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PACIFIC JOURNAL OF MATHEMATICS

Volume 187 No. 1 January 1999

PacificJournalofM

athematics

1999Vol.187,N

o.1

PacificJournal ofMathematics

Volume 187 No. 1 January 1999

PACIFIC JOURNAL OF MATHEMATICS

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PACIFIC JOURNAL OF MATHEMATICSVol. 187, No. 1, 1999

COMPONENTS OF THE SPACE PARAMETRIZINGGRADED GORENSTEIN ARTIN ALGEBRAS

WITH A GIVEN HILBERT FUNCTION

Mats Boij

We give geometric constructions of families of graded Gor-enstein Artin algebras, some of which span a component of thespace Gor(T ) parametrizing Gorenstein Artin algebras with agiven Hilbert function T . This gives a lot of examples whereGor(T ) is reducible. We also show that the Hilbert functionof a codimension four Gorenstein Artin algebra can have anarbitrarily long constant part without having the weak Lef-schetz property.

1. Introduction.

In R = k[x, y] and in R = k[x, y, z], the parameter space Gor(T ) of gradedGorenstein Artin quotients of a given Hilbert function T is irreducible forall T . The codimension two case is implied by F. S. Macaulay’s result thatGorenstein height two ideals in k[x, y] are complete intersections [8], and thefact that the family of graded complete intersection ideals of given generatordegrees is irreducible. The codimension three case was proved S. Diesel [4].A. Iarrobino and V. Kanev [7] have recently shown that in codimension fiveand higher, Gor(T ) can be reducible. In this paper, we will give a large classof examples where Gor(T ) is reducible, in codimension four and higher.

A. Iarrobino and V. Kanev also proved that there is a close relationbetween graded Gorenstein Artin algebras of codimension three and finitelength Cohen-Macaulay sub-schemes of P2. They show that whenever theHilbert function T is equal to s for at least three degrees, there is a fibrationGor(T ) → H(T ) ⊆ Hilbs(P2) which takes the form f to the initial part ofthe ideal annR(f). We prove that there can be no such results in higher codi-mensions, by showing that for r ≥ 4, there are codimension r GorensteinArtin algebras with arbitrary long constant part in the Hilbert function,whose initial ideal does not define a sub-scheme of length s in Pr−1. Theway this is done is by exploring the weak Lefschetz property.

Setup 1.1. Let k be an infinite field and let R = k[x1, x2, . . . , xn] be thepolynomial ring in n variables. Denote by E the injective hull of k over R.If R is graded in non-negative degrees with deg xi = 1, then E is naturally

1

http://pjm.math.berkeley.edu/pjmhttp://dx.doi.org/10.2140/pjm.1999.187-1

2 MATS BOIJ

graded in non-positive degrees, E = E0 ⊕ E−1 ⊕ · · · . By Matlis duality,the space P(E−c) parametrizes graded Gorenstein Artin quotients of R withsocle in degree c. For a given symmetric Hilbert function T = (t0, t1, . . . , tc),we consider the space of forms f in E such that the Gorenstein Artin algebraR/ annR(f) has Hilbert function T and we denote this space by Gor(T ) (cf.A. Iarrobino [6]). Each element f in E−c gives a map f : Rc −→ k, byf(g) = g ◦ f ∈ E0 ∼= k. Let md,c : Rd×Rc−d → Rc denote the multiplicationmap. Then we have that

Gor(T ) = {f ∈ E−c∣∣ rank f ◦md,c = td, for d = 0, 1, . . . , c}.

Thus there is a natural scheme structure on Gor(T ) as a locally closedsubscheme of P(E−c) given by the minors of catalecticant matrices. Thetangent space to the affine cone over Gor(T ) at a point f in E−c is given byannE(annR(f)2)−c (cf. A. Iarrobino and V. Kanev [7]).

2. Points on rational normal curves.

In this section we will construct families of Gorenstein Artin algebras relatedto sets of points on rational normal curves. We will compute the dimensionof these families, as well as the dimension of the tangent space to Gor(T ) at aspecial point in each family. In codimension four, this point is non-singular,and the dimension of the tangent space coincides with the dimension ofthe family, which shows that the family in this case spans an irreduciblecomponent of Gor(T ).

Setup 2.1. For a positive integer n, let

Tn = {C |C is rational normal curve in Pn}.(2.1.1)

This is an irreducible open subset of the Hilbert scheme Hilbnx+1(Pn) para-metrizing curves in Pn of Hilbert polynomial nx+ 1. The dimension of Tnis (n+ 1)2 − 4 = (n+ 3)(n− 1). Denote by R the homogeneous coordinatering of Pn.

Definition 2.2. For integers n and s, let

Pn,s = {(X,C) | IC ⊆ IX} ⊆ Hilbs(Pn)× Tn.(2.2.1)

Let Pn,s ⊆ Pn,s be the open subset on which the Hilbert function is maxi-mal.

Lemma 2.3. For non-negative integers n and s, we have that:i) The set Pn,s is closed and irreducible of dimension (n+ 3)(n− 1) + s.ii) If s ≥ 2n+ 1, the induced map Pn,s −→ Hilbs(Pn) is injective.

Proof. Since Tn is irreducible, Hilbs(Pn) is proper and all the fibers over Tnare irreducible of the same dimension s, we have that Pn,s is irreducible ofdimension (n+ 3)(n− 1) + s, by Lemma 5.1.

THE SPACE OF GRADED GORENSTEIN ARTIN ALGEBRAS 3

A general element of Pn,s is a set X of s distinct points on a specifiedrational normal curve C in Pn. For an element (X,C) in Pn,s with s ≥2n + 1, the set X determines the curve C, since the ideal of the curve isgenerated in degree 2 and there are no other generators of IX in degree 2.Thus, for s ≥ 2n + 1, we get an injective map from Pn,s to the Hilbertscheme Hilbs(Pn) parametrizing length s sub-schemes of Pn. �

Definition 2.4. For integers n, s and c, let

PFn,s,c = {(f,X,C) | f ∈ annE(IX)} ⊆ P(E−c)×Pn,s.(2.4.1)

Let PFn,s,c be the open subset of PFn,s,c on which the Hilbert function ofthe Gorenstein quotient R/ annR(f) is maximal.

Lemma 2.5. For non-negative integers n, s and c, we have that:i) The set PFn,s,c is closed and irreducible of dimension

(n+ 3)(n− 1) + s+HX(c)− 1.

ii) If HX(b(c− 1)/2c) = s and s ≥ 2n+ 1, then the induced map

PFn,s,c −→ P(E−c) is injective.

iii) If (f,X,C) is in PFn,s,c, then the Hilbert function of the GorensteinArtin algebra A = R/ annR(f) is given by

HA(d) = min{nd+ 1, s, n(c− d) + 1}, for d = 0, 1, . . . , c.

Proof. Here we have that Pn,s is irreducible, P(E−c) is proper, and all thefibers are projective spaces of the same dimension HX(c)− 1. Hence we getthat PFn,s,c is irreducible of dimension (n+ 3)(n− 1) + s+HX(c)− 1, byLemma 5.1.

Since a general element (f,X,C) of PFn,s,c corresponds to a reduced setof points X on a rational normal curve C together with a general form fannihilating the ideal of the points, we have that the Hilbert function of thequotient A = R/ annR(f) is given by

HA(d) = min{HX(d),HX(c− d)}, for d = 0, 1, . . . , c.(2.5.1)

If HX(b(c−1)/2c) = s, we have that the generators of IX can be in no higherdegree than b(c−1)/2c+1 = b(c+1)/2c. In that case we get that HA(b(c+1)/2c) = HA(b(c− 1)/2c) = s and annR(f)/IX has no generators in degreeslower than b(c+ 1)/2c+ 1. Hence the Gorenstein ideal annR(f) determinesthe ideal IX . If in addition s ≥ 2n + 1, we get, by the previous lemma, aninjective map from an open set in PFn,s,c into P(E−c) by projection ontothe first factor. �

Definition 2.6. Denote by Fn,s,c the image of PFn,s,c in P(E−c) when s ≥2n+ 1 and HX(b(c− 1)/2c) = s.

4 MATS BOIJ

Proposition 2.7. For positive integers n, s and c with 2n+1 ≤ s ≤ nb(c−1)/2c+ 1, we have that:

i) The family Fn,s,c is irreducible of dimension (n+ 3)(n− 1) + 2s− 1.ii) Let f be an element of Fn,a,c, then the Hilbert function of A =

R/ annR(f) is given by

HA(d) = min{nd+ 1, s, n(c− d) + 1}, for d = 0, 1, . . . , c.(2.7.1)

Proof. The proposition is an immediate consequence of Lemma 2.5 since thecondition HX(b(c− 1)/2c) = s is equivalent to nb(c− 1)/2c+ 1 ≥ s. �

Definition 2.8. Denote by Tn,s,c the Hilbert function given in Proposi-tion 2.7.

Remark 2.9. If f is a general element of Fn,s,c, and bc/2cn+1 ≥ s, then theGorenstein Artin algebra R/ annR(f) satisfies the weak Lefschetz property(cf. §3.1). Indeed, the ideal is annR(f) = IX + J , where J is generatedin degree at least bc/2c, and since IX is the defining ideal of a finite set ofpoints, R/IX has a non-zero divisor.

2.1. The tangent space to Gor(T). We will now compute the tangentspace of Gor(T ) at special point of the family Fn,s,c. In fact, the point wewill pick will not even be in the family, strictly speaking, but in the closure.However, it will have the same Hilbert function. Thus the dimension of thetangent space at this point will give an upper bound for the dimension ofthe tangent space at a general point of the family Fn,s,c.

We now identify E with the polynomial ring k[X0, X1, . . . , Xn], where Racts on E by contraction, i.e., xi ◦Xj = δij .

Lemma 2.10. For integers 2 ≤ a1 ≤ a2, · · · ≤ an, such that an − a1 ≤ 1,and c ≥ 2an + 2, let F =

∑ni=1X

ai0 X

c−aii and let A = R/ annR(F ). Then

the following hold:i) The dimension of annE(annR(F )2)−c is 2n2 + 2

∑ni=1 ai.

ii) The Hilbert function of A is

HA(d) = min

{nd+ 1, n+

n∑i=1

ai, n(c− d) + 1

}, for d = 0, 1, . . . , c.

Proof. The ideal I = annR(F ) is generated by the elements

xixj , for 1 ≤ i < j ≤ nxan+10xix

ai+10 , for 1 ≤ i ≤ n

xc+1−annxc−aii − xc−ann x

an−ai0 , for 1 ≤ i ≤ n.

(2.10.1)


Denote by IC the ideal ({xixj}1≤i an, we have that {xdi , x

d−1i x0, . . . , x

d−aii x

ai0 }1≤i≤n

is a k-basis of (R/IX)d. Since there are no generators of annR(F )/IX indegrees lower than c − an ≥ c/2 + 1, we have that the Hilbert function ofA = R/ annR(F ) is HA(d) = min{nd + 1, n +

∑ni=1 ai, n(c − d) + 1}, for

d = 0, 1, . . . , c. �

Proposition 2.11. Given positive integers n, s, c, write s = pn + q, where1 ≤ q ≤ n. Let

F =n−q∑i=1

Xp−10 Xc+1−pi +

n∑i=n−q+1

Xp0Xc−pi .

If c ≥ 2ds/ne and s ≥ 3n, then we have that:i) R/ annR(F ) has Hilbert function Tn,s,c.ii) The tangent space to Gor(Tn,s,c) at F has dimension 2n(n−1)+2s−1.iii) F lies in the closure of Fn,s,c.

Proof. We apply Lemma 2.10 with

ai =

{p− 1, for 1 ≤ i ≤ n− q,p, for n− q < i ≤ n.

6 MATS BOIJ

We have that s ≥ 3n implies that a1 ≥ 2. Furthermore, an = p = b(s −1)/nc = ds/ne − 1, which shows that c ≥ 2ds/ne implies that c ≥ 2an + 2.Thus the assumptions of the lemma are fulfilled and the first two assertionsfollow directly from the lemma.

To prove the third assertion, we look at the ideals IC and IX from theproof of Lemma 2.10. The geometric picture is that C is the union of n linesthrough one point and X is a scheme of length s concentrated at one pointof each line, with multiplicity p − 1 at n − q points and multiplicity p atthe remaining q points. This curve C does not belong to the family Tn ofrational normal curves, but it has the right Hilbert function, and it is in theclosure of Tn in the Hilbert scheme Hilbnx+1(Pn). In fact, we can deformthe ideal of C into a family of rational normal curves by looking at the idealof 2× 2-minors of the matrix(

αx0 + x1 αx1 + 2x2 . . . αxn−1 + nxnx1 x2 . . . xn

),

where α is a parameter. When α = 0, we get the ideal IC , while for allbut possibly finitely many values of α, we get the ideal of a rational normalcurve. When we construct the family Pn,s over the component of the Hilbertscheme Hilbnx+1(Pn) that contains Tn, we will no longer get irreduciblefibers over all points. However, we can take the closure of the family Pn,sin Hilbs(Pn) × Hilbnx+1(Pn) to get an irreducible family. Then we willget a family that contains (X,C), since the support of X is away from thesingularity of C. We can then continue the construction to get a slightlylarger family that contains Fn,s,c as an open dense subset. This family alsocontains F , which concludes the proof. �

Corollary 2.12. For n = 3, s ≥ 9, and b(c − 1)/2c ≥ ds/3e the familyF3,s,c spans a component of Gor(T3,s,c).

Proof. In this case, we have that the dimension of the tangent space is11 + 2s, which coincides with the dimension of the family F3,s,c. Thus thepoint we have chosen is, in fact, a smooth point of Gor(T3,s,c), and thefamily F3,s,c must span an irreducible component. It remains to check thatthe assumptions of Propostion 2.7 and Proposition 2.11 apply. The conditions ≥ 9 is equivalent to s ≥ 3n, and the condition b(c− 1)/2c ≥ ds/3e impliesthat c ≥ 2ds/ne and that nb(c− 1)/2c+ 1 ≥ s. �

3. Families of Gorenstein Artin algebras not satisfying the weakLefschetz property.

In this section we give a construction of families Gn,c of Gorenstein Artinalgebras not satisfying the weak Lefschetz property. The algebras in thefamilies Gn,c are generalizations of the examples given by R. Stanley [10]


and by D. Bernstein and A. Iarrobino [2] showing the existence of non-unimodal Hilbert functions of Gorenstein algebras. In addition, the Hilbertfunctions of the algebras in Gn,c are given by Tn,c+2,c. We will use this factto prove that Gor(Tn,c+2,c) is reducible for most choices of n and c. Thefamily Gn,c consists of Gorenstein ideals whose initial part is the square ofthe ideal of a line.

Setup 3.1. For a positive integer n, let G(1, n) be the grassmannian oflines in Pn. It is an irreducible variety of dimension 2(n − 1). For integersn and c, let

LGn,c = {(f, L) | f ∈ annE(I2L)} ⊆ P(Ec)×G(1, n).(3.1.1)

Let LGn,c be the open subset of LGn,c where the Hilbert function is maximal.

Proposition 3.2. For positive integers n and c we have that:i) The set LGn,c is irreducible of dimension 2(n− 1) + nc.ii) For (f, L) in LGn,c, the Hilbert function of A = R/ annR(f) is given

by

HA(d) = min{nd+ 1, c+ 2, n(c− d) + 1},(3.2.1)for d = 0, 1, . . . , c.

iii) If c ≥ 2n− 1, the map LGn,c −→ P(E−c) is injective.

Proof. The map LGn,c −→ G(1, n) is proper and surjective, G(1, n) is irre-ducible of dimension 2(n− 1) and all the fibers are irreducible of dimensionnc. The latter since the Hilbert function of R/I2L is nd + 1. Hence the setLGn,c is irreducible of dimension 2(n− 1) + nc, by Lemma 5.1.

To prove the second assertion, we note that we can change variables sothat the line L is given by the ideal (x2, x3, . . . , xn). Let HA(d) denote theHilbert function of A = R/ annR(f). Then HA(d) is given by the dimensionof Rd ◦ f . The condition that f annihilates I2L, shows that

Rd ◦ f = [(x0, x1)d]d ◦ f + [(x0, x1)d−1(x2, x3, . . . , xn)]d ◦ f.Thus we get HA(d) ≤ d+ 1 + d(n− 1). Since the latter term is contained inannE(x2, . . . , xn)∩Ed−c, we also get HA(d) ≤ d+1+c−d+1 = c+2. Sincethe HA(d) is symmetric around d = c/2, we get that the Hilbert function inthe proposition is an upper bound for the H(d).

It now suffices to show that there is one f that gives this maximal Hilbertfunction. This is can be done by choosing a compressed Artin level algebraB in two variables with socle of dimension n− 1 in degree c− 1, i.e., B is alevel algebra with Hilbert function HB(d) = min{d+ 1, (c− d)(n− 1)}, andtake A = B × Bˇ – the trivial extension of A by its dual.This Gorensteinalgebra has the right Hilbert function, (cf. D. Bernstein and A. Iarrobino [2],or M. Boij and D. Laksov [3]), and the square of the ideal generated by thegenerators of Bˇis zero in A.

8 MATS BOIJ

For the third assertion, we need to prove that if c ≥ 2n − 1 and (f, L)is in GLn,c then the ideal annR(f) determines the line L. The ideal I2Lis generated in degree 2, and looking at the Hilbert function, we see thatannR(f) has no other generators in degree 2. Thus the line is determinedby annR(f). �

Definition 3.3. Denote the image of LGn,c in P(E−c) by Gn,c, for c ≥2n− 1.

3.1. The weak Lefschetz property. Recall that a Gorenstein Artin al-gebra A with socle in degree c satisfies the weak Lefschetz property if themultiplication maps ` : Ad → Ad+1, for d = 0, 1, b(c − 1)/2c, are injectivefor a general linear form ` ∈ A1. We will use the following characterizationof the weak Lefschetz property, due to J. Watanabe [11, Prop. 3.4].

Proposition 3.4. The Gorenstein Artin algebra A = R/ annR(f) satisfiesthe weak Lefschetz property if and only if the Hilbert function of R/ annR(`◦f) agrees with the Hilbert function of A in degrees less than c/2 for a generallinear form ` ∈ R.

Proposition 3.5. Suppose that n ≥ 3, and c = 5 or c ≥ 7, then for anelement f ∈ Gn,c the Gorenstein Artin quotient A = R/ annR(f) does notsatisfy the weak Lefschetz property.

Proof. Under the assumptions of the proposition, the Hilbert function of theGorenstein Artin algebra A = R/ annR(f) for an element f of Gn,c attainsthe value c + 2 in degree b(c − 1)/2c. On the other hand, for any linearform ` ∈ R, we have that ` ◦ f is in Gn,c−1, but by Proposition 3.2, we havethat the Hilbert function of R/ annR(` ◦ f) is at most c+ 1. Thus it cannotagree with the Hilbert function of R/ annR(f) in degree b(c−1)/2c, and theproposition follows from Proposition 3.4. �

A. Iarrobino and V. Kanev [7] have shown that for codimension threeGorenstein algebras there is a strong connection to subschemes of P2. Inparticular, they show that if a codimension three Gorenstein Artin algebrahas Hilbert function containing a subsequence (s, s, s), there is a uniquelydetermined Cohen–Macaulay subscheme of P2 – the annihilating scheme –which ideal is generated by the lower degree part of the Gorenstein ideal.In particular, this means that the Gorenstein algebra must have the weakLefschetz property. The following theorem shows that a similar result isimpossible in codimension four and higher.

Theorem 3.6. In codimension four and higher, there are graded GorensteinArtin algebras not satisfying the weak Lefschetz property with arbitrarily longconstant part in the Hilbert function.


Proof. By Proposition 3.5 the algebras in Gn,c do not have the weak Lef-schetz property. However, their Hilbert function have a constant subse-quence (c+2, c+2, . . . , c+2), of length c+1−2b(c+1)/nc ≥ (c+1)(1−2/n).For n ≥ 3 this can be arbitrarily large. �

4. Components of Gor(T).

We can now use the results from the previous sections to conclude that inmany cases, we get two distinct components of Gor(T ). The idea is thatthe families Fn,c+2,c and Gn,c both parametrize Gorenstein Artin algebraswith Hilbert function Tn,c+2,c while the latter family, for large c, has higherdimension than the dimension of the tangent space of the first family, so itcannot lie in the same component.

Theorem 4.1. For n ≥ 3 and c ≥ max{11, 3n − 2} the parameter spaceGor(Tn,c+2,c) is reducible with one component containing Fn,c+2,c and an-other component containing Gn,c.

Proof. For c ≥ 2d(c+2)/ne and c+2 ≥ 3n the tangent space of Gor(Tn,c+2,c)at a general point of Fn,c+2,c has dimension at most 2n(n − 1) + 2(c +2) − 1 according to Proposition 2.11. Thus the irreducible component ofGor(Tn,c+2,c) that contains Fn,c+2,c has at most this dimension.

For c ≥ 2n− 1 the dimension of Gn,c is 2(n− 1) +nc, by Proposition 3.2.Any component of Gor(Tn,c+2,c) that contains Gn,c and some element withthe weak Lefschetz property must have dimension at least 2n−1+nc, sincethis is an open property.

One can easily check that c ≥ max{11, 3n−2} implies that c ≥ 2dc+2/ne,c+2 ≥ 3n and c ≥ 2n−1. Furthermore, 2n−1+nc > 2n(n−1)+2(c+2)−1,which implies that the family Gn,c cannot be in the same component ofGor(Tn,c+2,c) as Fn,c+2,c. �

Remark 4.2. The smallest example of reducible Gor(T ) that is given byTheorem 4.1 is n = 3, c = 11, which gives Hilbert function

T = (1, 4, 7, 10, 13, 13, 13, 13, 10, 7, 4, 1).

In this case we have that both families F3,13,11 and G3,11 have dimension 37,and by computer calculations that the tangent space of Gor(T ) at a generalpoint of G3,11 has dimension 37, so the family spans its component in thiscase.

By means of the computer algebra system, Macaulay [1], we have checkedthat the dimension of the tangent space of Gor(T3,c+2,c) at a general pointof G3,c coincides with the dimension of G3,c for c = 11, 12, . . . , 34.

10 MATS BOIJ

5. Appendix.

In proving the irreducibility of the families Fn,s,c and Gn,c we needed thefollowing lemma. This is stated in the projective case in Shafarevich [9,Thm. 8, Chap. I], although the proof there uses only properness. Forconvenience, we give a brief proof.

Lemma 5.1. Let f : X −→ Y be a proper surjective morphism of schemesof finite type over k. Suppose that Y is irreducible of dimension dY and thatall the fibers of f are irreducible of dimension d. Then X is irreducible ofdimension d+ dY .

Proof. Firstly, we may assume that both X and Y are reduced. Thus wecan use R. Hartshorne [5, Ch. II, Ex. 3.22] on the restrictions of f to thecomponents of X that surject onto Y . Since f is proper and surjective andY is irreducible each component of X must either map onto Y or outsidean open dense subset U of Y .

For each component X ′ of X surjecting onto Y there is an open set in Yover which the fibers of f |X′ have minimal dimension. Let y be a point inthe intersection of all such open sets and U . Since the fiber Xy is irreducible,it must lie completely within at least one of the fibers X ′y. Hence, the fibersof the map X ′ −→ Y must all have dimension d, and since the fibers areirreducible, we must have thatX ′y′ = Xy′ for all points y

′ in Y . ThusX = X ′

is irreducible of dimension d+ dY . �

References

[1] D. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geome-try and commutative algebra, Source and object code available for Unix and Macin-tosh computers. Contact the authors, or download from zariski.harvard.edu viaanonymous ftp.

[2] D. Bernstein and A. Iarrobino, A nonunimodal graded Gorenstein Artin algebra incodimension five, Comm. Algebra, 20(8) (1992), 2323-2336.

[3] M. Boij and D. Laksov, Nonunimodality of graded Gorenstein Artin algebras, Proc.Amer. Math. Soc., 120(4) (1994), 1083-1092.

[4] S.J. Diesel, Irreducibility and dimension theorems for families of height 3 Gorensteinalgebras, Pacific J. Math., 172(2) (1996), 365-397.

[5] R. Hartshorne, Algebraic geometry, Vol. 52 of Graduate Texts in Mathematics,Springer-Verlag, New York-Heidelberg-Berlin, 1977.

[6] A. Iarrobino, Associated graded algebra of a Gorenstein Artin algebra, Vol. 514 ofMemoirs of the American Mathematical Society, American Mathematical Society,Providence, 1994.

[7] A. Iarrobino and V. Kanev, The length of a homogeneous form, determinantal loci ofcatalecticants, and Gorenstein algebras, preprint, 1996.


[8] F.S. Macaulay, The algebraic theory of modular systems, Cambridge MathematicalLibrary, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916original.

[9] I.R. Shafarevich, Basic algebraic geometry 1, Springer-Verlag, Berlin, second edition,1994. Varieties in projective space.

[10] R. Stanley, Hilbert functions of graded algebras, Adv. in Math., 28 (1978), 57-83.

[11] J. Watanabe, The Dilworth number of Artinian rings and finite posets with rankfunction, in ‘Commutative algebra and combinatorics’ (Kyoto, 1985), Vol. 11 of Adv.Stud. Pure Math., 303-312, North-Holland, Amsterdam, 1987.

Received July 9, 1997 and revised September 12, 1997.

KTHS-100 44 StockholmSwedenE-mail address: [email protected]

mailto:[email protected]

PACIFIC JOURNAL OF MATHEMATICSVol. 187, No. 1, 1999

THE STOKES PHENOMENON IN EXACT ASYMPTOTICS

B.L.J. Braaksma, G.K. Immink, and Y. Sibuya

Dedicated to Prof. H.L. Turrittin on the occasion of his ninetieth birthday

As an introduction we present a new, elementary and con-structive proof of the multisummability properties of formalsolutions of linear ODE’s at irregular singular points. Thisserves to illustrate the geometric approach to multisumma-tion. Basic properties of multisums and the associated sheavesare derived. Next, we study Cauchy-Heine transforms in re-lation to multisummation and the Stokes phenomenon. Weshow how to construct multisums with a prescribed Stokesphenomenon, using the Malgrange-Sibuya isomorphism.Starting from the Stokes automorphisms we introduce thealien derivations of J. Ecalle and derive Ecalle’s bridge equa-tion for the general integral of linear ODE’s. The main ideasare illustrated with some very simple examples.

Introduction.

Consider the differential equation (D): Dy = 0, where D is a linear differ-ential operator of order n, with coefficients in C{z}. It has n independentformal solutions of the form:

ŷl(z) = eql(z)zρl ĥl(z)(1)

where ρl ∈ C, ql ∈ z−1/pC[z−1/p], and ĥl ∈ C[[z1/p]][log z] for some p ∈ N.According to the main asymptotic existence theorem (cf. [26]) there

exists � > 0 such that for every interval I ⊂ R with length |I| < � and everyl ∈ {1, . . . , n}, there exists a solution yl,I of (D) with the property thatyl,I ∼ ŷl on I, i.e.

e−ql(z)z−ρlyl,I(z) ∼ ĥl(z) as z → 0, arg z ∈ I.In general, however, this solution is not unique. The object of exact asymp-totics is to associate a unique “sum” with a given divergent power series.

Part of this work has been performed in the University of Southern California. Theauthors are indebted to the Department of Mathematics of USC and in particular ProfessorW.-A. Harris, Jr, for their hospitality and support. After the revision of this paper welearned that Professor Harris passed away. We express our deep feelings of sorrow at hisdecease.

13

http://pjm.math.berkeley.edu/pjmhttp://dx.doi.org/10.2140/pjm.1999.187-1

14 B.L.J. BRAAKSMA, G.K. IMMINK, AND Y. SIBUYA

For the class of divergent power series arising in the theory of analytic dif-ferential equations this can be achieved by means of a process called ‘multi-summation’. There exist two, essentially equivalent, approaches to the the-ory of multi-summability: The analytic approach, which is based on the useof Borel and Laplace transformations, and the geometric approach. In thelatter, an important part is played by so-called (k-precise) quasi-functions:Analytic functions which are defined modulo functions with exponential de-crease of some order (k). The term k-precise quasi-function was introducedby J.-P. Ramis in [24]. In this paper, we explain and develop the secondapproach.

The paper is organized as follows. In §1 we give an elementary andconstructive proof of the multi-summability of the formal series ĥl in (1).This section serves as an introduction to and motivation of the geometricapproach. In §2 we derive some basic properties of quasi-functions andmulti-sums. §3 deals with Cauchy-Heine transforms in relation to multi-summability and the Stokes phenomenon. Here, also the inverse problem isconsidered: To construct multi-sums with a prescribed Stokes phenomenon.In §4 we discuss the Stokes phenomenon in a fixed direction. In §5 weintroduce Stokes automorphisms and give a definition of the alien derivationsof J. Ecalle in terms of quasi-functions. The main ideas are illustrated intwo very simple examples of linear differential equations.

1. Introductory example.

We begin by introducing some notation.

Definition 1.1. Let I ⊂ R be an open interval. A(I) is the set of allfunctions f with the property that, for any closed interval I ′ ⊂ I, thereexists a positive number r such that f is holomorphic on the sector S(I ′, r) :={z : |z| < r, arg z ∈ I ′} of the Riemann surface of log z. With the naturalrestriction mapping, the sets A(I) form the sheaf A on R of holomorphicfunctions in a sector with vertex at O. For any θ ∈ I, the germ at θ off ∈ A(I) is the equivalence class of functions f ′ ∈ A(I ′), where I ′ 3 θ, suchthat f ′|I′′ = f |I′′ for some open interval I ′′ ⊂ I ∩ I ′ containing θ. This germis denoted by fθ. The stalk Aθ of A is the set of all germs at θ of functionsf ∈ A(I), where I is any open interval containing θ.

Let k > 0. By A≤−k, A

THE STOKES PHENOMENON IN EXACT ASYMPTOTICS 15

r, arg z ∈ I ′} of the Riemann surface of log z, and

supz∈S(I′,r)

|f(z)|ec|z|−k 0, for all c > 0, for some c < 0 and for all c < 0, respectively.

We consider the differential equation (D) with formal solutions (1). Forl,m ∈ {1, . . . , n}, let qml = qm − ql. If ql 6≡ qm, there exist κml ∈ 1pN andωml ∈ C∗ such that

qml(z) = ωmlz−κml + o(z−κml).

Definition 1.2. The numbers κml are called the levels of the differentialequation (D). We say that m ≺ l on the interval I if eqml(z) decreasesexponentially of order κml as z → 0, arg z ∈ I (so if cos(argωml−κmlθ) < 0if θ ∈ I). θ is a Stokes direction of level κml for the pair (m, l) ifcos(argωml − κmlθ) = 0. Let the increasing sequence of Stokes directionsbe denoted by {θh}h∈Z. If m ≺ l on an interval (θσ, θµ) of length π/κmlthen {θσ, θµ} is called a (m ≺ l)-Stokes pair or a Stokes pair of levelκml and we denote θµ by θml. If Ij , j = 1, . . . , r, are open intervals suchthat Ir ⊂ Ir−1 ⊂ . . . ⊂ I1 then (I1, . . . , Ir) will be called a nested multi-interval. If I and I(ν), ν = M, . . . ., N, are open intervals where M andN are integers such that M < N , ∪Nν=MI(ν) = I and I(ν) ∩ I(µ) 6= ∅ iff|ν − µ| ≤ 1 then {I(ν)}Nν=M is called a finite covering of I.

Suppose that (D) has r distinct levels: k1, . . . , kr in order of increasingmagnitude. It is known that one can associate a unique sum to ŷl and anynested multi-interval (I1, . . . , Ir), if |Ij | > π/kj and Ij does not containStokes pairs of level kj (cf. [22], [2], [5], [21]). Here we give another proofof this property based on the main asymptotic existence theorem mentionedabove. Another proof based on this theorem has been given by M. van derPut (to appear). Similar ideas can be found in B. Malgrange [19]. From themain asymptotic existence theorem we first derive a fundamental system ofnormalized solutions, due in the generic case to G.D. Birkhoff [3] and in thegeneral case to W. Balser, W.B. Jurkat and D.A. Lutz (cf. [13, p. 85]).

Lemma 1.3. Let l ∈ {1, . . . , n}, ν0 ∈ Z. Then there exist solutions{y(ν)l }ν∈Z of (D) such that y

(ν)l ∼ ŷl on (θν−1, θν+1) and

y(ν)l − y

(ν−1)l =

∑θhl=θν

αh(l, ν)y(ν−1)h if ν > ν0,(2)

y(ν)l − y

(ν−1)l =

∑θlh=θν−1

αh(l, ν)y(ν)h if ν ≤ ν0.(3)


Proof. Let I := (a, b) be an interval that contains a Stokes direction θν andwith |I| < � as in the Introduction. Then it is well known and easy to seethat yl,I ∼ ŷl on (θν−1, θν+1). Denote yl,I by ỹ

(ν)l .

Let y(ν0)l := ỹ(ν0)l for l = 1, . . . , n. Let ν ≥ ν0 + 1 and suppose the

construction of y(ν)l satisfying (2) has been performed up to ν − 1. LetI = (θν−1, θν+1). Then there exist constants bh such that

ỹ(ν)l − y

(ν−1)l =

∑θhl=θν

bhy(ν−1)h +

∑h≺l on I

bhỹ(ν)h ,

since the solutions y(ν−1)h with θhl = θν and ỹ(ν)h with h ≺ l on I can be

extended to a fundamental system by solutions ỹ(ν)h which dominate thelefthand side on (θν−1, θν). Therefore we choose

y(ν)l = ỹ

(ν)l −

∑h≺l on I

bhỹ(ν)h = y

(ν−1)l +

∑θhl=θν

bhy(ν−1)h ,

so that y(ν)l ∼ ŷl on I and (2) holds up to ν. Thus we recursively obtain ally

(ν)l with ν ≥ ν0. Similarly the case with ν < ν0 can be handled. �

From this lemma we deduce:

Theorem 1.4. Let l ∈ {1, . . . , n} and (I1, . . . , Ir) be a nested multi-intervalsuch that |Ij | > π/kj and Ij does not contain any (h ≺ l)-Stokes pair of levelκhl = kj for j = 1, . . . , r. Let kr+1 := ∞. Then there exist a finite covering{I(ν)} of I1 and solutions w(ν)l of (D) such that the endpoints of all Ij belongto the set of endpoints of all I(ν) and such that

w(ν)l ∼ ŷl on I

(ν),(4)

e−ql(w

(ν)l − w

(ν−1)l

)∈ A≤−kj+1

(I(ν−1) ∩ I(ν)

)if I(ν−1) ∪ I(ν) ⊂ Ij .(5)

Thus e−ql{w

(ν)l

}I(ν)⊂Ij

defines an element

e−qlwlj ∈ A/A≤−kj+1(Ij) with e−qlwlj (mod A≤−kj ) = e−qlwl,j−1|Ij .(6)

Moreover, if j ∈ {1, . . . , r} then wlj is uniquely determined by ŷl and themulti-interval (I1, . . . , Ij). In particular, wlr is an ordinary function whichis a uniquely determined solution of (D) on Ir.

Proof. It is sufficient to give the proof for the case that the intervals Ijhave Stokes directions as endpoints since otherwise we can enlarge Ij tosuch intervals with the other assumptions remaining satisfied. Let Ij =(aj , bj), j = 1, . . . , r and I(ν) := (θν−1, θν+1), ν ∈ Z. We use Lemma 1.3


with ν > ν0 where θν0 < a1. Let the linear operator M(ν)j on the solution

space of (D) be defined by

M(ν)j y

(ν)m = y

(ν−1)m +

∑θhm=θν ,κhm≤kj

αh(m, ν)y(ν−1)h ,m = 1, . . . , n,(7)

except if

θν−1 = ap ∈ Ij (so p > j) and κml = kp,m ≺ l on I(ν)(8)

in which case we replace the righthand side of (7) by 0. Then (2) impliesthat

e−ql(1−M (ν)j

)y(ν)m ∈ A≤−kj+1

(I(ν−1) ∩ I(ν)

)if m = l or m ≺ l on I(ν).

(9)

Define w(ν)l = y(ν)l ,∀ν ∈ Z with θν+1 ≥ b1 and and from these by downward

recursion in ν with a1 < θν−1 < θν < b1:

w(ν−1)l = M

(ν)j w

(ν)l if j := max{t : [θν−1, θν ] ⊂ It}.(10)

We will show that there exist constants cm(ν) such that if θν > a1:

w(ν)l − y

(ν)l =

∑m∈J(ν)

cm(ν)y(ν)m(11)

if

J(ν) := {m ∈ {1, . . . , n} : m 6= l, κml = kp, θν ∈ Ip,m ≺ l on (ap, θν+1)}.(12)

If m ∈ J(ν) then ap ≤ θν−1 and m ≺ l on I(ν). Hence (4) follows from (11).Since w(ν)l − w

(ν−1)l =

(1−M (ν)j

)w

(ν)l we obtain (5) from (11), (12) and

(9).

Proof of (11). If θν+1 ≥ b1 then w(ν)l = y(ν)l and (11) follows. Next suppose

(11) holds for some ν with [θν−1, θν ] ⊂ I1. From (10) and (7) it follows that

w(ν−1)l − y

(ν−1)l =

∑θλl=θν ,κλl≤kj

αλ(l, ν)y(ν−1)λ +

∑m∈J(ν)

cm(ν)M(ν)j y

(ν)m ,(13)

where

j := max{t : [θν−1, θν ] ⊂ It},(14)

κλl = ks, s ≤ j, κml = kp, θν ∈ Ip,m ≺ l on (ap, θν+1).(15)

We only have to show that the righthand side of (13) is a linear combinationof y(ν−1)h with h ∈ J(ν − 1). As [θν−1, θν ] ⊂ Ij ⊂ Is and {θν − π/ks, θν}


is a (λ ≺ l)-Stokes pair which by assumption is not included in Is we haveθν − π/ks ≤ as and therefore λ ≺ l on (as, θν) and λ ∈ J(ν − 1).

Remains to consider the termsM (ν)j y(ν)m in (13), so withm ∈ J(ν). If j < p

then since θν ∈ Ip, we have θν−1 6∈ Ip (cf. (14)). Therefore θν−1 = ap, m ≺ lon I(ν) (cf. (15)) and M (ν)j y

(ν)m = 0 (cf. (8)). Finally suppose p ≤ j. Then

θν−1 ∈ Ij ⊂ Ip and m ∈ J(ν − 1). We have to show that for h in the sumin (7), so if θhm = θν , κhm = ki, i ≤ j, we have h ∈ J(ν − 1). Now h ≺ mon (θν − π/ki, θν). From this and (15) we get h ≺ m ≺ l on (a, θν), wherea := max{θν − π/ki, ap} < θν . Since qhl = qhm + qml we have κhl = kµwhere µ = max{i, p}. Then µ ≤ j, ap ≤ aµ ≤ aj < θν < bj ≤ bµ. Hence thecondition on Stokes pairs implies aµ ≥ θν−π/kµ ≥ θν−π/ki and so aµ ≥ a.Therefore h ≺ l on (aµ, θν), i.e. h ∈ J(ν − 1). Thus (11) holds in general.

The uniqueness statement follows from the next Proposition 1.5. �

Proposition 1.5. Let j ∈ {1, . . . , r}, k := kj ,m := kj+1. Let I be an openinterval of length > πk . Suppose there exist a finite covering {I

(ν)}Nν=1 of Iand solutions u(ν)l and v

(ν)l of (D), ν = 1, . . . , N , l = 1, . . . , n, satisfying

the folowing conditions

1) u(ν)l (z) ∼ ŷl(z) on I(ν),

2) e−ql(u

(ν)l − u

(ν−1)l

)∈ A≤−m

(I(ν) ∩ I(ν−1)

)if ν > 1,

3) v(ν)l satisfies the same conditions as u(ν)l and e

−ql(u

(ν)l − v

(ν)l

)∈

A≤−k(I(ν)).

Then e−ql(u

(ν)l − v

(ν)l

)∈ A≤−m(I(ν)), ν = 1, . . . , N .

Proof. There exist constants ch,ν such that

v(ν)l − u

(ν)l =

∑h≺l on I(ν),κhl=k

ch,νu(ν)h (mod e

qlA≤−m(I(ν))).(16)

Hence if ν ≥ 2 then on I(ν) ∩ I(ν−1) we have

v(ν−1)l = v

(ν)l (mod e

qlA≤−m(I(ν) ∩ I(ν−1)))

= u(ν)l +∑

h≺l on I(ν),κhl=kch,νu

(ν)h (mod e

qlA≤−m(I(ν) ∩ I(ν−1))).

In the last sum we have u(ν)h = u(ν−1)h (mod e

qhA≤−m(I(ν) ∩ I(ν−1))), h ≺ lon I(ν) and κhl = k. So using (16) with ν replaced by ν − 1 we see thatch,ν = ch,ν−1 and ch,ν 6= 0 only if h ≺ l on I(ν) ∪ I(ν−1). By varying ν we seethat ch,ν 6= 0 only if h ≺ l on I which is impossible since κhl = k, |I| > π/k.The assertions now follow from (16) with ch,ν = 0. �


Remark 1.6. e−qlwlr is called the multi-sum of e−ql ŷl associated withthe multi-interval (I1, . . . , Ir). A generalization of Proposition 1.5 is Wat-son’s lemma (cf. Proposition 2.3). Note that any interval Ij in Theorem1.4 contains a Stokes direction of level kj . The construction in the proof ofTheorem 1.4 only uses coverings by intervals I(ν) = (θν−1, θν+1) and inter-vals Ij bounded by Stokes directions. It could also be given with upwardrecursion by choosing θν0 > b1 in Lemma 1.3. If y

(ν)l is a normalized solu-

tion in the sense of Jurkat (cf. Lemma 1.3) with θν+1 − θν−1 > π/kr thenthe proof of Theorem 1.4 implies that e−qly(ν)l is the multisum of e

−ql ŷlon nested multi-intervals (I1, . . . , Ir) where |Ij | > π/kj , Ij does not containany (h ≺ l)-Stokes pair of level κhl = kj and either Ij = (aj , θν+1), j =1, . . . , r, ar = θν−1 or Ij = (θν−1, bj), j = 1, . . . , r, br = θν+1.

Next we consider the Stokes phenomenon for multi-sums of formal solu-tions of (D). As we have seen above it is sufficient to consider multisums onintervals bounded by Stokes directions.

Proposition 1.7. Let l ∈ {1, . . . , r} and let (I1, . . . , Ir) and (Ĩ1, . . . , Ĩr) benested multi-intervals such that if j ∈ {1, . . . , r} then Ij and Ĩj have length> π/kj and they do not contain any Stokes pair of level kj. Moreover,assume that there exist i ∈ {1, . . . , r} with the property Ij = Ĩj if j 6= i, andIi ∩ Ĩi = (θσ, θµ) such that θσ ∈ Ii and {θσ, θµ} is a (s ≺ l)-Stokes pair oflevel κsl = ki for some s ∈ {1, . . . , r}. Let Λ := {h : θhl = θµ, κhl = ki}and let ulj and vlj , j = 1, . . . , r satisfy the properties of Theorem 1.4 withrespect to (I1, . . . , Ir) and (Ĩ1, . . . , Ĩr) respectively.

Then there exist constants γh, h ∈ Λ such that

vlj − ulj =∑h∈Λ

γhuhj if j ≥ i.(17)

Proof. We may assume that u(ν)l and v(ν)l are constructed as in the proof

of Theorem 1.4. Let Ii = (θα, θµ) and Ĩi = (θσ, θβ). Then α < σ <µ < β, (θα, θβ) ⊂ Ii−1 and if i < r also Ii+1 ⊂ (θσ, θµ) and σ < µ − 1.Let j := max{s : θν−1, θν ∈ Is}, j̃ := max{s : θν−1, θν ∈ Ĩs}. If ν ≥ βthen j < i, j̃ < i and therefore j = j̃. From (10) it now follows thatv

(ν)l = u

(ν)l if ν ≥ β − 1.

Let ν := min{λ ≥ µ : v(λ)l = u

(λ)l

}. So µ ≤ ν ≤ β − 1, θν ∈ Ĩi \ Ii. Then

j = i− 1, and except in case ν = µ = σ + 1 moreover j̃ = i. From (10) and


(11) it follows that except in case ν = µ = σ + 1:

v(ν−1)l − u

(ν−1)l =

(M

(ν)i −M

(ν)i−1

)u

(ν)l

=(M

(ν)i −M

(ν)i−1

)y(ν)l + ∑m∈Jg(ν)

cm(ν)y(ν)m

with g = max{t : θν , θν+1 ∈ It}. Then g < i as θν 6∈ Ii. From (7) we get(

M(ν)i −M

(ν)i−1

)y(ν)m =

∑θhm=θν ,κhm=ki

αh(m, ν)y(ν−1)h .

If m ∈ Jg(ν), κml = kp, κhm = ki, θhm = θν then p ≤ g < i and m ≺ l on(ap, θν+1). As qhl = qhm + qml we have κhl = ki, θhl = θν . Since θν ∈ Ĩi wehave θν − π/ki 6∈ Ĩi = (θµ − π/ki, θβ) and as ν ≥ µ we get ν = µ. Hencev

(ν)l = u

(ν)l if ν ≥ µ. So if µ > σ + 1 we get:

v(µ−1)l − u

(µ−1)l =

∑h∈Λ

δhy(µ−1)h .(18)

Next we consider (11) for u(µ−1)h :

u(µ−1)h = y

(µ−1)h +

∑m∈Jhi(µ−1)

cmh(µ− 1)y(µ−1)m

where Jhi(µ− 1) := {m : m ≺ h on (as, θµ), κmh = ks, s ≤ i}. If h ∈ Λ,m ∈Jhi(µ− 1) then m ∈ Λ by the same argument as above. Since the solutionsu

(µ−1)h , h ∈ Λ, are linearly independent and the same holds for y

(µ−1)h , we

obtain

v(ν)l − u

(ν)l =

∑h∈Λ

γhu(ν)h (mod e

qlA≤−kj+1(I(ν)))(19)

for ν = µ− 1, j = i. From the construction of u(ν)l and v(ν)l for ν < µ− 1 by

means of (10) we now deduce (19) for I(ν) ⊂ Ij ∩ Ĩj , j ≥ i if µ > σ + 1 and(17) follows.

In case ν = µ = σ + 1 we have i = r, j = r − 1 and thus v(µ)l − u(µ−1)l =(

1−M (µ)r−1)u

(µ)l =

(1−M (µ)r−1

)(y

(µ)l +

∑m∈Jr−1(µ) cm(µ)y

(µ)m

). From this

and (7) we deduce in the same way as above v(µ)l − u(µ−1)l =

∑h∈Λ γhu

(µ−1)h

which implies (17) since vlr = v(µ)l and ulr = u

(µ−1)l . �

Remark 1.8. Proposition 1.7 shows that the only contributions to theStokes phenomenon come from solutions which have as exponential fac-tor exp qh such that {θσ, θµ} is a (h ≺ l)-Stokes pair of level κhl = ki. Soexp qhl has maximal descent near 0 of order ki in the direction θµ−π/(2ki),


a so-called singular or anti-Stokes direction of level ki (cf. [5], [17]). Forcalculations of the Stokes multipliers see [5], [6] and Example 5.10.

The Stokes phenomenon of level ki occurs in general if in Proposition 1.7the condition on Ii ∩ Ĩi is replaced by: Ii ∪ Ĩi contains exactly one Stokespair {θσ, θµ} of level ki and this is a (s ≺ l)-Stokes pair for some s, θσ ∈ Ii.Without affecting the multisums we may replace Ii by (θα, θµ) and Ĩi by(θσ, θβ) with suitable α, β such that α < σ, µ < β, in view of Proposition1.5. Now again Proposition 1.7 is applicable. The case where there are morethan one i with Ii 6= Ĩi can be reduced to the case with a single such i.

2. Some basic properties of quasi-functions and multisums.

In this section we introduce some notations that will be used throughoutthe paper. Unfortunately, there is no uniformity of notation in the existingliterature on Gevrey series and functions and the like. Furthermore, werecall a number of known properties of quasi-functions and multi-sums andprove some new results of a rather technical nature (cf. [1], [21], [20], [24]).

Definition 2.1. By A(0) we denote the sheaf on R of holomorphic functionsadmitting an asymptotic expansion ∈ C[[z]] as z → 0 in some sector withvertex at O and by A(k) the subsheaf of functions that are Gevrey of order1k . If f ∈ A

(0) its asymptotic expansion will be denoted by f̂ or J(f). ByA 0 and let I be an interval of R, open, half-open or closed. Ak-precise quasi-function f on I is a section on I of the quotient-sheaf Qk :=A/A≤−k, i.e. f ∈ Qk(I) = A/A≤−k(I). A k-precise quasi-function f canbe represented by a collection of functions {fj}j∈J defined and analytic onan open covering {Ij}j∈J of I. Two such collections {fj}j∈J defined on acovering {Ij}j∈J and {f ′j}j∈J ′ defined on a covering {I ′j}j∈J ′ are consideredequivalent if fj |Ij∩I′h − f

′h|Ij∩I′h ∈ A

≤−k(Ij ∩ I ′h) for all j ∈ J and h ∈ J ′such that Ij ∩ I ′h 6= ∅. We say that f is bounded, has exponential growthof order l > 0, etc. if the functions fj , j ∈ J have this property. If, for allj ∈ J , fj ∈ A(k)(Ij) then the asymptotic expansion of fj is independent ofj. This asymptotic expansion will again be denoted by f̂ or J(f).

k-precise quasi-functions with at most subexponential growth of order k,i.e. sections of the sheaf Q


holds if we replace the condition that f vanishes in θ by: On a neighborhoodI0 of θ f is represented by a function which has exponential decrease of orderk in the direction θ.

Proposition 2.3. Let f ∈ A≤−k/A 0 and I is an intervalwith length > π/k. Then f = 0.

This proposition extends a result for ordinary functions (i.e. f ∈A≤−k(I)), due to Watson. In the present form it can be found in Malgrange-Ramis [21] (lemme de Watson relatif) and Kostov [15] (cf. also Il’yashenko[11]). We give another proof using the following proposition:

Proposition 2.4. Letf be holomorphic and of exponential type in the neigh-borhood of ∞ in a sector S with vertex 0. Let the extended indicator func-tion Hf : S → [−∞,∞) be defined by Hf (z) := lim supr→∞ r−1 log |f(rz)| ifz ∈ S. Then Hf is homogeneous of order 1 and convex on S. In particular:If the segment [z1, z2] ⊂ S and Hf (zj) < Aj where Aj ∈ R, j = 1, 2, then

f(r(λz1 + (1− λ)z2)) = O(1) exp{r(λA1 + (1− λ)A2)}

as r →∞, uniformly in λ ∈ [0, 1].

This result can be deduced from the Phragmén-Lindelöf theorem as hasbeen shown by E.G.F. Thomas (private communication). It also followsfrom the subharmonicity and homogeneity of Hf (cf. [10, p. 313]). Therestriction of Hf to the unit circle is the usual indicator function.

Next consider the case that f ∈ A≤1/A


Proof of Proposition 2.2. We may assume that k = 1. Suppose that fis represented by a function f0 on I0. Let S0 be the sector with z ∈ S0iff arg z ∈ I0. Then H̃f0 is negative in the direction −θ and since it isnonpositive on S0 (by the assumption) and convex it has to be negative onS0. Hence f |I0 ∈ A≤−1(I0). Repeating this reasoning we obtain a chain ofsubintervals of I on which the representatives of f are sections of A≤−1. Sof = 0. �

Lemma 2.5 (“Borel-Ritt”, cf. [21]). The mapping J : A(0)/A 1, let 0 < k1 < · · · < kr and k =(k1, . . . , kr). For j = 1, . . . , r, let Ij be an open, half-open or closed in-terval of R. If Ir ⊂ · · · ⊂ I1 we call I := (I1, . . . , Ir) a nested multi-interval.Let fj be a kj-precise quasifunction on Ij : fj ∈ Qkj (Ij), j ∈ {1, . . . , r}. If,for all j ∈ {2, . . . , r},

fj (mod A≤−kj−1) = fj−1|Ij

we call f := (f1, . . . , fr) a k-precise (or (k1, . . . , kr)-precise) quasifunctionon the nested multi–interval I. The set of all k-precise quasifunctions onI will be denoted by Q(k1,... ,kr)(I1, . . . , Ir) or Qk(I). By Q

πkj , or a half-open or closed interval of length ≥

πkj

, such that

Ir ⊂ . . . ⊂ I1. If there exists f = (f1, . . . , fr) ∈ Q(k1)(k2,... ,kr+1)(I1, . . . , Ir)such that f1 (mod A≤−k1) = f0|I1 , the formal power series f̂ is (k1, . . . , kr)-summable on (I1, . . . , Ir) with (k1, . . . , kr)-sum (or multi-sum) (f1, . . . , fr).The set of all (k1, . . . , kr)-sums on (I1, . . . , Ir) will be denoted by Σk(I),where k = (k1, . . . , kr) and I = (I1, . . . , Ir).

A formal series f̂ of the form

f̂(z) =∑α∈J

p∑l=0

f̂αl(z)zα(log z)l

where p ∈ N, J is a finite subset of C and f̂αl ∈ C[[z]] for all l and α, will becalled k-summable on the multi-interval I, with k-sum (f1, . . . , fr), if f̂αl isk-summable on the multi-interval I for each α ∈ J and each l ∈ {1, . . . , p},


with k-sum (fαl1 , . . . , fαlr ), such that

fj(z) =∑α∈J

p∑l=0

fαlj (z)zα(log z)l, j = 1, . . . , r.

Let dj ∈ R, j = 1, . . . , r. d := (d1, . . . , dr) is called an admissible multi-direction with respect to k if, for j = 2, . . . , r,

|dj − dj−1| ≤π

2kj−1− π

2kj.

f̂ is k-summable in the multi-direction d if f̂ is k-summable on the multi-interval (I1, . . . , Ir), where Ij = [dj − π2kj , dj +

π2kj

]. By C{z}k,d we denotethe set of all formal power series that are k-summable in the multi-directiond.

An admissible multi-direction d ∈ Rr is called a singular direction off̂ if f̂ 6∈ C{z}k,d.

Let f̂ ∈ C[[z]]k1 and f0 = J−1(f̂) and suppose there exist s ∈ N and,if s > 1, positive numbers k1 < · · · < ks, an admissible multi-direction(d1, ...,ds) with respect to (k1, ...,ks) and (f1, ...,fs−1)∈Q(k2,...,ks)(I1, ...,Is−1),where Ij = [dj− π2kj , dj+

π2kj

], j = 1, . . . , s−1, such that f1 (mod A≤−k1) =f0|I1 . ds is called a singular direction of level ks of f̂ if f̂ 6∈ C{z}k′,d′for any k′ = (k′1, . . . , k

′r) with r ≥ s and any admissible multi-direction

d′ = (d′1, . . . , d′r) such that k

′j = kj and d

′j = dj for j = 1, . . . , s.

Let f̂ ∈ C[[z]], k > 0 and θ ∈ R. The pair {θ, θ + πk } will be called aStokes pair of level k for f̂ if the number θ+ π2k is a singular direction oflevel k for f̂ . By C{z}k we denote the set of all f̂ ∈ C[[z]] with the propertythat f̂ ∈ C{z}k,d for every admissible multi-direction d = (d1, . . . , dr) ∈ Rr,except for a finite number of values of each single direction d1, . . . , dr.

If f̂ is (k1, . . . , kr)-summable with (k1, . . . , kr)-sum f = (f1, . . . , fr), wesay that f̂ (mod C{z}) is (k1, . . . , kr)-summable with (k1, . . . , kr)-sum f(mod C{z}).

The above definition of k-summability is easily seen to be equivalent tothe following.

Definition 2.8. Let k := (k1, . . . , kr) with 0 < k1 < . . . < kr and I :=(I1, . . . , Ir) a nested multi-interval and let I0 = R. The formal power seriesf̂ ∈ C[[z]]k1 is k-summable on I if f0 = J−1(f̂) has a representative {f0,ν}defined on a covering {I0,ν} of R, with the property that f0,ν − f0,ν+1 ∈A≤−kj+1(I0,ν ∩ I0,ν+1) for all ν such that I0,ν ∪ I0,ν+1 ⊂ Ij .

Remark 2.9. Obviously, if (f1, . . . , fr) is a (k1, . . . , kr)-sum on (I1, . . . , Ir),then fj ∈ Q(k1)kj+1(Ij) for all j ∈ {1, . . . , r}. Suppose that (f

′1, . . . , f

′r−1, fr)


is another (k1, . . . , kr)-sum on (I1, . . . , Ir). Then we have, for all j ∈{0, . . . , r},

fj |Ir = f ′j |Ir = fr (mod A≤−kj+1).

By Proposition 2.2, this implies that fj = f ′j for all j ∈ {0, . . . , r}. Thusthe quasi-functions f0, . . . , fr−1 are uniquely determined by fr. Therefore,we will sometimes simply denote the multi-sum (f1, . . . , fr) by fr.More generally, suppose that I = (I1, . . . , Ir) is a nested multi-interval,k = (k1, . . . , kr) with 0 < k1 < · · · < kr and (f1, . . . , fr) ∈ Q πkj for j = 1, . . . , r. Aformal power series f̂ is (k1, . . . , kr)-summable on (I1, . . . , Ir) if and onlyif, for j = 1, . . . , r, Ij does not contain any Stokes pair of level kj for f̂ .

Lemma 2.11. Let r ∈ N, let 0 < k1 < · · · < kr+1 = ∞. Let Ij, j =0, . . . , r, be open intervals of R with the property that Ir ⊂ Ir−1 ⊂ . . . ⊂ I0.For j = 0, . . . , r, let {Iij}i∈J be an open covering of Ij such that Iir ⊂Iir−1 ⊂ . . . ⊂ Ii0 for all i ∈ J , where J is an index set. Moreover, assumethat Ii11 ∩ I

i21 6= ∅ for some i1, i2 ∈ J implies Ii1r ∩ Ii2r 6= ∅. Suppose there

exist f i = (f i0, . . . , fir) ∈ Q


Hence it follows that

(f i2j−1 − fi1j−1)|Ii1j ∩Ii2j = 0 (mod A

≤−kj )

for all i1, i2 ∈ J such that Ii1r ∩ Ii2r 6= ∅. In view of Proposition 2.2 thisimplies that

(f i2j−1 − fi1j−1)|Ii1j−1∩Ii2j−1 = 0 (mod A

≤−kj ).

Consequently, there exists a unique fj−1 ∈ Q


for all i ∈ Z. Let Ij := ∪i∈ZIij. Then there exists a unique f = (f1, . . . , fr) ∈Q


In various applications (e.g. in the theory of linear differential equations,cf. §1) one deals with multi-summable formal power series having a finitenumber of singular directions of each level in any finite interval. In thatcase, each quasi-function in a multi-sum of the formal series possesses a finiterepresentative, consisting of multi-sums of the same formal power series.

Lemma 2.15. Let r ∈ N, 0 < k1 < k2 < · · · < kr+1 = ∞. Let f̂ ∈ C[[z]]k1and suppose that f̂ possesses a (k1, . . . , kr)-sum (f1, . . . , fr) on the multi-interval (I1, . . . , Ir) where I1, . . . , Ir are open. Assume that the intervalI1 only contains a finite number of Stokes pairs of level kj for each j ∈{2, . . . , r}. Then the quasi-function fj has a finite representative of theform {f (ν) : ν ∈ {Mj , . . . , Nj}}, where M1 ≤ M2 ≤ · · · ≤ Mr ≤ Nr ≤ · · · ≤N2 ≤ N1 and f (ν) is a (k1, . . . , kr)-sum of f̂ for each ν ∈ {M1, . . . , N1}.

Proof. There exists a covering of I1 by intervals I(ν) = (a(ν), b(ν)), ν =M1, . . . , N1, where M1 and N1 are integers with M1 < N1, such that |I(ν)| >π/kr, I(ν) does not contain any Stokes pair of level kj for j = 2, . . . , r,a(ν) < a(ν+1) < b(ν) < b(ν+1), and for all j ∈ {1, . . . , r} there exist integersMj and Nj such that Ij = (a(Mj), b(Nj)). Hence M1 ≤ M2 ≤ · · · ≤ Mr ≤Nr ≤ · · · ≤ N2 ≤ N1. We may assume that Mr = Nr.

Let ν ∈ {M1, . . . , N1} and let j be the largest integer such that I(ν) ⊂ Ij .Then we define I(ν)h := Ih if h ∈ {1, . . . , j} and I

(ν)r := I(ν). If j < r − 1,

there exist open intervals I(ν)h of length > π/kh, h ∈ {j+1, . . . , r− 1}, suchthat I(ν)h does not contain any Stokes pair of level kh and (I

(ν)1 , . . . , I

νr ) is

a nested multi-interval. Then, by Corollary 2.10, f̂ is multi-summable onthis multi-interval with multisum (f (ν)1 , . . . , f

(ν)r ). Note that Iν1 = I1 and

fν1 = f1. Let f(ν) := f (ν)r .

If Mj ≤ ν ≤ Nj and h ∈ {1, . . . , j}, then I(ν)h = Ih and from Proposition2.3 it then follows that f (ν)h = fh. Furthermore, fj |I(ν) = f

(ν)j |I(ν) = f (ν)

(mod A≤−kj+1) which implies that fj can be represented by {f (ν), ν =Mj , . . . , Nj}. �

3. Multisummability and Cauchy-Heine transforms.

Suppose we are given two open intervals I1 and I2 such that I1∩I2 6= ∅, andtwo analytic functions f1 ∈ A(0)(I1) and f2 ∈ A(0)(I2), admitting the sameasymptotic expansion f̂ as z → 0. Obviously, the difference f1 − f2 belongsto A


Now define f1 and f2 by

f1(z) := CH−ρ (f) :=∫ ρ

0

f(ζ)2πi(ζ − z)

dζ − 2π < arg z − arg ρ < 0

and

f2(z) := CH+ρ (f) :=∫ ρ

0

f(ζ)2πi(ζ − z)

dζ 0 < arg z − arg ρ < 2π.

CH−ρ (f) and CH+ρ (f) are so-called Cauchy-Heine transforms of f . Note that

CH−ρ (f)(z) = CH+ρ (f)(ze

2πi). By deformation of the path of integration,the functions f1 and f2 can be continued analytically to elements of A((α−2π, β)) and A((α, β + 2π)), respectively. With the aid of the expansion∑∞

n=0 ζ−n−1zn for (ζ − z)−1, it can be shown that f1 and f2 admit an

asymptotic expansion ĈHρ(f) =∑∞

n=0 anzn as z → 0, where

an =1

2πi

∫ ρ0f(ζ)ζ−n−1dζ.

It is easily seen that f1−f2 = f . A different choice of ρ leads to a pair of func-tions f̃1 and f̃2 such that f̃1−f1 = f̃2−f2 and this difference is regular at O.The construction described above is extremely useful in many applications.For example, suppose we are given a formal power series f̂ ∈ C[[z]]. f̂ definesan element f0 ∈ A(0)/A


Definition 3.1. Let f ∈ A 0 and let a, b ∈ R : a < b. Iff ∈ A≤−k((a, b)) then

CH+(f) ∈ A(k)((a, b+ 2π))/C{z}

andCH−(f) ∈ A(k)((a− 2π, b))/C{z}.

For a proof of this lemma cf. [19, pp. 114, 115].

Corollary 3.3. Let k > 0, let a, b be real numbers such that a < b, andf ∈ A≤−k((a, b)). Then F := ĈH(f) ∈ C{z}k,d/C{z} for all d ∈ (a −2π + π2k , b −

π2k ) mod 2π. CH

+(f) is the k-sum of F in the directionsd ∈ (a + π2k , b + 2π −

π2k ) and CH

−(f) is the k-sum of F in the directionsd ∈ (a − 2π + π2k , b −

π2k ). In particular, if a = −

π2k and a < b + 2π −

πk ,

then CH+(f) is a k-sum in the directions d ∈ (0, �) for some � > 0, andif a < − π2k then it is a k-sum in the direction 0. Similarly, if b =

π2k and

a < b + 2π − πk , then CH−(f) is a k-sum in the directions d ∈ (−�, 0) for

some � > 0, and if b > π2k it is a k-sum in the direction 0.

Lemma 3.4. Let I0 = (a0, b0) ⊂ I = (a, b), 0 < k < l, f0 ∈ A≤−k(I0), f ∈A≤−k/A≤−l(I), such that f |I0 = f0 (mod A≤−l). Then there exists F ∈A(k)/A≤−l(a, b+ 2π) with the property that

F |(a0,b0+2π) (mod C{z}) = CH+(f0) (mod A≤−l).(20)

If there exist b+ ∈ (a, b), b+ < b0 + 2π and f+ ∈ A≤−k(b+, b) such thatf |(b+,b) = f+ (mod A≤−l) then there exists g+ ∈ A(l)(b+, b0 + 2π) with theproperty that

CH+(f0)|(b+,b0+2π) = CH+(f+)|(b+,b0+2π) + g

+ (mod C{z}).(21)

Similarly, if there exist a− ∈ (a, b), a− > a0 − 2π and f− ∈ A≤−k(a, a−)such that f |(a,a−) = f− (mod A≤−l) then there exists g− ∈ A(l)(a0, a−+2π)with the property that

CH+(f0)|(a0,a−+2π) = CH+(f−)|(a0,a−+2π) + g

− (mod C{z}).(22)


Proof. Let f be represented by {fj}j∈J on a covering {Ij}j∈J of I where Jis an interval of Z containing 0, and Ij = (aj , bj) with aj < aj+1 < bj < bj+1.Thus fj ∈ A≤−k(Ij), fj − fj+1 ∈ A≤−l(Ij ∩ Ij+1).

According to Lemma 3.2, CH+(fj) possesses a representative φj ∈A(k)(aj , bj + 2π). Let ψj := φj − φj+1. Then, again by Lemma 3.2,ψj ∈ A(l)(aj+1, bj + 2π) with an asymptotic expansion ψ̂j ∈ C[[z]]l.

Let N ≥ 0, N ∈ J and define ρN :=∑N−1

j=0 J−1ψ̂j (cf. Lemma 2.5). Thus

ρN ∈ A(l)/A≤−l(R). If aN < b0 + 2π we define χN =∑N−1

j=0 ψj , so that

χN ∈ A(l)(aN , b0 + 2π) and φ0|(aN ,b0+2π) = φN |(aN ,b0+2π) + χN(23)

and

ρN |(aN ,b0+2π) = χN (mod A≤−l).

Define

FN = (φN + ρN |(aN ,bN+2π)) (mod A≤−l).

Then FN ∈ A(k)/A≤−l(aN , bN+2π) and if aN < b0+2π then FN |(aN ,b0+2π) =φ0|(aN ,b0+2π) (mod A

≤−l).Assume that also N + 1 ∈ J . Let ρN be represented by ρ̃N ∈ A(l)(Ĩ)

on some interval Ĩ ⊂ (aN+1, bN + 2π). Then FN may be represented on Ĩby φN + ρ̃N = φN+1 + ρ̃N+1 where ρ̃N+1 := ψN + ρ̃N , and so ρ̃N+1 is arepresentative of ρ̂N+1 on Ĩ. Hence FN and FN+1 have a representative incommon and so by quasi-analyticity FN |(aN+1,bN+2π) = FN+1|(aN+1,bN+2π).The sheaf property now implies that the functions FN with N ≥ 0 determinean element F ∈ A(k)/(A≤−l(a0, b+ 2π)) such that (20) holds.

In a similar way we may extend φ0 to quasi-functions on intervals whichstart to the left of a0 by choosing N ≤ 0, N ∈ J and

χN = −−1∑j=N

ψj if a0 < bN + 2π and ρN :=−1∑j=N

ψ̂j .

The corresponding FN with N < 0 can be pasted with those with N ≥ 0 toobtain F ∈ A(k)/A≤−l(a, b+ 2π).

If the condition on f+ is fulfilled then we may choose N such that aN =b+, bN = b, fN = f+, φN is a representative of CH+(f+) and thus we obtain(21) from (23) with g+ := χN . Similarly if the condition on f− is fulfilled.

�

From this lemma we will derive:

Corollary 3.5. Let r ∈ N, 0 < k1 < · · · < kr < kr+1 = ∞, I := (I1, . . . , Ir)be a nested multi-interval with Ij =(aj , bj), j = 1, . . . , r and f := (f1, . . . , fr)∈ Q≤−k1(k2,... ,kr+1)(I). Let Ĩj := (aj , bj + 2π) and Ĩ := (Ĩ1, . . . , Ĩr).


Then there exists F = (F1, . . . , Fr) ∈ Q(k1)(k2,... ,kr+1)(Ĩ) such thatFr (mod C{z}) = CH+(fr). If |Ĩj | > π/kj for j = 1, . . . , r – so in par-ticular if k1 ≥ 1/2 – then F is the (k1, . . . , kr)-sum of F̂0 on Ĩ where F̂0(mod C{z}) = ĈH(fr).

If, in addition, for all j ∈ {1, . . . , r − 1} there exist b+j ∈ (aj , bj) andf+j ∈ A≤−k1(b

+j , bj) such that b

+j < br + 2π and fj is represented by f

+j

on (b+j , bj) then for j = 1, . . . , r there exists g+j ∈ A(kj)(b

+j−1, bj + 2π) with

b+0 := b+1 such that

CH+(fr) =r∑j=1

g+j (mod C{z})(24)

where in both sides we take the restriction to the intersection of the intervalson which the terms are well-defined.

Similarly, if for all j ∈ {1, . . . , r − 1} there exists a−j ∈ (aj , bj) suchthat ar < a−j + 2π and fj is represented by a function on (aj , a

−j ) then for

j = 1, . . . , r there exists g−j ∈ A(kj)(aj , a−j−1 + 2π] with a

−0 := a

−1 such that

CH+(fr) =r∑j=1

g−j (mod C{z})(25)

where in both sides we take the restriction to the intersection of the intervalson which the terms are well-defined.

Proof. We may apply the previous lemma with I0 := Ir, f0 := fr, k :=k1, f := fj , l := kj+1, I := Ij . Hence there exist Fj ∈ A(k1)/A≤−kj+1(Ĩj)such that

Fj |Ĩr = φr (mod A≤−kj+1),

where φr is a representative of CH+(fr). Hence Fj−1|Ĩr=Fj (mod A≤−kj )|Ĩr .

With the aid of Proposition 2.2 we now deduce the first assertion.Next let the assumptions concerning (24) be satisfied. Let fj |(b+j ,bj) be

represented by f+j ∈ A≤−k1(b+j , bj), j = 1, . . . , r − 1. Let f+r := fr, b+r :=

ar. Then we may apply the preceding lemma with k := k1, l := kj , f0 :=f+j , I0 := (b

+j , bj), f := fj−1, I := Ij−1, f

+ := f+j−1, b+ := b+j−1 where j =

2, . . . , r. It follows that there exists g+j ∈ A(kj)(b+j−1, bj + 2π) such that

CH+(fj)+|(b+j−1,bj+2π)= g+j + CH

+(f+j−1)|(b+j−1,bj+2π) (mod C{z}), j = 2, . . . , r.

With g+1 := CH+(f+1 ) ∈ A(k1)(b

+1 , b1 + 2π) (cf. Lemma 3.2) we obtain (24)

by addition. The proof of (25) is similar. �


Remark 3.6. Similar results hold for CH− since CH−(f)(z) =CH+(f)(ze2πi). The corollary may be applied to the case that f = hgwhere g ∈ Q πkj ,- {Iir}Ni=1 is a covering of [0, 2π),- αij < β

i+1j < α

i+1j < β

ij < β

N+1j for all i ∈ {1, . . . , N} and j ∈

{1, . . . , r},- |Ii+11 ∩ Ii1| < 2π for all i ∈ {1, . . . , N}.

Let IN+1j = I1j + 2π. Suppose there exist g

i ∈ Q≤−k1(k2,... ,kr+1)(Ii+1 ∩ Ii), i ∈

{1, . . . , N}. Then there exists a unique F̂ ∈ C[[z]]k1, (k1, . . . , kr)-summableon Ii with sum f i for all i ∈ {1, . . . , N+1}, such that fN+1r (z) = f1r (ze−2πi)and f i+1r − f ir = gir for all i ∈ {1, . . . , N}. Moreover, F̂ (mod C{z}) =∑N

i=1 ĈH(gir).

Proof. For i ∈ {1, . . . , N + 1}, let f ir =∑N

h=1CHdih(ghr ), where dih = +

if h < i and dih = − if h ≥ i. From Corollary 3.5 we deduce that f iris a (k1, . . . , kr)-sum on (Ii1, . . . , I

ir). Moreover, f

i+1r − f ir = CH+(gir) −

CH−(gir) = gir and

fN+1r (z) =N∑h=1

CH+(ghr )(z) =N∑h=1

CH−(ghr )(ze−2πi) = f1r (ze

−2πi).

�

Theorem 3.8. Let 12 < k1 < · · · < kr, kr+1 = ∞ and f̂ ∈ C[[z]]. f̂ ∈C{z}(k1,... ,kr) if and only if there exist di ∈ [0, 2π), f ir ∈ A

((di− π2kr , di+1 +

π2kr

)), i = 1, . . . , N , where dN+1 := d1 + 2π, and gij ∈ Q

≤−k1kj+1

((di+1 −

π2kj, di+1 + π2kj

)), j = 1, . . . , r, i = 1, . . . , N , with the following properties:

f ir(z) ∼ f̂(z) as z → 0, arg z ∈ (di − π2kr , di+1 +π

2kr), i = 1, . . . , N ,

gij (mod A≤−kj )=gij−1|„di+1− π2kj−1 ,di+1+

π2kj−1

«, j=2, ..., r, i=1, ..., N ,


f i+1r − f ir = gir, for all i ∈ {1, . . . , N}, where fN+1r is defined byfN+1r (z) = f

1r (ze

−2πi).

4. The Stokes phenomenon in the direction 0.

In the two remaining sections we shall restrict ourselves to the study of(k1, . . . , kr)-sums on multi-intervals (I1, . . . , Ir), where Ij = [− π2kj ,

π2kj

), orIj = (− π2kj ,

π2kj

], or Ij = [− π2kj ,π

2kj]. Therefore, we introduce the following

conventions. If r ∈ N and 0 < k1 < · · · < kr are given, we write

I0j =[− π

2kj,π

2kj

], I+j =

(− π

2kj,π

2kj

]and I−j =

[− π

2kj,π

2kj

), j ∈ {1, . . . , r}.

We say that (f1, . . . , fr) is a (k1, . . . , kr)-sum in the multi-direction (d1, . . . ,dr) ∈ {+, 0,−}r if (f1, . . . , fr) is a (k1, . . . , kr)-sum on (I1, . . . , Ir), whereIj = I

djj .

Definition 4.1. By Σ± we denote the differential algebra of all multi-sumsin the direction ±, i.e. the set of f ∈ A0 (A0 denotes the stalk in O ofthe sheaf A) with the following property: There exist a positive integerr, positive numbers k1, . . . , kr such that k1 < · · · < kr < kr+1 = ∞ and(f1, . . . , fr) ∈ Σ(k1,... ,kr)(I

±1 , . . . , I

±r ) such that fr = f .

By Σ0 we denote the differential algebra of all multi-sums in the direction0.

By Σ


Proof. The first two statements follow immediately from Lemma 2.12 (cf.also [22, Lemma 10]). Let f̂ ∈ Σ̂. It is easily verified that, for j = 1, . . . , r,

f+j − f−j ∈ Q

≤−k1kj+1

(− π

2kj,π

2kj

)and

f+j − f−j (mod A

≤−kj ) = (f+j−1 − f−j−1)|„− π

2kj, π2kj

«

and thus σ+(f̂)−σ−(f̂) ∈ Σ


f̂ in the multi-direction θl±. Then we have

f lj = f±j +

j∑i=1

F ij if j ≤ l,

f lj = f±j +

l∑i=1

F ij if j > l.

Proof. We prove the statements of the Proposition in the case of the uppersigns. Suppose that f̂ is (k1, . . . , kr)-summable in every multi-directionθl+, with sum (f l1, . . . , f

lr). Let F

ij := f

ij − f

i−1j , j = i, . . . , r. We have

f ij = fi−1j ∈ A(k1)/A≤−kj+1(I

−j ) for all j < i, f

ii ∈ A(k1)/A≤−ki+1(I

−i )

and f i−1i ∈ A(k1)/A≤−ki+1(I+i ). For all j > i, both f

ij and f

i−1j belong to

A(k1)/A≤−kj+1(I+j ). From Lemma 2.14 we deduce that the quasi-functionsF ij have the desired properties. In order to prove the uniqueness of F

ii it

suffices to show that∑r

i=1 Fir = 0 implies F

ij = 0 for all i ∈ {1, . . . , r} and

all j ≥ i. Writing

F 1r = −r∑i=2

F ir

we see that

F 1r |(− π2kr , π2kr ) ∈ A≤−k2

((− π

2kr,π

2kr

)).

Hence it follows that F 11 |(− π2kr , π2kr ) = 0. By Proposition 2.2 this implies thatF 11 = 0. Using (26) and Proposition 2.3 one easily verifies that F

1j = 0 for

all j ≥ 1. By means of induction on i it can now be shown that F ij = 0 forall i ∈ {1, . . . , r} and all j ≤ i.

Now suppose that 2) holds. Let l ∈ {1, . . . , r}. Let f lj := f+j +

∑li=1 F

ij

if j ∈ {l + 1, . . . , r} and f lj := f−j for j ∈ {1, . . . , l}. Obviously, f lj ∈

A(k1)/A≤−kj+1(I+j ) for j > l and f lj ∈ A(k1)/A≤−kj+1(I−j ) for j ≤ l. Further-

more, it is easily seen that f lj (mod A≤−kj ) = f lj−1|I+j for all j > l + 1, andf lj (mod A≤−kj ) = f lj−1|I−j for all j ≤ l. Noting that f

lr = f

−r −

∑ri=l+1 F

ir

and F ii (mod A≤−ki) = 0 for all i we find that, for all j > l,

f lj |“− π2kr

, π2kr

” =(f−j −

j∑i=l+1

F ij

)∣∣∣∣“− π

2kr, π2kr

”.In particular,

f ll+1 (mod A≤−kl+1)|“− π2kr

, π2kr

” = f−l |“− π2kr

, π2kr

”


and, by Proposition 2.2, this implies that f ll+1 (mod A≤−kl+1) = f−l |I+l+1 =

f ll |I+l+1 . It remains to be proved that, for j ≤ l,

f−j = f+j +

j∑i=1

F ij .

This follows easily from (26), (27) and the fact that F ir (mod A≤−kj+1) = 0for all i > j. �

Corollary 4.5. Let f ∈ Σ l, for everyl ∈ {1, . . . , r}, iff f can be written in the form

f =r∑i=1

F ir

where the F ir are functions with the properties mentioned in Proposition 4.4.Moreover, if these conditions are satisfied, then the multi-sum glr of ĈH(f)in the multi-direction θl is given by

glr = CH±(f) +

l∑i=1

F ir .

Proof. According to Corollary 3.5 and Remark 3.6, ĈH(f) ∈ Σ̂ andσ±(ĈH(f)) = CH±(f). Hence

σ±(ĈH(f))− σ∓(ĈH(f)) = ±f (mod C{z}).The above statements now follow immediately from Proposition 4.4. �

5. Stokes automorphisms and alien derivations.

A multi-summable formal power series gives rise to a non-trivial Stokesphenomenon if and only if it is divergent. Stokes phenomena can be usedin the analytic classification of certain local objects, e.g. locally analyticdifferential equations. In many cases of interest, a detailed analysis of theStokes phenomenon associated with such an object will yield a complete setof analytic invariants of this object (i.e. quantities that are invariant underan analytic change of variable).

There is another, essentially equivalent way to obtain these invariants.Suppose, for example, that f̂ =

∑∞n=1 anz

n is a 1-summable power se-ries. Then its formal Borel transform with respect to the variable z−1:∑∞

n=1 antn−1

Γ(n) defines the germ at O of an analytic function. This functioncan be continued analytically in any direction which is not a singular direc-tion of f̂ . Similarly, with a (k1, . . . , kr)-summable power series f̂ , one can


associate r such analytic functions ϕ1, . . . , ϕr. ϕ1 is the formal Borel trans-form, with respect to the variable z−k1 , of f̂ and the functions ϕ2, . . . , ϕrare so-called accelerates of ϕ1. In many applications, the function ϕj can becontinued analytically along paths avoiding a certain discrete set of singularpoints whose arguments coincide with the singular directions of level kj off̂ . The singularities of the functions ϕ1, . . . , ϕr determine the type of Stokesphenomenon and, in an appropriate context, the analytic invariants associ-ated with f̂ . In order to describe the nature of these singularities, J. Ecalleintroduced the so-called alien derivations (cf. [8]). In this section we givean alternative definition derived directly from the Stokes phenomenon. Inthe sequel we restrict ourselves to the Stokes phenomenon in the direction0.

Example 5.1. Consider the locally analytic differential equation

z2dy

dz= f(z, y)

where f ∈ C{z, y} such that f(0, 0) = 0 and ∂f∂y (0, 0) = ω > 0. It possessesa unique formal solution ĥ ∈ C{z}1 with singular direction 0. There existformal power series ĥn ∈ C{z}1, n ∈ N, such that

σ+(ĥ)− σ−(ĥ) =∞∑n=1

e−nω/zσ−(ĥn)

where the series in the right-hand side converges if Re (1/z) is sufficientlysmall (cf. [7]). σ+ and σ− denote the multi-sums defined in the previoussection (cf. Definition 4.1).

Example 5.2 (Laplace integrals). Let �1, �2 and r be positive numbers,�j < 2π, j = 1, 2 and U := S(0, �1) ∪ S(2π − �2, 2π) ∪ S((0, 2π), r) (cf.Definition 1.1). Let φ be an analytic function on U . Then the Laplacetransform of φ will be defined as follows (cf. [7]). Let C be a contour in Ufrom∞ exp(η1i) to∞ exp{(2π−η2)i} where 0 < ηj < �j , j = 1, 2, η1+η2 < π.Here and in the following we apply a lemma of Ecalle which says that if afunction ψ is analytic on a union Γ of finitely many unbounded contoursin C which have asymptotes at ∞ then there exists an entire function Esuch that |ψ − E| is bounded on Γ (cf. [7]; cf. also a lemma of Sato in[14]). Choosing Γ = C and ψ = φ we see that

∫C exp(−t/z){φ(t)−E(t)}dt

exists on S(η1− π/2, π/2− η2) and it belongs to A


Next suppose that there exists ω > 0 such that φ possesses analyticcontinuations φ− and φ+ on U− := U ∪ (0, ω) ∪ S(−�2, 0) and U+ := U ∪(0, ωe2πi) ∪ S(2π, 2π + �1) respectively.

Let C− be a contour in U− from ∞ exp(−η2i) to ∞ exp{(2π−η2)i}. Thensimilarly as above we may define LC−φ− ∈ A


Definition 5.3. By T 0 we denote the set of formal series f of the form

f =∑ω∈C

e−ω/zfω,

where fω ∈ A


S1 is a differential subalgebra of A


Furthermore, we define

S∞ =∞⋂n=1

Sn, T ∞ =∞⋂n=1

T n,

S =

{f ∈ S∞ :

∞⋃n=1

{|ω| : ω ∈ Ωn(f)} is a discrete set

}and

T =

f ∈ T ∞ : ⋃ω0∈Ω0(f)

∞⋃n=1

{ω0 + |ω| : ω ∈ Ωn(fω0)} is discrete

.By means of induction on n, one easily proves the following lemma.

Lemma 5.7. Let n ∈ N. f ∈ T n if and only if fω ∈ Sn for all ω ∈ Ω0(f).Similarly, f ∈ T if and only if fω ∈ S for all ω ∈ Ω0(f).

Note that, for a given ω,

((S− id)nf)ω =∑

ω1,... ,ωn>0

Sωn · · ·Sω1(fω−(ω1+···+ωn))

and this sum equals 0 for all but a finite number of integers n. Therefore, theinfinite sum

∑∞n=0(−1)n(S−id)nf, so S−1f is well-defined for all f ∈ T and

its support is discrete. (Alternatively, we could define T to be the largestsubalgebra of T 1 that is invariant under S.) Thus S is a differential algebraautomorphism of T . We define the ‘directional alien derivation’ ∆ by

∆ := log S =∞∑n=1

(−1)n+1

n(S− id)n.

∆ is a derivation of T (cf. [7]). We have

(∆f)ω =∞∑n=1

(−1)n+1

n

∑ω′∈C

∑ω′′∈(0,∞)n:|ω′′|+ω′=ω

Sω′′(fω′).

∆ can be decomposed as follows: ∆ =∑

ω>0 ∆̇ω, where ∆̇ω is defined by

(∆̇ωf)ω′ =∞∑n=1

(−1)n+1

n

∑ω′′∈(0,∞)n:|ω′′|=ω

Sω′′(fω′−ω).(31)

∆̇ω is a derivation of T which transforms an element of T with support Ω intoan element with support Ω+ω. As S is a differential algebra automorphism,


∆ and ∆̇ω commute with ddz . Furthermore, ∆ω := eω/z∆̇ω is a derivation of

S. We have

∆ω =∞∑n=1

(−1)n+1

n

∑ω′′∈(0,∞)n:|ω′′|=ω

Sω′′ .

For further details we refer the reader to [7, 12].

Definition 5.8. For all α > 0, ρα will denote the ‘change of variable’:ραf(z) = f(zα). We put: Sα = ραSρ−1α , ∆̇

αω = ρα∆̇ωρ

−1α and ∆

αω =

ρα∆ωρ−1α . Obviously,

∆̇αω = e−ωz−α∆αω.

Let r ∈ N, let k = (k1, . . . , kr), where 0 < k1 < . . . < kr < kr+1 = ∞ andlet d = (d1, . . . , dr) ∈ {+,−}r. By Sk;d we denote the set of all f ∈ ρk1Swith the following property:

For all n ∈ N and all ω = (ω1, . . . , ωn) ∈ Ωn(ρ−1k1 f), and all j ∈ {1, . . . , r},there exist quasi-functions fω,j ∈ Q


Proposition 5.9. Let r ∈ N, 0 < k1 < · · · < kr+1 = ∞. Let i ∈{1, . . . , r} and let d,d′ ∈ {+,−}r such that dj = d′j for all j 6= i, di =

−d′i. For each j ∈ {1, . . . , r}, let Ij := Idjj and I

′j := I

d′jj . Suppose

that there exist (y11, . . . , y1r ) ∈ Q

i, whereas y2i − y1i ∈ Q≤−kiki+1

((− π2ki ,π

2ki)). On the other hand, there

exist complex numbers u1, . . . , un such that

y2r − y1r =n∑l=1

uleql(z)zρlhdl,r.

As y2r − y1r ∈ A≤−ki(Ir) and eql is of order κl, it follows that ul = 0 for alll such that κl < ki. From the fact that cos(argωl − κl arg z) changes signon Ir if κl = kr we infer that ul = 0 for all l such that κl = kr as well.Consequently,

y2r−1 − y1r−1|Ir =∑

l:ki≤κl


For every d ∈ {+,−}r and every l ∈ {1, . . . , n}, hdl,r satisfies a linearD.E. of order n, with formal solutions eqmlzρml ĥm, where qml = qm − ql,ρml = ρm−ρl, m = 1, . . . , n. We fix a multi-direction d and write hl,j := hdl,jfor all j ∈ {1, . . . , r}. As ĥl is multi-summable in every multi-direction ∈{+,−}r, so in particular in every multi-direction ∈ {+,−}r whose first j−1components coincide with those of d, there exists, for every j ∈ {1, . . . , r},an element h±l,j of Q

0

e−ωmlz−kj (umleq̃ml(z)zρmlh+m,j)

where uml ∈ C. This shows that hl,j−1|»− π

2kj, π2kj

– ∈ ρkjS1 and

ρkjSωρ−1kjhl,j−1|»

− π2kj

, π2kj

– = ∑m:κml=kj ,ωml=ω

umleq̃ml(z)zρmlhm,j−1|»

− π2kj

, π2kj

–.(32)

Let kj = (kj , . . . , kr) and dj = (dj , . . . , dr). It is easily seen thathl,j−1|»

− π2kj

, π2kj

– ∈ Skj ;dj . It can also be considered to be an element ofTkj ;dj with support {0}. Let ω > 0 and define

J(j)l,ω := {m ∈ {1, . . . , n} : deg(qml + ωz

−kj ) < pkj}

(here qml + ωz−kj is considered a polynomial in z−1/p) and

J(j,s)l,ω = {(m1, . . . ,ms) ∈ {1, . . . , n}

s : κm1l = · · · = κmsl = kj ,0 < ωm1l < · · · < ωmsl < ω}.

From (31), (32) and Definition 5.8 we deduce that

∆̇kjω hl,j−1|»− π

2kj, π2kj

– = ∑m∈J(j)l,ω

Al,m(ω)eqml(z)zρmlhm,j−1|»− π

2kj, π2kj

–(33)

where

Al,m(ω) = uml +∞∑s=1

(−1)s

s+ 1

∑(m1,... ,ms)∈J(j,s)l,ω

um1lum2m1 . . . umms .


Defining

Φ(j)(u; z) =n∑l=1

uleql(z)zρlhl,j−1|»

− π2kj

, π2kj

–

and extending ∆̇kjω to Tkj ;dj [eq1 , . . . , eqn ], we can summarize the above re-lations in one formula, for each j ∈ {1, . . . , r} (“the bridge equation oflevel kj”):

∆̇kjω Φ(j)(u; z) = A(j)ω (u)∇uΦ(j)(u; z)(34)

where A(j)ω (u) is a row vector with components

A(j)ω,m(u) =∑

l:deg(qml+ωz−kj )


form aĥ+ be−1/z, where a, b ∈ C. Let ĥ2 and ĥ3 denote the (unique) formalpower series solutions of D3(e−1/zy) = e−1/z, or, equivalently, of(

z3d

dz− 1 + z

)y = 1(35)

and D3y = ĥ, respectively. The equation Dy = 0 has a formal integral ofthe form

Φ(u; z) = u1e−12z−2 + u2e−z

−1ĥ2(z) + u3ĥ3(z).

It has two levels: k1 = 1 and k2 = 2. One easily verifies that

J(1)3,1 = {2}, J

(2)

2, 12

= {1}, J (2)3, 1

2

= {1}

and J (j)l,ω = ∅ otherwise. Hence there exist complex numbers A3,2, A2,1,A3,1,+ and A3,1,− such that

∆̇11Φ(u; z) = (0, u3A3,2, 0)∂

∂uΦ(u; z)

and

∆̇212,±Φ(u; z) = (u2A2,1 + u3A3,1,±, 0, 0)

∂

∂uΦ(u; z).

(Note that the 2-summability of ĥ2 implies that A2,1 is independent of d1.)In this simple example the invariants A3,2, A2,1, A3,1,+ and A3,1,− can beeasily calculated. The difference between the 1-sums h+ and h− of ĥ in thedirections + and − is

h−(z)− h+(z) = 2πie−1z .

This implies that ∆̇11ĥ = 2πie− 1

z . Hence we deduce the relation(z3

d

dz− 1)

∆̇11ĥ3 = ∆̇11

(z3

d

dz− 1)ĥ3 = ∆̇11ĥ = 2πie

− 1z .

It follows that ∆̇11ĥ3 = 2πie− 1

z ĥ2 and thus A3,2 = 2πi.It can be shown that the 2-sums h+2 and h

−2 of ĥ2 in the directions + and

− are given by

h±2 (z) = e− 1

2z−2+z−1

∫γ±(z)

e12ζ−2−ζ−1ζ−3dζ

where γ±(z) is a path from O to z, tangent at O to a half line with directionθ such that ±θ ∈ (π4 ,

3π4 ) (the expression in the right-hand side of the above

formula represents the unique solution of (35) which is bounded on the halfline arg z = ±π2 ). It is easily seen that

h−2 (z)− h+2 (z) = i

√2πe−

12z−2+z−1− 1

2


and thus A2,1 = i√

2πe .

For any multi-direction d ∈ {+,−}2 let hd3 denote the function definedby

hd3 (z) = e− 1

2z−2∫γd2 (z)

e12ζ−2ζ−3hd1(ζ)dζ.

By means of elementary asymptotic methods (cf. [26]) it can be sh

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