latter [12,Corollary 10.11], while Γ↾ (0,+∞) is definable in the latter [12,Example 8.1].At the time, it was unknown whether Γ↾ (0,+∞) was definable in the former.
This state of affairs thus left the following question unanswered: is there an o-minimal expansion of the real field in which both Γ↾ (0,+∞) and ζ↾ (1,+∞) are definable?Based on additional information gained from Rolin and Servi's paper [7] about the structures (R an * , exp) and (R G , exp), we show in a separate paper (in preparation) that Γ↾ (0,+∞) is not definable in (R an * , exp) either.So to answer the question in the affirmative, we need to come up with an o-minimal structure that properly expands both the expansion of the real field by Γ↾ (0,+∞) and the expansion of the real field by ζ↾ (1,+∞) .
Indeed, we construct here an o-minimal expansion of the real field that expands (R G , exp) and in which ζ↾ (1,+∞) is definable (see the Main Corollary below).
To recap, for an indeterminate X = (X 1 , . . ., X n ), we denote by C [[X * ]] the set of all generalized power series of the form F (X) = α∈[0,∞) n a α X α , where each a α ∈ C and the support supp(F ) := {α ∈ [0, ∞) n : a α = 0} is contained in a product A 1 × • • • × A n of sets A i ⊂ [0, ∞) that are well-ordered with respect to the usual ordering of the real numbers (see [11,Section 4] for details).The series F (X) converges if there exists r > 0 such that F r := α |a α |r α < ∞; we denote by C {X * } the set of all convergent generalized power series [11,Section 5].
The generalized power series that we extend the notion of multisummability to have special support: we call a set A ⊆ R natural if A ∩ (−∞, a) is finite, for every a ∈ R; and we call a set Restricting our attention to generalized power series with natural support allows us to use such objects as asymptotic expansions of germs (see Proposition 1.16).This has already been expoited in [4], where the o-minimality of the expansion of the real field by certain Dulac germs is proven.
In Sections 1 and 2 below, we define a notion of multisummability in the positive real direction for generalized power series of natural support, appropriately named generalized multisummability in the positive real direction (or simply generalized multisummability in the real direction when working in the logarithmic chart of the Riemann surface of the logarithm, as we do throughout this paper).We verify that the resulting system G * of algebras (both of functions and of germs) satisfies the axioms in [7], leading to the following: let the language L G * and the structure R G * be as in [7,Definition 1.21] for our system G * of algebras in place of A there.

Main Theorem.
(1) The structure R G * is model complete, o-minimal and polynomially bounded and has field of exponents R.
By construction, all functions defined on compact polydisks by convergent generalized power series with natural support are definable in R G * ; and we show in Section 1 that the same holds for all functions defined on compact sets by standard power series that are multisummable in the positive real direction.Recall that, for x ∈ [0, e −2 ], ζ (− log x) is the sum of the generalized power series ∞ n=1 x log n , which has natural support.In particular, both exp↾ [0,1] and ζ(− log x)↾ [0,e −2 ] are definable in R G * , as is the function log Γ(x) − (x − 1 2 ) log x on the interval (1, +∞) (see [12,Example 8.1]).Therefore, Theorem B of [12] gives: Main Corollary.
(1) The structure (R G * , exp) is model complete and o-minimal, and it admits quantifier elimination in the language L G * ∪ {exp, log}.
As we rely on [7] for the proof of o-minimality of R G * , the main contribution of this paper is the generalization of multisummability in the positive real direction to generalized power series of natural support and the establishment of the axioms in [7] for the corresponding system G * of algebras of functions and germs.
As in [12], our starting point here is a characterization, due to Tougeron [9], of multisummable power series in terms of infinite sums of convergent power series of decreasing radii of convergence.Thus, we move to the logarithmic chart of the Riemann surface of the logarithm, since we are working with arbitrary real exponents.Then we define a multisummable generalized power series (in the real direction) as the infinite sum of a sequence of convergent generalized power series with decreasing radii of convergence and support contained in a fixed natural set (Section 1.4).
The corresponding theory of multisummability in one variable, developed in Section 1, differs from the classical one in that there is no origin around which we can use contour integration.One example of a classical result that we cannot generalize is the following: every classical multisummable power series can be decomposed into a sum of singly summable series; we do not know if this is the case in the generalized setting (see Section 1.7 for details).However, we do obtain the crucial quasianalyticity for our system of algebras (Section 1.6).
Also, as in [12], this approach lends itself naturally to define generalized multisummability in the positive real direction in several variables, and we follow the corresponding steps in [12] as closely as possible (Section 2).In Sections 3 and 4, we establish the axioms of [7].

Remarks.
(1) To our knowledge, this is the first time [7] was used to prove the o-minimality of a structure that was previously unknown to be o-minimal.The same procedure could be used to obtain the o-minimality (and related results) of the structures R an * [11], R G [12] and R C [8].The resulting quantifier elimination given by [7, Theorem B] is new in each of these cases, and it is used in our forthcoming paper to show that Γ↾ (0,+∞) is not definable in (R an * , exp).
(2) The only closure property needed in [7] but not established in [11], [12] or [8] is closure under infinitesimal substitutions in the convergent variables (Proposition 3.9).The proof of this in the structures discussed in the previous remark is similar to the proof given here for G * .Finally, from the point of view of generalized multisummability, as in the classical theory, there is nothing special about the positive real direction.Our generalized notion works in any other direction, and one could correspondingly come up with a notion of "generalized multisummability" as done in the classical situation.This raises some interesting questions in their own right (see Section 1.7), which we do not address in this paper.
1. Generalized multisummable functions of one variable 1.1.Preliminaries.We denote by C = C ∪ {−∞} the logarithmic chart of the Riemann surface of the logarithm, with the additional "origin" of C represented by "−∞", where we convene that Re(−∞) = −∞.For r ∈ R, we let and a log-line is a set 2 , denote by ∂S 0 the directed path following the boundary of S 0 from the "lower left end" to the "upper left end".We define the logarithmic Borel transform We leave it as an exercise to check that B d ′ f only depends on d ′ , but not on the other parameters of S 0 (as long as they are in the prescribed range).More is true: Remark 1.1.If θ ′ < π and g(z) := f (log d z), for z ∈ exp(S 0 ), then the change of variables z = e w gives that where B d ′ g denotes the Borel transform of g in the direction d ′ as defined in [5] (see also Section 5.2 of [2]).Thus, the following is obtained from Propriétés 1-3 on p. 38 of [5]: For later use, we make the bound in Proposition 1.2(2) more precise: )) be a closed subsector of S with θ ′ ∈ π 2 , θ , and set where L d g denotes the Laplace transform of g in the direction d as defined in [5] (see also Section 5.1 of [2]).Thus, the following is obtained from Propriétés 1-2 on pp.41-42 of [5]: Proposition 1.5.Let ϕ > 0 and set S := S(d, ∞, ϕ).Let f : S −→ C, and assume that for every closed log-subsector S 0 of S, the restriction f↾ S 0 is log-holomorphic and there exist C, D > 0 such that |f (w)| ≤ Ce De Re w for w ∈ S 0 .Then: (1) For each θ ∈ (0, ϕ), there exists In view of the previous proposition, we call the union V := θ∈(0,ϕ) V (d, R(θ)) a logsectorial domain, and we refer to the common extension Lf : V −→ C of L 0 f of L 0 f given by Proposition 1.5 as the log-Laplace transform of f .Note that, in practice, we shall usually restrict the domain of Lf to a sector S d, log R, θ + π 2 for suitable θ ∈ (0, ϕ) and R > 0 on which it is log-holomorphic.
For f : S(d, r, θ) −→ C as in Section 1.2.1, Proposition 1.2 implies that L(Bf ) is defined and log-holomorphic on every closed log-subsector S 0 of S(d, r, θ) ∩ V .Indeed, L is the inverse operator to B (see page 44 of [5]): 1.3.Generalized power series with complex coefficients.Let now F (X) = α≥0 a α X α ∈ C {X * } be such that F r < ∞, for some r > 0. We explain here how such a series defines a log-holomorphic function on some log-disk.Denoting by log the principle branch of the logarithm on C \ (−∞, 0], we set Then, for w ∈ S(0, ∞, π), we have that It follows, in particular, from [11,Lemma 5.5] that F extends continuously to −∞ and satisfies F (−∞) = F (0). Below, we refer to the log-holomorphic function F thus defined on H(log r) as the log-sum of F (X).
1.3.1.Logarithmic Borel transform of convergent generalized power series with natural support.We again fix F (X) = a α X α ∈ C {X * } and r > 0 such that F r < ∞.In addition, we assume that the support of F (X) -a subset of [0, ∞) by definition -is natural.Since F is defined on H(log r) = S(0, log r, ∞), we obtain from Proposition 1.2 that its log-Borel transform BF is log-holomorphic on C.
In view of Example 1.7, we set called the formal Borel transform of F (X).Note that, for σ > 0, we have by Binet's second formula (see [14]) that Thus, for any σ > 0, we have that Since F has natural support, the sum is finite for all σ; so the series BF (X) has infinite radius of convergence, and its log-sum BF is also log-holomorphic on C. In summary: Proposition 1.8.Let F (X) be a convergent generalized power series with natural support.Then both BF and BF are log-holomorphic on C, and we have BF = BF .
Proof.Since F (X) has natural support, we write F (X) = N n=0 a n r αn with either N ∈ N, or N = ∞ and lim n→∞ α n = +∞.For w ∈ C, let S be the closure of the log-sector S(Im w, log r, π), and define K : ∂S −→ C by K(η) := 1 2πi e w−η e e w−η .
For n ∈ N with n ≤ N, let u n : ∂S −→ C be defined by as claimed.
1.4.Generalized multisummable functions.We now define generalized multisummable functions inspired by Tougeron's characterization of multisummable functions [9] and by their presentation in [12].However, while it was possible in [12] to refer to the existing literature for summability, it is not the case in our setting.More precisely, our aim is to show a quasianalyticity result for our functions analogous to that in [12,Proposition 2.18].To this end, we need to introduce suitable Borel and Laplace transforms adapted to the generalized multisummable framework (see Section 1.5).The presentation turns out to be more readable in this setting by replacing the usual "Gevrey order" k by 1/k.This leads to the following definitions.
For R, k ≥ 0, θ > π/2 and p ∈ N, we set Let K ⊆ [0, ∞) be a nonempty finite set and r > 1 (note that the situation studied in [12] corresponds, in the current notation, to π/2 < θ < π and K ⊆ [0, 1], in order to avoid dealing with the logarithmic chart), and set Moreover, we fix a natural set ∆ ⊆ [0, ∞) and set τ := (K, R, r, θ, ∆) (note that ∆ = N in [12]).We define and, for p ∈ N, Remark 1.9.If 0 ∈ K and K ′ := K \ {0} is nonempty, then S τ = S τ ′ and S τ p = S τ ′ p for all p, where τ ′ = (K ′ , R, r, θ, ∆).Definition 1.10.For each p ∈ N, let f p : S τ p −→ C be log-holomorphic, that is, there exists a log-domain D p ⊇ S τ p and a log-holomorphic g p : D p −→ C such that f p = g p ↾ S τ p .Moreover, we assume that there are generalized power series F p (X) ∈ C {X * } with support contained in ∆ such that F p ρ τ p < ∞ and where f p S τ p := sup w∈S τ p |f p (w)| denotes the sup norm of f p on S τ p .The second of these finiteness assumptions implies that p f p converges uniformly on S τ \{−∞} to a holomorphic function g : S τ \ {−∞} −→ C, while the first implies that this g extends continuously to −∞, so that the resulting g : S τ −→ C is log-holomorphic.From now on, we abbreviate this situation by writing Thus, for a log-holomorphic function f : S τ −→ C, we set We set G τ := {f : S τ −→ C : f is log-holomorphic and f τ < ∞} .It is immediate from this definition that G τ is a C-vector space under pointwise addition; moreover, if ∆ is closed under addition, then G τ is closed under multiplication of functions, making G τ a C-algebra.
Convention 1.11.If ∆ is natural, then so is its closure under addition; so we assume from now on that ∆ is closed under addition.
(1) We call a function f generalized multisummable in the real direction if f ∈ G τ for some τ as above.
(2) We call a function f generalized K-summable in the real direction if there exist Example 1.14.In terms of Example 1.12, Tougeron's characterization of multisummable functions implies that if f is K-summable in the positive real direction, then f • exp is generalized K-summable in the real direction.
Let f ∈ G τ with associated functions f p and series F p be as in Definition 1.10 such that p F p ρ τ p r p < 2 f τ and p f p S τ p r p < 2 f τ .In Proposition 1.16 below, we show that f has asymptotic expansion F (e w ) at −∞, for the generalized power series F (X) with support contained in ∆ (and hence natural), defined in (1.3) below.To do so, say F p (X) = a p,α X α for each p, where each a p,α ∈ R, and write ρ p for ρ τ p .Then for each p and α, and for arbitrary s ∈ (1, r), we have Therefore, for each α and arbitrary s ∈ (1, r), we get for each α, and which has support contained in ∆.
Lemma 1.15.There exist D, E > 0 such that for all p ∈ N and all β ≥ 0, we have Proof.Fix p ∈ N and β ≥ 0, and let w ∈ S τ p .We distinguish two cases: which proves the estimate in this case.
This completes the proof of Case 2 and therefore of the lemma.
Proposition 1.16 (Gevrey estimates).For every closed log-subsector S 0 of S τ , there exist D, E > 0 such that, for each β ≥ 0, Proof.Let D, E > 0 be obtained from Lemma 1.15, and let ), the proposition now follows from Stirling's formula for Γ (see [1]).Proposition 1.16 implies that F (e w ) is an asymptotic expansion of f at −∞; hence, it is uniquely determined by f (and is, in particular, independent of the particular sequence {f p }), and we write T f (X) := F (X).The map T : Remark 1.17.Standard methods for proving topological completeness of function spaces (see e.g.Rudin's Real and Complex Analysis) show that the normed algebra (G τ , • τ ) is complete; we leave the details to the reader.
with a α = p a p,α as above.Since ρ τ p = R for each p, the assumption p T f p R r p < ∞ implies that p,α |a p,α |R α < ∞.This implies that the family {a p,α e αw } is summable on cl H(log R); in particular, the order of summation can be changed.Thus, we have for ] has natural support and satisfies F R < ∞, then (the appropriate restriction of) the function F belongs to G τ , where τ = (K, R, r, θ, ∆) with K, r, θ and ∆ ⊇ supp(F ) arbitrary.To see this, simply take f 0 := F and f p := 0, for p > 0.
To see the claim, note that the function x → f α (x) := (x + 1) α r ′ r x attains its maximum at x α = α log(r/r ′ ) − 1; so this maximum is = (e log(r/r ′ )) −1 independent of α.From Binet's second formula we get a constant C ′′ > 0 such that, for all α ≥ 0, C ′′ e φ(α) Γ(α), where φ(x) is the Stirling function.Since the latter is bounded at ∞, there is a constant where C := C ′ r r ′ , because our assumptions on r ′ and R ′ imply that Ae 2 ≤ R R ′ .This proves the claim.
It follows from the claim that

5) also proves that BF is convergent and that
Bf p H(log R ′ ) (r ′ ) p < ∞; this settles the assertion in this case.So assume that µ K > 1 or |K| > 1.By (1.5), it suffices to show that for some constant C > 0 that only depends on θ and θ ′ .
For the second estimate: for some constant C > 0 independent of p. Now note that our assumptions on r ′ and Finally, it remains to show that Bf = τ ′ p Bf p and T (Bf ) = B(T f ): set g := p Bf p .The above estimates show that g ∈ G τ ′ with T g = p BF p = B(T f ), so we need to show that Bf = g.Since f = p f p uniformly in S τ , it follows from integration theory that Bf = B p f p = p Bf p = g, as required.
We immediately obtain the following from Proposition 1.2 and Example 1.7.
We immediately obtain the following from Proposition 1.5 and Example 1.7.
In view of the above, we define the formal λ-Borel transform and the formal λ-Laplace transform We get the following from Proposition 1.8: Corollary 1.25.Let F be a convergent generalized power series with natural support.Then both B λ F and B λ F are log-holomorphic on C, and we have B λ F = B λ F .
So we assume that l > 1 and the proposition holds for lower values of l.Then by Corollary 1.26, the function ) and some appropriate R ′ > 0 and r ′ > 1.
From the inductive hypothesis applied to as claimed.
(1) Let f be generalized K-summable in the real direction, where K = (k 1 , . . ., k l ).Do there exist generalized (k i )-summable functions g i , for i = 1, . . ., l, such that This question is motivated by the following: let f be a K-summable function in the positive real direction (in the classical sense, at the origin; to avoid branching, let's assume k 1 > 1 2 ).Then by Example 1.14, the function f • exp belongs to G τ for some τ = (K, R, r, θ, N); indeed, Tougeron's characterization implies that f • exp = τ p f p • exp for functions f p that are holomorphic at the origin.This property of being holomorphic at the origin can be used, via Cauchy integration, to show that there exist (k i )-summable functions g i , for i = 1, . . ., l, such that f = g 1 + • • • + g l .However, for general g ∈ G τ with g = τ p g p , the functions g p • log have essential singularities at the origin, so the Cauchy integration argument used for f does not work for g • log to write g as a sum of a generalized (k i )-summable functions.
(2) From the point of view of multisummability, there is nothing special about the real direction chosen here.(We are only interested in the real direction here, because we are aiming to construct algebras of real functions.)Indeed, one can similarly define generalized K-summable functions in any direction d.However, it is not clear to us what the right generalization of "generalized multisummable" (without specified direction) should be: in the classical case, all multisummable functions are 2πiperiodic in the logarithmic chart, and they are defined to be multisummable if they are multisummable in all but finitely many directions in R/2πZ.In contrast, the logarithmic sums of generalized convergent power series are not ai-periodic for any a > 0 in general (take, for instance, the function e αw + e βw with α and β linearly independent over Q), so generalized multisummable functions in the real direction aren't either.Possibly, the right way to define "generalized multisummable" would be to look for something like Stokes phenomena in differential equations over (quotients of) convergent generalized power series.(3) Is there a Ramis-Sibuya theorem (see [6]) for ordinary differential equations involving quotients of convergent generalized power series?As hinted at in Question 2, such a theorem might inform the correct definition of the term "generalized multisummable".

Generalized multisummable functions in several variables
The extension of the notion "generalized multisummable in the real direction" to several variables roughly follows the treatment in [12, Section 2] of the notion "multisummable in the positive real direction", keeping in mind that we work in the logarithmic chart.Since we will not need to work with the ramified Borel and Laplace operators any more, we will revert here to the classical notation for Gevrey orders used in [12].
It will be useful for the definitions below to set Im(−∞) = arg 0 = 0. where log denotes the standard branch of the logarithm.Finally, if α ∈ [0, ∞) m and r ∈ (0, ∞) m , we put If X ⊆ C m and 1 ≤ ν < m, and if a ∈ C ν and b ∈ C m−ν , then we let be the fibers of X over a and b, respectively.If r, r ∈ R m , we write r ≤ r if r i ≤ r i for each i (and similarly with "<" in place of "≤").
2.1.Convergent generalized power series.Let X = (X 1 , . . ., X m ).Similar to the onevariable case, we denote by C [[X * ]] the set of all generalized power series of the form F (X) = α∈[0,∞) m a α X α , where each a α ∈ C and the support is contained in a cartesian product of well-ordered subsets of [0, ∞) (see [11,Section 4] for details).The series F (X) converges if there exists a polyradius r ∈ (0, ∞) m such that F r := α |a α |r α < ∞; we denote by C {X * } the set of all convergent generalized power series [11,Section 5].
For r ∈ R m , we let be the log-disk of log-polyradius r.For an set D ⊆ C m , we set  Note that if m = 1, then S k (r, θ) = S(0, r, θ/k), where the latter is the sector defined in Section 1.1.The reason for allowing k i = 0 is that we need our class of log-polysectors to be closed under taking cartesian products with log-disks; for instance, if m > 1 and k . Finally, our polystrips are "in the real multidirection"; they can easily be defined in any multidirection1 , but we shall not do this here as we do not need to for our purposes.Next, for p ∈ N we put, by adapting [12, Section 2] to the logarithmic chart, Note that if z ∈ S K p (r, θ) and t ≤ 0, then t + z ∈ S K p (r, θ); in particular, S K p (r, θ) is connected.
We need to introduce the following norms for generalized power series: let U ⊆ C m be an open neighbourhood of the origin such that |z| ∈ U for every z ∈ U.For a generalized power series F ∈ C {X * }, we set It follows from the previous section that, if F U < ∞, then F is convergent and the log-sum of F extends to a log-holomorphic function F : U −→ C such that F U ≤ F U , where F U denotes the sup norm of F on U.
Similar to Section 1.4, we now define generalized multisummable functions in several variables.The role of the usual norm • ρ on generalized power series there is taken on here by the norm • U as defined above, where U = D k p (R); note indeed that z ∈ D k p (R) implies |z| ∈ D k p (R), as required.Below, we denote this particular norm • U by • R,k,p .Thus, let f p : S τ p −→ C be log-holomorphic and bounded, and assume that there is a natural set ∆ ⊆ [0, ∞) m and, for each p, a convergent generalized power series we have that p f p converges uniformly on S τ to a log-holomorphic function f : As before, we abbreviate this situation by writing and we set Thus, as in Section 1.4, we obtain a Banach algebra over C of functions in m variables with norm • τ .Note that if m = 1, G τ as defined here is the same as G τ ′ defined in Section 1.4, where τ ′ = (K ′ , R, r, θ, ∆) with K ′ obtained from K by replacing all nonzero k ∈ K by 1/k (see the introductory remarks at the beginning of Section 2).
Arguing as in the one-variable case, if T (f p )(X) = a α,p X α for each p, we obtain a generalized power series T f (X) = a α X α , where a α = p a α,p for each α.
2.4.Quasianalyticity.In view of proving our o-minimality result in Section 4, we show in this section that the map T : ] is injective.First, we explain the reason why we define the log-k-polysectors and the log-k-polydisks via scalar products rather than just taking Cartesian products of the corresponding objects in one variable.This is done because scalar product behaves well with respect to fibers, as shown in Remark 2.2.
The next lemma shows that, although the log-k-polydisks are not themselves log-disks, the set S τ p contains a suitable log-disk.
Lemma 2.1.We have Proof.The first and third inclusions are straightforward.For the second, let k, l ∈ (0, ∞) m be such that l ≤ k, p ∈ N and ρ ∈ R; it suffices to show that Since Re w < ρ as well, it follows that k • Re w < l • ρ − log(1 + p), as required.
where Π m : R m −→ R is the projection on the last coordinate.Note that if | Im a| is sufficiently small, then θ(a) > π/2.If θ(a) > 0, then S τ (a) is contained in the fiber (S τ ) a of S τ over a.Moreover, if Re a < R ′ then, for each p ∈ N, the set H km p (log R m ) is contained in the fiber of the set H k p (log R) over a.Therefore, if θ(a) > 0 and Re a < log R ′ , then S τ (a) p is contained in the fiber S τ p a , for each p. Proof.Choose f p , for p ∈ N, such that f = τ f p .For p ∈ N, let f a,p : S τ (a) p −→ C be defined by f a,p (w) := f p (a, w); these functions are well defined by Remark 2.2, and f a (w) = p f a,p (w) for every w ∈ S τ (a) .
For each p we set F a,p (X m ) := T (f p )(e a , X m ), a generalized power series in the indeterminate X m .Since T (f p ) R,k,p < ∞ and since, by Remark 2.2, the polyradius ρ(b in particular, F a,p Rm,km,p ≤ T (f p ) R,k,p and f a,p = F a,p , for each p. Therefore f a = τ (a) f a,p , that is, f a ∈ G τ (a) .Since the inequality F a,p Rm,km,p r p ≤ T (f p ) R,k,p r p holds for all choices of {f p }, we also get f a τ (a) ≤ f τ .

Claim. The series
m ]] and is equal to T (f a )(X m ).To see this claim, since for each p, it follows from Lemma 2.1 that for each α m ∈ [0, ∞) and each p, for any s > 1.In particular, for s ∈ (1, r) we have, for each α m ∈ [0, ∞), that (2.1) It follows from (2.1) that for all α m , (2.2) Proposition 2.4 (Quasianalyticity).The map T : Proof.For the injectivity of T , let f ∈ G τ be such that T (f ) = 0; we need to show that 3) and f is holomorphic on its (connected) domain, it follows that f = 0, as claimed.
As in [12, Cor.2.19], we now obtain For i ∈ {1, . . ., m}, we also consider F as an element of ], and we denote by ord i (F ) the corresponding order of F in the indeterminate X i .Note that ord i (F ) > 0 implies ord(F ) > 0, for each i.
Proof.For simplicity, we ρ p = ρ τ p = ρ τ ′ p and S p = S τ p = S τ ′ p , for each p; recall that ρ p ∈ cl D k p (R) for each p. Say f = τ p f p with p T f p R,k,p r p ≤ 2 f τ and p f p Sp r p ≤ 2 f τ ; since each T f p is convergent we may assume, after replacing each f p by , it follows that G p has radius of convergence at least ρ p , and that f p (w) = e γwm g p (w) for w ∈ H(log ρ p ), where g p := G p for each p.We extend g p to all of S p by setting g p (w) := f p (w)/e γwm for w ∈ S p \ H(log ρ p ).
Since T f p ρp = ρ γ p,m T g p ρp for each p, we get for some C = C(s/r) > 0 depending only on s r .It follows, on the one hand, that p g p H(log ρp) s p ≤ C g τ as well.On the other hand, since T g p Sp\H(log ρp) ≤ ρ −γ p,m T f p Sp\H(log ρp) , the same argument as above also proves that p g p Sp\H(log ρp) s p ≤ C g τ .Therefore, the function g : S τ ′ −→ C defined by g := p g p belongs to G τ ′ and satisfies g τ ′ ≤ C g τ and f (w) = e γwm g(w) for w ∈ S τ ′ , as claimed.
2.6.Generalized multisummable germs.Similar to Section 2 of [12], we let (T m , ≤) be the directed set of all tuples τ = (K, R, r, θ, ∆) as above, where τ whenever τ ′ ≤ τ and in this situation, for f ∈ G τ , the restriction f↾ S τ ′ belongs to G τ ′ .The directed limit of the directed system (G τ : τ ∈ T m ) under these restrictions is the set G m of germs at −∞ of functions in G τ , as τ ranges over T m .This G m is a C-algebra containing the germs at −∞ of the functions e γz i , for i = 1, . . ., m and γ ≥ 0, and we extend each norm • τ to all of G m by setting (1) For g ∈ G m and τ ∈ T m , we have f g τ ≤ f τ g τ .
Similar to [12, Section 3], we set C {X * } τ := T (G τ ), for τ ∈ T m , and C {X * } G := T (G m ).We refer to the latter as the C-algebra of all generalized multisummable series in the indeterminates X; note that Extending our notation of log-sum of convergent generalized power series, we also shall write F := T −1 (F ), for F ∈ C {X * } τ .2.7.Mixed series.We now consider additional indeterminates Y = (Y 1 , . . ., Y n ), and we defined mixed series similar to [12,Section 3].Thus, for τ ∈ T m and ρ ∈ (0, ∞) n , and for The latter is a Banach C-algebra with respect to the former norm.We sometimes refer to the indeterminates X i as the generalized Gevrey variables and to the indeterminates Y j as the convergent variables.Each As in Section 2.6, we set and we extend each norm Ordering the product T m × (0, ∞) n by the product order, we obtain the following generalization of Lemma 2.7: (2) If F (−∞, 0) = 0, then lim (τ,ρ) F τ,ρ = 0, where the limit is taken over the downward directed set Proof.Parts (1)-( 4) follow from Lemma 2.7.Part ( 5) is just a restatement of the fact that each C {X * ; Y } τ,ρ is a Banach algebra.Part ( 6) is proved along the lines of [12,Lemma 3.5].
We start with an elementary lemma similar to [12,Lemma 4.3].The essential difference between the proofs here and those in [12,Section 4] is that we cannot use Taylor expansion to compute the series after substitution as in [12,Lemma 4.2]; instead, we have to rely on our additional assumptions built into the norms • τ .Lemma 3.2.Let X ′ := (X 1 , . . ., X m−1 ), and let σ : {X, Y } −→ R[X ′ , Y ] be the substitution given by σ(X i ) := X i if i < m, σ(X m ) := X m−1 and σ(Y j ) := Y j for each j.Then for every
Case 1: n = 0. Choose convergent F p ∈ C {X * , Y } such that F = τ p F p ; as in Case 1 of the proof of [12,Lemma 4.3], it follows that p F p S τ ′ p r p ≤ F τ and σF = F • σ.Also, let k ′ be defined for K ′ as k is defined for K; it remains to show that p σF p R ′ ,k ′ ,p r p < ∞.Let s ∈ D k ′ p e R ′ ; then we have σF p s = F p σ(s) , for each p ∈ N, by the definition of these norms.Since σ(s) ∈ D k p e R by the above, we obtain Since s ∈ D k ′ p e R ′ was arbitrary, it follows that p σF p R ′ ,k ′ ,p r p ≤ p F p R,k,p r p , so that σF ∈ C {(X ′ ) * , Y } τ ′ ,ρ .Moreover, since this argument works for all sequences of convergent F p ∈ C {X * , Y } such that F = τ p F p , we also get σF τ ′ ≤ F τ in this case.
Case 2: n > 0; this case literally follows the proof of Case 2 of [12, Lemma 4.3], which we reproduce here for the convenience of the reader.We let F By the same argument as before each σF d belongs to C {(X ′ ) * , Y } τ ′ ,ρ , and since each F d has finite support (as a series in Y ), we get for all (w ′ , y) ∈ S τ ′ × D(ρ), which finishes the proof.
We now proceed to proving closure under each of the substitutions in Examples 3.1.
Permutations.Let σ be a permutation of {1, . . ., m} and, for x ∈ C m , we denote by σ(x) Proof.The proof follows the general strategy of the proof Lemma 3.2, but is easier and left to the reader.
The argument there shows that, for each ν, we have , and it follows from Taylor's Theorem that So it remains to show that each f ν belongs to G τ ′ ; to do so, we define k ′ for K ′ as k was defined for K, and we establish the following Claim.Each f p,ν is given by a generalized power series F p,ν with support in ∆ ′ such that To see the claim, we use [11, Lemma 6.5]-or more precisely, the following modification of it: the stated hypotheses there, namely, that τ ≤ ρ, τ m < λ and τ γ m−1 (λ + τ m ) < ρ m , were sufficient for the purposes of that paper, but not quite necessary to obtain the same conclusion from the proof of that lemma.Indeed, it suffices to assume that τ i ≤ ρ i for all i = m, that τ m < λ and that τ γ m−1 (λ + τ m ) ≤ ρ m to obtain the same conclusion, and we shall verify these weaker hypotheses below (with γ = 1) in order to apply that lemma here, without further mention of this discrepancy.
On the one hand, by Taylor's Theorem, we have for each p that On the other hand, using the binomial formula, we have for each p that was arbitrary, we finally get Finally, we get from [11, Lemmas 5.9(2,3) and 6.3(4)] that f p,ν = F p,ν , for each p and ν.This finishes the proof of the claim.
It follows from the claim that f ν = τ ′ p f p,ν .Moreover, since the claim holds for all sequences F p such that F = τ p F p , we also get that which finishes the proof of the proposition in Case 1.
Ramifications.Let σ be a ramification of the Gevrey variable X i 0 as in Examples 3.1 (3).Permuting coordinates, we may assume that i 0 = 1.Fix τ ∈ T m and ρ ∈ (0, ∞) n .We define (3) For γ ∈ N n ′ and k ∈ N, denote by N(γ, k) the number of ways to write γ as the sum of exactly k many nonzero elements in N n ′ .Then for any nonzero β ∈ N n , we have N γ j , β j .
Proof.A first computation (left to the reader) shows that, for β ∈ N n , we have where H β,γ (X ′ ) = where C > 0 is an absolute constant.Finally, since ( ρ) γ = 2 (n ′ +1) γ • (ρ ′ ) γ , we obtain (3.5) Multiplying by (ρ ′ ) γ and summing over γ therefore yields for some absolute constant D > 0, as required.As the results of [7] do not make use of the Weierstrass Preparation Theorem (which is in general not available in the quasianalytic setting), we may dispense with proving this property here.
Proof of Main Theorem.The previous discussion implies that our system A of algebras satisfies Conditions (1) and ( 4) of [7,Proviso 1.20].Moreover, Condition (2) is implied by Proposition 3.6, while Condition (3) follows from Proposition 2.4.So the theorem follows from [7, Theorems A and B].Finally, note that it suffices to add the reciprocal function to obtain quantifier elimination, as all real powers with nonnegative exponents are already in the language L G * .

( 1 . 2 .
We shall mainly focus on the direction d = 0 in this paper.)We extend the standard topology on C to C by declaring the log-disks as basic open neighbourhoods of −∞.Note that the usual covering map of the Riemann surface of the logarithm is represented in the logarithmic chart by the exponential function, and we extend it to a continuous function on C by setting e −∞ := 0. For each d ∈ R, the restriction of e w to S(d, ∞, π) \ {−∞} is injective; its inverse is the branch of the logarithm log d in the direction d.We are mostly interested in partial functions on C with values in C. In this spirit, we call a set D ⊆ C a log-domain if D ∩ C is a domain (in particular, every domain in C is a log-domain).If D ⊆ C is a log-domain, a log-holomorphic function on D is a continuous function f : D −→ C such that the restriction of f to D ∩ C is holomorphic.For example, every holomorphic function on a domain in C is log-holomorphic, and the exponential function is log-holomorphic on C. The logarithmic Borel and Laplace transforms.1.2.1.Logarithmic Borel transform.Let d, r ∈ R and θ > π/2, and write S = S(d, r, θ).Let f : S −→ C be such that f↾ S 0 is bounded and log-holomorphic, for every closed log-subsector S 0 of S. Given a closed log-subsector

) 2 )( 3 )
The function Bf : S ′ −→ C defined by Bf (w) := (B Im w f )(w) is log-holomorphic on every closed log-subsector S 0 of S ′ .(For every closed log-subsector S 0 of S ′ , there exist C, D > 0 such that |Bf (w)| ≤ Ce De Re w for w ∈ S 0 .Let α ≥ 0, and assume that for every closed log-subsector S 0 of S, we have |f (w)| = O e α Re w as w → −∞ in S 0 .Then for every closed log-subsector S 0 of S ′ , we have |Bf (w)| = O e α Re w as w → −∞ in S 0 .Accordingly, we call the function Bf defined in the proposition above the log-Borel transform of f .For D ⊆ C and g : D −→ C, we set g D := sup {|g(z)| : z ∈ D} .

( 2 )
Let α ≥ 0, and assume that for every closed log-subsector S 0 of S, we have |f (w)| = O e α Re w as w → −∞ in S 0 .Then, in the situation of part (1), for every closed log-subsector S 0 of V (d, R(θ)), we have |Lf (w)| = O e α Re w as w → −∞ in S 0 .

Corollary 1 .( 1 ) 2 )
23.Let f : S −→ C be a function, and assume that for every closed logsubsector S of S, the restriction f↾ S is log-holomorphic and there exist C, D > 0 such that |f (w)| ≤ Ce De (Re w)/λ for w ∈ S. Let also θ ∈ (0, ϕ), and let r ≤ log(D)λ be as above and setS ′ := S d, r, θλ + πλ 2 .Let α ≥ 0,and assume that for every closed log-subsector S contained in S, we have |f (w)| = O e α Re w as w → −∞ in S.Then, for every closed log-subsector S contained in S ′ , we have |(L λ f )(w)| = O e α Re w as w → −∞ in S. (For α ≥ 0, we have L λ p α = Γ(αλ)p α .For f : S(d, r, θλ) −→ C as in Section 1.5.1,Corollary 1.22 implies that L λ (B λ f ) is defined and log-holomorphic on every closed log-subsector S contained in S(d, r, θλ).From Proposition 1.6, we therefore obtain: Corollary 1.24.For f : S(d, r, θλ) −→ C as in Section 1.5.1, we have L λ (B λ f ) = f .1.5.3.Formal Borel and Laplace transforms.Let now

Lemma 2 . 3 .
Let f ∈ G τ and a ∈ C m−1 be such that θ(a) > 0 and Re a < log R ′ .Then the function f a : S τ (a) −→ C, defined by f a (w) := f (a, w), belongs to G τ (a) and satisfies f a τ (a) ≤ f τ and T (f a )(X m ) = T (f )(e a , X m ).
this is done in Proposition 1.27, so we assume m > 1.Let a ∈ C m−1 , and define θ(a) and τ (a) as in Remark 2.2.Assume that θ(a) > π/2 and Re a < R ′ ; by Lemma 2.3 and the assumption T f = 0, we obtain f a ∈ G τ (a) with T (f a ) = 0.It follows from quasianalyticity of G τ (a) that f a = 0. Since the set of a ∈ C m−1 for which the latter holds contains an open set (by Lemma 2.

Proposition 2 . 9 .
higher degree, with u ∈ C nonzero.Using Lemma 2.8, the proof of [9, Proposition 4.1] now establishes the following, where Y ′ = (Y 1 , . . ., Y n−1 ): Let f ∈ C {X * ; Y } G , and assume that n > 0 and F is regular in Y n of order d.Then the series F factors uniquely as F = GH, where G ∈ C {X
, where p αn is defined as in Example 1.7.Proceeding as in the proof of Lemma 1.3, we obtain a C > 0 such that ∂S |u n (η)|dη ≤ C|a n |r αn , for each n.Since F r < ∞, it follows that n ∂S |u n (η)|dη < ∞.If follows from analysis that the functions u n , for each n, as well as n u n and n |u n | are integrable on ∂S and that 1and set s := (s ′ , 2ρ 0 ) and t := σ(s).By Claim 2 of [12, Lemma 4.7], we have t ∈ cl D k p