On Christol’s theorem. A generalization to systems of PDE’s with logarithmic singularities depending upon parameters

by Francesco Baldassarri and Bruno Chiarellotto

§0 Introduction

This article is dedicated to a better understanding of the beautiful theorem of Christol proved in [CH1] (cf. the generalization in [A2, Résultat 1]) and to its extension (cf. theorem 4.9 below) to first-order systems of relative PDE’s defined on a tubular neighborhood of an affinoid space, with logarithmic singularities at the center. This extension is necessary in view of our program of presenting a proof of a general comparison theorem between the algebraic and the rigid cohomologies of a smooth scheme of characteristic p > 0, with suitable coefficients. These coefficients should be algebraically defined overconvergent isocrystals, with nice logarithmic singularities. For more details, we refer the reader to [BE] and [BA-CT3].

Christol’s theorem deals with first-order systems of ordinary differential equations of the type

(0.1)  (δ − G)y = 0

where δ = x\frac{d}{dx} and G is an n × n matrix of functions analytic in the p-adic open unit disk D(0, 1^{−}). He, as well as subsequent authors (André, Christol-Dwork), writes a formal matrix solution of (0.1) at the origin as Y = Y_{G}x^{C(G)} where Y_{G} is an n×n matrix with formally meromorphic entries and C(G) is a constant matrix whose eigenvalues may be assumed to be among those of G(0). He then gives conditions for the actual p-adic meromorphy of Y_{G} in the disk D(0, 1^{−}). The matrix Y_{G} is well-known to be itself a formally meromorphic solution of a system of the type (0.1) (of order n^{2}), so Christol’s theorem really gives conditions for a formally meromorphic solution of such a system to be p-adically meromorphic in the disk D(0, 1^{−}). This viewpoint turns out to be more easily generalized to the case of several variables (also with dependence upon parameters) and in fact to give a slightly stronger result even in the ordinary case. This is why we need to review Christol’s proof. The proof we present here seems in fact to be simpler than the original one, even in the ordinary case, partly because of the obviously simpler statement we use and partly because we take advantage of

This paper is in final form and no version of it will be submitted for publication elsewhere.

1991 Mathematics Subject Classification. Primary 12H25. new arguments of Christol and Dwork [CH-DW]. In that paper, those authors prove, among other results, only a special case of [CH1], but assert that their arguments could be used to simplify the proof of the full result of [CH1], as we actually do.

In the part of the present article that deals with ordinary differential equations, we follow step by step the proofs in [CH-DW] indicating the necessary modifications. Our theorem 1.6 provides a small improvement on André’s Résultat 1 [A2].

A formulation of a theorem for systems of PDE’s precisely in the form given by Christol [CH1] in the one dimensional case, naturally requires a detailed study of the formal theory for such systems. This theory, developed by Gérard and Levelt in [G-L] in the absolute case, was extended to systems depending upon parameters in [BA2], following a suggestion of N. Katz. We feel however that that theory deserves a more extensive treatment than the one it previously received. We refer to the forthcoming [BA-CT2] for such a treatment, that will include as an application the expected formulation of the generalized Christol theorem.

We replaced the notion of “p-adic non-Liouville” numbers (see (1.2) below) by the weaker notion of number “of positive type ρ”, for ρ ∈ (0, 1] (loc. cit.). This also represents an improvement on previous notions introduced by André [A2]. We are grateful to Professor Dwork for suggesting this definition and for the subsequent gain in generality and clarity of exposition.

We hope that our reconsideration of this important work of Christol, Dwork and André will be useful to the reader.

The methods of the present article may also be applied to free André’s transfer theorem ([A1, V.4.2], [A2, Résultat 2] and [CH-DW, sect.6,7,8]) from some restrictions on the exponents. For clarity of exposition, we decided to postpone the exposition of the generalized André’s theorem, to a subsequent article [BA-CT1]. Our most immediate geometric needs ([BA-CT2],[BA-CT3]) are filled by the present discussion, but André’s theorem has important number theoretic and geometric applications. In particular, it should in the end allow an extension of our theorem of comparison of cohomology [BA-CT3] to a more general type of coefficients. Our interest in the case of irrational exponents is instead mainly motivated by variation of p-adic cohomology and the so-called Boyarski Principle [DW2].

§1 The one-dimensional result.

1.1 We recall the notation of [CH-DW].

p= a prime number;

k= an algebraically closed field of characteristic zero, complete under a nonarchimedean absolute value

| | : k ⟶ R_{≥0}

extending the p-adic absolute value of Q_{p} and normalized by |p| = p^{−1}; V= ring of integers of k; D(0, ρ^{−})={x ∈ k | |x| < ρ}, the open disk of radius ρ ≥ 0 centered at 0; π= a fixed (p − 1)-st root of −p in k, π^{p−1} = −p; E= the completion of k(x) under the Gauss norm which we will still indicate

by | − |;

E_{∘}= the ring of analytic elements on D(0, 1^{−}). The elements of E_{∘} are uniform limits on D(0, 1^{−}) of sequences of rational functions without poles on D(0, 1^{−}) ; therefore E_{∘} ⊆ E and the sup-norm on D(0, 1^{−}) coincides, for elements of E_{∘}, with the Gauss norm | − |.

E′_{∘}= the quotient field of E_{∘}.

Finite dimensional spaces over E with a natural basis are provided with a norm derived from |−| in the standard way: we still denote such a norm by |−|. This applies to n-tuples and to matrices. In particular an element of GL(n, E) is said to be unimodular if it and its inverse are bounded by 1.

1.2 For an element λ ∈ k , we define the type of λ as the radius of convergence ρ(λ) of the series

(1.2.1)  g_{λ}(x) = ∑_{{s=0 \atop λ−s≠0}}^{∞} \frac{x^{s}}{λ − s}.

We therefore have:

(1.2.2)  ρ(λ) = lim inf_{s} |λ − s|^{\frac{1}{s}}

and ρ(λ) ∈ [0, 1]. The type of a number is clearly invariant under translations by integers.

Let now C be a square matrix with entries in k, C ∈ M_{n}(k), and ∏_{λ}(x−λ)^{e_{λ}} be its characteristic polynomial det(x − C). We define the type ρ(C) of C as the product

(1.2.3)  ρ(C) = ∏_{λ} ρ(λ)^{e_{λ}}.

The type of a matrix therefore only depends upon the types of its eigenvalues; in particular two matrices having the same number (counting multiplicities) of eigenvalues in each class of k/Z, have the same type. For any G ∈ M_{n}(k[[x]]) we will also say that ρ(G(0)) is the type of the system (0.1) at 0 and, more abusively, of G.

We recall (cf. [CH1], [A2]) that an element λ ∈ k is said to be p-adically Liouville (in the weak sense) if there exists an r ∈ R_{>0} such that for an infinite set of n ∈ Z one has

(1.2.4)  |λ − n| ≤ inf(r^{n} , r^{−n}).

Therefore a number λ is p-adically non-Liouville iff both λ and −λ are of type 1. It is clear that p-adically Liouville numbers are in Z_{p}. It is known by the p-adic Roth theorem [LA, VI.1, Theorem1] that if λ is an algebraic number the series g_{λ}(x) has logarithmic growth δ in D(0, 1^{−}) [CH2, 2.3] for any δ > 2. In particular, algebraic numbers are p-adically non-Liouville. In [A2], André considers numbers λ ∈ k which “ are not p-adically Liouville in the strong sense” and associates to such a λ a positive real number r. To us this means that λ and −λ have positive type greater or equal to r. We now have:

Lemma 1.3. If λ ∈ Z_{p} is a number of type ρ then an element λ′ ∈ Z_{p} such that pλ′ − λ = n ∈ Z, is of type ρ^{p}.

Proof. We can assume that n = 0 and that λ ∉ N. The series

∑_{s∈N} \frac{x^{s}}{λ − s}

converges in D(0, ρ^{−}). But:

∑_{s∈N} \frac{x^{s}}{λ − s} = ∑_{s∈N} \frac{x^{s}}{pλ′ − s} = ∑_{{s≢0 \atop mod.p}} \frac{x^{s}}{pλ′ − s} + ∑_{{s≡0 \atop mod.p}} \frac{x^{s}}{pλ′ − s}.

But, by hypothesis, λ′ ∈ Z_{p}, so the series

∑_{{s≢0 \atop mod.p}} \frac{x^{s}}{pλ′ − s}

converges in D(0, 1^{−}). The second series ( in the last term of the previous equation ) may be re-written as:

∑_{r∈N} \frac{x^{pr}}{p(λ′ − r)}

and it converges for x ∈ D(0, ρ^{−}). It follows that λ′ is of type ρ^{p}. But the argument could be reversed. Q.E.D.

1.4 For any differential ring extension (R, δ) of (k[x], δ), for G ∈ M_{n}(R) and H ∈ GL(n, R) we define

(1.4.1)  G_{[H]} = (δH)H^{−1} + HGH^{−1}.

Therefore the substitution z = Hy transforms the system δy = Gy into δz = G_{[H]}z. Under these circumstances we will say that the two previous systems are equivalent over R. For u ∈ E we define by u^{ϕ} to be the composition x → u(x^{p}).

Let (Ω, | − |) be an algebraically closed and complete valued extension field of (k, | − |), containing an element t of absolute value 1 whose residue class is trascendental over the residue field of k. We call t a generic point (for k). For ρ ≥ 0 we will denote by D(t, ρ^{−}) (resp. D(t, ρ)) the open (resp. closed) disk of radius ρ centered at t, in Ω:

D(t, ρ^{−}) = {x ∈ Ω | |x − t| < ρ}

(resp.

D(t, ρ) = {x ∈ Ω | |x − t| ≤ ρ}).

Let S be the set of all G ∈ M_{n}(E) satisfying the following two conditions: S1. G ∈ M_{n}(E_{∘}). S2. The system (δ − G)y = 0 has a solution matrix U_{G,t} at t (normalized

by U_{G,t}(t) = I_{n}) which converges in D(t, 1^{−}) = {x ∈ Ω | |x − t| < 1}.

We define, for any differential extension ring (R, δ) of (k[x], δ) and G ∈ M_{n}(R), G_{0} = I_{n}, G_{1} = G and for m ∈ N:

(1.4.2)  G_{m+1} = δG_{m} + G_{m}G − mG_{m}.

Condition S2 is then equivalent to the following

S′2. For any η ∈ (0, 1):

(1.4.3)  lim_{m→+∞} |\frac{G_{m}}{m!} |η^{m} = 0.

This is clear since

(1.4.4)  U_{G,t}(x) = ∑_{m=0}^{∞} \frac{G_{m}(t)}{m!}(\frac{x − t}{t})^{m}.

On the other hand condition S2 is known to imply [CH2, Proposition 6.2.9] that the eigenvalues of G(0) are in Z_{p}.

We consider also the conditions:

S3. |G| ≤ 1.

S4. If an eigenvalue of G(0) is an integer then it is zero.

If condition S1 is satisfied and the difference between two eigenvalues of G(0) is never a non-zero integer, the equation (δ − G)y = 0 has a formal solution matrix at 0 which may be written in the form Y_{G}x^{G(0)}, where

(1.4.5)  Y_{G} ∈ GL(n, k[[x]]) , Y_{G}(0) = I_{n}.

This is best understood within the framework of Gérard-Levelt’s theory [G-L], according to which, under the previous assumptions, the systems (δ − G)y = 0 and (δ − G(0))y = 0 are equivalent over k[[x]]. In the general case, under the assumption S1 we can only assume the existence of a formal solution matrix at 0 in the form: Y_{G}x^{C(G)}, where Y_{G} ∈ GL(n, k((x))) and C(G) ∈ M_{n}(k). One could also assume that the eigenvalues of C(G) lie among those of G(0) and that two of them never differ by a non-zero integer ([G-L]).

Remark 1.5 We already recalled that the matrix Y_{G} viewed as a column vector of length n^{2} satisfies a system of differential equations:

(1.5.1)  (δ − H)Y_{G} = 0,

where H ∈ M_{n^{2}}(E_{∘}) if G belongs to M_{n}(E_{∘}). The eigenvalues of H(0) are differences of two eigenvalues of G(0) : this explains the conditions (namely, the differences of exponents should be non-Liouville numbers, hence numbers of type 1) imposed by Christol [CH1] in order to estimate the radius of convergence of Y_{G}. Using the formulation S2, one can show that the system (δ−C(G))y = 0 satisfies S2 if (δ − G)y = 0 did, since C(G) = G_{[N]}(0) for a matrix N ∈ M(k[x, \frac{1}{x}]). We conclude that if the system (δ − G)y = 0 has property S2 then (δ − H)y = 0 also has it : in fact the two systems, (δ − C(G))y = 0 and (δ − G)y = 0, are equivalent to the constant system δy = 0 over the ring of functions analytic in the generic disk of radius 1.

We will prove:

Theorem 1.6. Let G ∈ S be of type ρ(G) . Let y ∈ k((x))^{n} be a column solution of the system

(1.6.1)  (δ − G)Y = 0.

Then the entries of y are meromorphic in D(0, ρ(G)^{−}) (with possible poles only at x = 0).

Corollary 1.6.2. Let G ∈ S be such that the exponents of G(0) are p-adically non-Liouville. Let y ∈ k((x))^{n} be as in the theorem. Then the entries of y are meromorphic in D(0, 1^{−}) (with possible poles only at x = 0) .

Notice that our theorem 1.6 and, in particular, corollary 1.6.2 are stronger than the main result of [CH1], while their proof will be easier.

Another immediate corollary is the following result. Let A_{r} (resp. B_{r}), for r ∈ (0, 1], denote the ring of analytic (resp. bounded analytic) functions on D(0, r^{−}):

A_{r} = {∑_{i=0}^{∞} a_{i}x^{i} ∈ k[[x]] | lim_{i→∞} |a_{i}|ρ^{i} = 0, ∀ρ ∈ (0, r)}.

B_{r} = {∑_{i=0}^{∞} a_{i}x^{i} ∈ k[[x]] | sup_{i∈N} |a_{i}|r^{i} < ∞}.

The sup-norm on D(0, r^{−}) for an element f(x) = ∑_{i=0}^{∞} a_{i}x^{i} of B_{r}, will be denoted by

|f(x)|(r) = sup_{i∈N} |a_{i}|r^{i}.

We will also denote by |−|(r) the norm (resp.semi-norm) derived in the standard way on matrices with entries in B_{r} (resp. A_{s}, for any s ∈ (r, 1]). Let S_{B_{r}} (resp.S_{A_{r}}) denote the set of matrices G ∈ M_{n}(B_{r}) (resp. G ∈ M_{n}(A_{r})) such that:

S_{B_{r}} For every η ∈ (0, 1)

lim_{m→+∞} |\frac{G_{m}}{m!} |(r)η^{m} = 0.

(resp.

S_{A_{r}} For every η ∈ (0, 1) and σ ∈ (0, r)

lim_{m→+∞} |\frac{G_{m}}{m!} |(σ)η^{m} = 0. )

Corollary 1.7. Let G ∈ S_{B_{r}} (resp. G ∈ S_{A_{r}}) be of type ρ and y ∈ k((x))^{n} be a column solution of the system (1.6.1). Then the entries of y are meromorphic in D(0, (rρ)^{−}) ( with possible pole only at x = 0).

Proof (of the corollary, granting the theorem). Let α ∈ k^{∗} be such that |α| < r and let H(x) = G(αx). Then H_{m}(x) = G_{m}(αx), so that H(x) ∈ S. So, if y(x) ∈ k((x))^{n} is a vector solution of (1.6.1), it follows that y(αx) is a vector solution of (δ − H)y = 0 and, by theorem 1.6, it is meromorphic in D(0, ρ^{−}). Therefore y(x) is meromorphic in D(0, (rρ)^{−}). Q.E.D.

Remark 1.7.1. The previous condition appeared in [CH1] as the condition of convergence of the solution matrix in the open generic disk of radius r.

We will denote by \overline{S} ⊂ S the set of G ∈ S satisfying also conditions S3 and S4. We have:

Lemma 1.8. Let F be an element of S. There exists a matrix Q ∈ GL(n, E_{∘}[\frac{1}{x}]) such that F_{[Q]} ∈ \overline{S} and ρ(F_{[Q]}) = ρ(F) .

Proof. We follow the procedure of section 4 of [CH-DW], up to line - 6 of that section (condition Ri of [loc.cit.] coincides with our condition Si for i = 1, 2, while S3 = R4). Thus, in the notation of [loc.cit.], F_{[H_{1}H]} also satisfies S1, S2, S3. We recall that H ∈ GL(n, E′_{∘}) and H_{1} is a unimodular matrix in GL(n, k(x)). By Lemma 1.10 below, the eigenvalues of F_{[H_{1}H]}(0) differ by integers from those of F (0). It follows that F_{[H_{1}H]} is also of type ρ(F). By (2.3) of [loc.cit.] there exists H_{2} ∈ GL(n, k[x, x^{−1}]) unimodular such that (F_{[H_{1}H]})_{[H_{2}]} = F_{[H_{2}H_{1}H]} satisfies S4. Conditions S1, S2, S3 are automatically valid for F_{[H_{2}H_{1}H]}, also of type ρ(F). We then put Q = H_{2}H_{1}H ∈ GL(n, E′_{∘}) and conclude that G = F_{[Q]} ∈ \overline{S}. We want to show that Q ∈ GL(n, E_{∘}[\frac{1}{x}]). This follows from the fact that the systems (δ − G)Y = 0 and (δ − F)Y = 0 have no singularities in D(0, 1^{−}) \ {0}. Q.E.D.

Our theorem 1.6 is therefore equivalent to the statement:

Theorem 1.9. Let G ∈ \overline{S} be of type ρ and let y ∈ k[[x]]^{n} be a column solution of the system (1.6.1). Then the entries of y are analytic in D(0, ρ^{−}).

We recall here the elementary :

Lemma 1.10. Let G, H ∈ M_{n}(k[[x]]) and P ∈ GL(n, k((x))) such that H = G_{[P]}. Then for any coset α + Z ∈ k/Z, the matrices G(0) and H(0) have the same number, counting multiplicities, of eigenvalues in α + Z.

Proof. We write P = AΔB, with A, B ∈ GL(n, k[[x]]) and Δ a diagonal matrix with elements of the form x^{r}, r ∈ Z. We then have:

H = G_{[AΔB]} = ((G_{[B]})_{[Δ]})_{[A]}

and we are reduced to the case P = Δ and H = G_{[Δ]}. We can subdivide our matrices H, G, Δ into rectangular blocks H = (H_{i,j}), G = (G_{i,j}), Δ = (Δ_{i,j}) for i, j = 1, …, s, such that Δ_{i,j} = 0 if i ≠ j, Δ_{i,i} = r_{i}I_{e_{i}} and r_{i} < r_{j} for i < j. We then have:

H = (δΔ)Δ^{−1} + ΔGΔ^{−1}
H_{i,j}(x) = r_{i}δ_{ij} + x^{r_{i}−r_{j}}G_{ij}(x) i, j = 1, …, s.

By evaluating at x = 0 we deduce that if i > j

H_{ij}(0) = 0

and if i < j

G_{ij}(0) = 0.

On the other hand, for i = 1, …, s:

H_{ii}(0) = r_{i} + G_{ii}(0)

Let P_{i}(λ) ∈ k[λ] be the characteristic polynomial of G_{ii}(0). We conclude that the characteristic polynomials of H(0) and G(0) are, respectively, ∏_{i=1}^{s} P_{i}(λ) and ∏_{i=1}^{s} P_{i}(λ − r_{i}). Q.E.D.

§2 The trivial estimate.

For a matrix C ∈ M_{n}(k) and s ∈ N, we define (C)_{0} = I_{n} and (C)_{s} = C(C +I_{n})(C +2I_{n}) … (C +(s−1)I_{n}). The following proposition replaces lemma 3.1 of [CH1].

Proposition 2.1. Let C ∈ M_{n}(k) be a matrix none of whose eigenvalues is a positive integer. Then the series

h_{C}(x) = ∑_{i=0}^{∞} \frac{x^{i}}{det(I_{n} − C)_{i}} = ∑_{i=0}^{∞} \frac{(−1)^{ni}x^{i}}{det(C − I_{n})(C − 2I_{n}) … (C − iI_{n})}

converges in D(0, (|π|^{n}ρ(C))^{−}).

Proof. We first notice that the series h_{C}(x) is the Hadamard product of the series h_{λ}(x) each taken with the multiplicity of λ as eigenvalue of C. We may then assume n = 1. By the formal Kummer identity [WA, 4.42]

1F_{1}(1; 1 − λ; x) = e^{x} _{1}F_{1}(−λ; 1 − λ; −x)

one obtains for λ ≠ 0, 1, 2 …:

(2.1.1)

∑_{i=0}^{∞} \frac{x^{i}}{(1 − λ)_{i}} = e^{x} ∑_{i=0}^{∞} \frac{(−x)^{i}(−λ)_{i}}{(1 − λ)_{i}i!} = e^{x} ∑_{i=0}^{∞} \frac{(−x)^{i}(−λ)}{(−λ + i)i!} = e^{x}λ ∑_{i=0}^{∞} \frac{(−x)^{i}}{(λ − i)i!},

while for λ = 0, h_{0}(x) = e^{x}. The last series in (2.1.1) is the Hadamard product of e^{(−x)} and g_{λ}(−x): its radius of convergence is therefore the product of the radii of convergence of those two series, namely |π| and ρ(λ) . Since e^{x} converges in D(0, |π|^{−}), we conclude that ∑_{i=0}^{∞} \frac{x^{s}}{(1−λ)_{s}} converges in D(0, (|π|ρ(λ))^{−}). Q.E.D.

We have:

Proposition 2.2. Let G ∈ M_{n}(E_{∘}) be of type ρ and satisfy S4. Let y ∈ k[[x]]^{n} be a column solution of (1.6.1). Then the entries of y are analytic in D(0, (max(1, |G|)^{−n}|π|^{n}ρ)^{−}). In particular, let G ∈ \overline{S} be of type ρ and y ∈ k[[x]]^{n} be a column solution of (1.6.1). Then the entries of y are analytic in D(0, (|π|^{n}ρ)^{−}).

Proof. We write

G = ∑_{j=0}^{+∞} G^{(j)}x^{j},

G^{(j)} ∈ M_{n}(k) and y(x) = ∑_{i=0}^{∞} y_{i}x^{i}, with y_{i} ∈ k^{n}. From (δ − G)y(x) = 0 we obtain

(2.2.1)  G(0)y_{0} = 0

and

(2.2.2)  (G(0) − i)y_{i} = − ∑_{j=0}^{i−1} G^{(i−j)}y_{j}, for i ≥ 1.

We assume with no restriction that |y_{0}| = 1, and we deduce :

|y_{i}| ≤ max(1, |G|)^{i}|[(G(0) − I_{n})(G(0) − 2I_{n}) … (G(0) − iI_{n})]^{−1}| ≤

(2.2.3)

≤ \frac{max(1, |G|)^{i}(max(1, |G(0)|))^{(n−1)i}}{|det(I_{n} − G(0))_{i}|} ,

hence

(2.2.4)  |y_{i}| ≤ \frac{1}{|det(I_{n} − G(0))_{i}|} max(1, |G|)^{ni}

(by assumption S4, G(0) − j is invertible whenever j ∈ N, j ≠ 0). We conclude that y is dominated by the Hadamard product of the series h_{G(0)}(x) and a series of radius of convergence max(1, |G|)^{−n} . Q.E.D.

Corollary 2.3. Let G ∈ M_{n}(E_{∘}) be of type ρ. Let y ∈ k((x))^{n} be a column solution of (1.6.1). Then the entries of y are meromorphic in

D(0, (max(1, |G|)^{−n}|π|^{n}ρ)^{−})

with poles at most at 0.

Proof. We need only to apply some shearing transformations (cf. [CH-DW, prop. 2.3]) to reduce to the situation of the previous statement (G ∈ S4).Q.E.D.

§3 Frobenius action.

We need some lemmas:

Lemma 3.0. Let G ∈ \overline{S} be of type ρ. There exists P ∈ GL(n, k[x, x^{−1}]) unimodular, such that G_{[P]} ∈ \overline{S} is of type ρ and the eigenvalues of G_{[P]}(0) are in pZ_{p}.

Proof. We apply the “shearing transformation” of (2.3) of [CH-DW]. Q.E.D.

Lemma 3.1. Let r ≥ 0.

(i) The map x → x^{p} sends the closed disk D(t, r) into D(t^{p}, max(r^{p}, p^{−1}r)). In particular, if r ≥ |π|, x → x^{p} sends D(t, r) into D(t^{p}, r^{p}).

(ii) The image of D(t, r) under the map x → x^{p} contains D(t^{p}, r^{p}) (without restriction on r).

(iii) For r ≥ |π|, the map x → x^{p} induces a surjection of D(t, r) onto D(t^{p}, r^{p}) each fiber of which has exactly p elements.

( We omit the easy proof).

The next lemma fills a gap in the proof of the theorem in section 3 (resp. of Lemma 8.2) of [CH-DW], where it is asserted that pF lies in M_{n}(E_{∘}) (resp. in M_{n}(E′_{∘})).

Lemma 3.2. Let r ≥ |π| and f be meromorphic in the closed disk D(t^{p}, r^{p}). (So, by the previous Lemma, f(x^{p}) is meromorphic on D(t, r)). Suppose f(x^{p}) coincides on D(t, r) with g ∈ E_{∘} (resp.E′_{∘}) . Then f extends to an element of E_{∘} (resp.E′_{∘}).

Proof. We may restrict ourselves to the case g ∈ E_{∘}. Let h(x) = \frac{1}{p} ∑_{z^{p}=x} g(z). If g ∈ E_{∘} ∩k(x), one sees as in [DW1, 3.5] that also h ∈ E_{∘} ∩k(x); it then follows from [DW1, 5.2] that for any g ∈ E_{∘}, h ∈ E_{∘}. We assert that h = f on D(t^{p}, r^{p}). For x ∈ D(t, r), h(x^{p}) = \frac{1}{p} ∑_{z^{p}=x^{p}} g(z) = \frac{1}{p} ∑_{ζ^{p}=1} g(xζ). Now xζ ∈ D(t, r), hence g(xζ) = f((xζ)^{p}) = f(x^{p}), i.e. h(x^{p}) = f(x^{p}) for all x ∈ D(t, r). Hence we have h = f on the image of D(t, r) under the p^{th} power map and this image is D(t^{p}, r^{p}) by the previous lemma. Q.E.D.

Theorem 3.3. Let G ∈ \overline{S} be such that the eigenvalues of G(0) are in pZ_{p}. Then there exist H ∈ GL(n, E_{∘}), H unimodular, and F ∈ M_{n}(E_{∘}) satisfying S1,S2 such that

(3.3.1)  pF^{ϕ} = G_{[H]}.

We have ρ(F) = ρ(G)^{p}.

Proof. We follow the proof of the theorem in section 3 of [CH-DW] up to formula (3.2.1.1) included. We have:

pH(0) = ∑_{s=0}^{∞} p\binom{G(0)}{s}z_{s}s! ,

where z_{s} denotes a p-adic integer. Therefore:

pH(0) = pI_{n} + p ∑_{s=1}^{∞} z_{s}G(0)(G(0) − 1) … (G(0) − s + 1).

We may now assume that G(0) is a triangular matrix whose diagonal entries are in pZ_{p}, to conclude that:

(3.3.2)  |detH(0)| = 1.

From this we deduce, as in [loc.cit.], that H ∈ GL(n, E_{∘}) and that it is unimodular. We still follow the proof in [loc.cit.], and, in particular, adopt definition (3.3) of V (x) and (3.3.1). We conclude, as in [loc.cit.] but with the explanations given in the previous lemma, that F satisfies S1 (=R1 in the notations of [loc.cit]). We are not interested in equation (3.0.2) of [loc.cit.], now replaced by the obvious:

(3.3.3)  pF (0) = H(0)G(0)H(0)^{−1}.

This relation shows, with the help of Lemma 1.3, that ρ(F) = ρ(G)^{p}. Property S2 is seen to hold for F as in [loc.cit.] We are not interested in the remaining part of the proof of [loc.cit.], which aims at the proof of relation (3.0.3) of that paper. Q.E.D.

Corollary 3.4. Let G ∈ \overline{S} . There exist G_{1} ∈ \overline{S} and U in GL(n, E_{∘}[\frac{1}{x}]), such that

(3.4.1)  (pG_{1}^{ϕ})_{[U]} = G

and such ρ(G_{1}) = ρ(G)^{p}.

Proof. By lemma 3.0, we find P ∈ GL(n, E_{∘}[\frac{1}{x}]), unimodular, such that G_{[P]} satisfies the conditions of applicability of the theorem (for G replaced by G_{[P]} and ρ(G_{[P]}) = ρ(G)). If F ∈ M_{n}(E_{∘}) satisfies S1, S2, is of type ρ(F) = ρ(G)^{p} and

pF^{ϕ} = (G_{[P]})_{[H]} = G_{[HP]}

for some H ∈ GL(n, E_{∘}), H unimodular, as in the theorem, we put L = HP. By lemma 1.8, there exists L_{1} ∈ GL(n, E_{∘}[\frac{1}{x}]) such that F = (G_{1})_{[L_{1}]} with G_{1} ∈ \overline{S} with ρ(G_{1}) = ρ(F). We then have

(3.4.2)  p(G_{1[L_{1}]})^{ϕ} = G_{[L]}

and therefore

(3.4.3)  (pG_{1}^{ϕ})_{[L_{1}^{ϕ}]} = G_{[L]}.

We then put U = L^{−1}L_{1}^{ϕ}. Q.E.D.

3.5 For H ∈ M_{n}(E_{∘}) let Sol(H, k[[x]]) be the k-vector space of column solutions y ∈ k[[x]]^{n} of (δ − H)y = 0. We also define R(H) ∈ [0, 1] to be the supremum of the set:

(3.5.1)  {r ∈ [0, 1] | every y(x) ∈ Sol(H, k[[x]]) is analytic in D(0, r^{−})}.

We have seen in section 2 that if H ∈ \overline{S} is of type ρ(H),

(3.5.2)  R(H) ≥ |π|^{n}ρ(H).

Lemma 3.6. Let G,G_{1},U be as in corollary 3.4 . The map

(3.6.0)  y ⟶ Uy^{ϕ}

induces an isomorphism of k-vector spaces

(3.6.1)  Sol(G_{1}, k[[x]]) ⟶ Sol(G, k[[x]]).

Proof. The map (3.6.0) is clearly an injection and is stable on k((x))^{n}. The fact that zero is the only integral exponent for both δ − G and δ − G_{1} shows that (3.6.0) restricts to a map (3.6.1), still an injection. On the other hand the matrices G_{1}(0) and pG_{1}^{ϕ}(0) = pG_{1}(0) have the same nullity, which coincides with the dimension of the k-vector space Sol(G_{1}, k[[x]]) and also with the dimension of Sol(pG_{1}^{ϕ}, k[[x]]) = Sol(pG_{1}^{ϕ}, k((x))) since pG_{1}^{ϕ}(0) has no non-zero integral eigenvalues (the eigenvalues of G_{1}(0) are in Z_{p}). Multiplication by U clearly gives an isomorphism between Sol(pG_{1}^{ϕ}, k((x))) and Sol(G, k((x))) = Sol(G, k[[x]]). Q.E.D.

Corollary 3.7. In the notation of the lemma, we have

(3.7.1)  R(G) = R(G_{1})^{\frac{1}{p}} .

3.8 We can now prove theorem 1.9. We know from section 2 that R(G) ≥ |π|^{n}ρ(G). Choose G_{1} ∈ \overline{S} as in theorem 3.3 for G. We can repeat the process in order to obtain a sequences of matrices in \overline{S}: G_{1}, G_{2}, …, G_{l}, … ( the matrix G_{l} is related to G_{l−1} as G_{1} is to G in theorem 3.3). We have:

R(G) = R(G_{l})^{\frac{1}{p^{l}}} .

and

ρ(G_{l}) = ρ(G)^{p^{l}}.

But from our weak estimate (§2) we obtain

R(G_{l}) ≥ |π|^{n}ρ(G_{l}) ≥ |π|^{n}ρ(G)^{p^{l}},

we conclude that:

R(G) ≥ |π|^{\frac{n}{p^{l}}} ρ(G).

For l → ∞, we conclude.Q.E.D.

§4 The case of several variables.

The present section deals with a generalization of the previous results to the case of several variables over an affinoid k-algebra A.

We need to introduce some more notation.

4.1 Let A be an integral affinoid k-algebra [BGR]. We denote by | − |_{A} its supremum norm, which is a Banach k-algebra norm on A ( it is actually equivalent to every complete k-algebra norm on A [BGR, 6.2.4 Theorem 1]).

By the Noether Normalization Lemma there exists a finite monomorphism of k-algebras [BGR, 6.1.2 Corollary 2] :

T_{d} ⟶ A

where T_{d} is the free Tate k-algebra in d indeterminates. It’s well-known that T_{d} is integrally closed. The supremum norm on A, | − |_{A}, is induced by the spectral norm on the quotient field of A deduced from the Gauss norm (actually, an absolute value) on T_{d} [BGR, 3.8.1 Proposition 7]. Hence there exist a finite number of inequivalent absolute values on A [BGR 3.3.3 Proposition 1]:

(4.1.1)  | − |_{1}, …, | − |_{l}

which extend the Gauss norm on T_{d} and such that for every a ∈ A :

(4.1.2)  |a|_{A} = max_{j=1,...,l} |a|_{j}.

We will denote by K = Q(A), the quotient field of A. For each j = 1, …, l, let K_{j} denote an algebraically closed extension of K which is complete with respect to an absolute value extending the absolute value | − |_{j} : we will retain the same notation, | − |_{j}, for the extended absolute value on K_{j}.

4.2 Let x_{1}, …, x_{s} be indeterminates over A. For α ∈ N^{s}, α ∈ (α_{1}, …, α_{s}) we define |α|_{R} to be the standard euclidean norm of α ∈ R^{s} and x^{α} = x_{1}^{α_{1}} … x_{s}^{α_{s}}. We will refer to the s-tuple (0, …, 1, … 0) (1 in the position i) as δ_{i}. Let r ∈ (0, 1]. We indicate by A{{x_{1} …, x_{s}}}_{r} = A{{x}}_{r} the ring of power series

(4.2.1)  { ∑_{α∈N^{s}} a_{α}x^{α}, a_{α} ∈ A | lim_{|α|_{R}→+∞} |a_{α}|_{A}η^{|α|_{R}} = 0 ∀η ∈ [0, r)}.

We define a subring A{{x}}_{r}^{b} of A{{x}}_{r} by :

(4.2.2)  A{{x}}_{r}^{b} = { ∑_{α∈N^{s}} a_{α}x^{α}, a_{α} ∈ A | sup_{α∈N^{s}} |a_{α}|_{A}r^{|α|_{R}} < +∞}.

This is a Banach k-algebra for the norm |−|(r) [BGR 2.1.5], defined for ∑_{α∈N^{s}} a_{α}x^{α} ∈ A{{x}}_{r}^{b}, as

| ∑_{α∈N^{s}} a_{α}x^{α}|(r) = sup_{α∈N^{s}} |a_{α}|_{A}r^{|α|_{R}}.

The same symbol will denote the obvious extension of | − |(r) to a norm on matrices with coefficients in A{{x}}_{r}^{b} (resp. in A{{x}}_{c}, for any c ∈ (r, 1]). We will also write:

(4.2.3)  A[[x]][\frac{1}{x}] = A[[x_{1}, …, x_{s}]][\frac{1}{x_{1} … x_{s}}]

and

(4.2.4)  A\{\{x\}\}_{r}[\frac{1}{x}] = A\{\{x_{1}, …, x_{s}\}\}_{r}[\frac{1}{x_{1} … x_{s}}].

We note, in view of future application, that A{{x}}_{r} may also be viewed as the ring of the analytic functions on the trivial bundle in open polydisks of radius r and dimension s over the affinoid base space SpA [BE], i.e. on :

SpA × D(0, r^{−})^{s}.

It follows that A{{x}}_{r}^{b} represents the ring of bounded analytic functions on the previous fiber space and, finally, that A\{\{x\}\}_{r}[\frac{1}{x}] is the ring of meromorphic functions on SpA × D(0, r^{−})^{s} with poles only along the divisor x_{1} … x_{s} = 0.

4.3 Given a finite dimensional vector space V over a field L and m commuting L-linear endomorphisms of V, (φ_{1}, …, φ_{m}), we define a multieigenvalue of (φ_{1}, …, φ_{m}) as an m-tuple of elements in the algebraic closure \overline{L} of L

(λ_{1}, …, λ_{m}) ∈ \overline{L}^{m}

such that if \overline{V} = V ⊗_{L} \overline{L} and \overline{φ}_{i}, i = 1, …, m, denotes the obvious \overline{L}-linear extension of φ_{i} to \overline{V}, one has:

(4.3.1)  ⋂_{i=1}^{m} Ker\overline{V} (\overline{φ}_{i} − λ_{i}) ≠ 0.

The dimension of the \overline{L}-vector space

(4.3.2)  ⋂_{i=1}^{m} Ker\overline{V} (\overline{φ}_{i} − λ_{i})^{dim V}

will be called the multiplicity of the multieigenvalue (λ_{1}, …, λ_{m}).

In case for any multieigenvalue (λ_{1}, …, λ_{m}) of (φ_{1}, …, φ_{m}) the sum λ_{1} + …+λ_{m} is in k, we will define the type of (φ_{1}, …, φ_{m}) to be radius of convergence of the series

(4.3.3)  g_{(φ_{1},...,φ_{m})}(x),

Hadamard product of the series

g_{λ_{1}+...+λ_{m}}(x)

each taken with the multiplicity of (λ_{1}, …, λ_{m}) as multieigenvalue of (φ_{1}, …, φ_{m}).

4.4 Let r ∈ (0, 1]. A standard system of PDE’s on A{{x}}_{r} relative to A with logarithmic singularities at x_{1} … x_{s} = 0 is an integrable system of partial differential equations of the type:

(4.4.1)  x_{i} \frac{∂}{∂x_{i}} y = H_{δ_{i}}(x)y  i = 1, …, s

where H_{δ_{i}}(x) ∈ M_{n}(A{{x}}_{r}). The integrability condition means: (4.4.2)

x_{j} \frac{∂}{∂x_{j}} H_{δ_{i}}(x)+H_{δ_{i}}(x)H_{δ_{j}}(x) = x_{i} \frac{∂}{∂x_{i}} H_{δ_{j}}(x)+H_{δ_{j}}(x)H_{δ_{i}}(x) for i, j = 1, …, s.

By evaluating the previous formulas at x_{1} = x_{2} = … = x_{s} = 0 , one gets:

H_{δ_{i}}(0)H_{δ_{j}}(0) = H_{δ_{j}}(0)H_{δ_{i}}(0), for i, j = 1, …, s

where H_{δ_{i}}(0) ∈ M_{n}(A). It follows that the system

(4.4.3)  x_{i} \frac{∂}{∂x_{i}} y = H_{δ_{i}}(0)y  i = 1, …, s

with H_{δ_{i}}(0) ∈ M(A), i = 1, …, s, is still of the same form as (4.4.1). If Y is a solution matrix of (4.4.1) and, for α ∈ N^{s} we put

\frac{∂^{α}}{∂x^{α}} = \frac{∂^{α_{1}}}{∂x_{1}^{α_{1}}} . . . . . . \frac{∂^{α_{s}}}{∂x_{s}^{α_{s}}} ,

we have

(4.4.4)  x^{α} \frac{∂^{α}}{∂x^{α}} Y = H_{α}(x)Y,

where H_{α} ∈ M_{n}(A{{x}}). The matrices H_{α}(x) may be constructed by recursive formulas. In fact for α ∈ N^{s}, H_{α}(x) as in (4.4.4), we have

(4.4.5)  H_{α+δ_{i}}(x) = x_{i} \frac{∂}{∂x_{i}} (H_{α}(x)) + H_{α}(x)(H_{δ_{i}}(x) − α_{i}).

One should note that the analogous recursion formulas for the derived system (4.4.3) give precisely the matrices of (4.4.5) evaluated at x_{1} = x_{2} = … = x_{s} = 0: i.e., if Y is a solution of (4.4.3), then for α ∈ N^{s}, we have

x^{α} \frac{∂^{α}}{∂x^{α}} Y = H_{α}(0)Y.

We may express the previous remark symbolically by the equation:

(4.4.6)  H_{α}(0) = H(0)_{α}

for α ∈ N^{s}.

One deduces also from (4.4.5) that if H_{δ_{i}}(x) ∈ M_{n}(A{{x}}_{r}^{b}) for i = 1, …, s, then the matrices H_{α} ∈ M_{n}(A{{x}}_{r}^{b}) for every α ∈ N^{s}.

4.5 We will indicate by _{A,s}S_{B_{r}} (resp. _{A,s}S_{A_{r}}) the set of standard systems (as in (4.4.1))

(4.5.1)  x_{i} \frac{∂}{∂x_{i}} y = G_{δ_{i}}(x)y

with G_{δ_{i}}(x) ∈ M_{n}(A{{x}}_{r}^{b}) (resp.G_{δ_{i}}(x) ∈ M_{n}(A{{x}}_{r}) for i = 1, …, s, which satisfy the following condition :

A,sS_{B_{r}} For each η ∈ R, 0 < η < 1, on setting α! = α_{1}! … α_{s}!, in the previous notation we get :

(4.5.2)  lim_{|α|_{R}→+∞} |\frac{G_{α}(x)}{α!}|(r)η^{|α|_{R}} = 0.

(resp.

A,sS_{A_{r}} For each η, σ ∈ R, 0 < η < 1,0 < σ < r , in the previous notation we have :

(4.5.2)′  lim_{|α|_{R}→+∞} |\frac{G_{α}(x)}{α!}|(σ)η^{|α|_{R}} = 0. )

Notice that

(4.5.3)  _{A,s}S_{A_{r}} = ⋂_{σ∈(0,r)}_{A,s}S_{B_{σ}}

Remark 4.5.4 The multieigenvalues (λ_{1}, …, λ_{s}) of (G_{δ_{1}}(0), …, G_{δ_{s}}(0)), are usually referred to as the multiexponents of system (4.5.1) (exponents if s = 1) and their multiplicity as the multiplicity as multiexponents. (Cf. [BA2, 2.24], [BA-CT2]).

Remark 4.5.5 We will see later (5.2) that condition _{A,s}S_{B_{r}} (resp. _{A,s}S_{A_{r}}) implies that any eigenvalue λ_{i} of G_{δ_{i}}(0) is in Z_{p} for i = 1, …, s.

Definition 4.5.6 For a standard system in _{A,s}S_{A_{r}}, as (4.5.1), we define its type to be the type of (G_{δ_{1}}(0), …, G_{δ_{s}}(0)) (cf. the previous remark)

4.6 For A = k, s = 1 we have the equalities

k{{x}}_{r} = A_{r}
k{{x}}_{r}^{b} = B_{r}

(cf. §1). Moreover, condition _{k,1}S_{B_{r}} coincides with S_{B_{r}}, and condition _{k,1}S_{A_{r}} coincides with S_{A_{r}}.

4.7 It should be noted that if a system (cf. (4.5.1))

(4.7.0)  x_{i} \frac{∂}{∂x_{i}} y = G_{δ_{i}}(x)y  i = 1, …, s

is in _{A,s}S_{B_{r}} (resp. _{A,s}S_{A_{r}}), then

(4.7.1)  x_{i} \frac{∂}{∂x_{i}} y = G_{δ_{i}}(0)y  i = 1, …, s

is also in _{A,s}S_{B_{r}} (resp. _{A,s}S_{A_{r}}).

4.8 Finally, by use of the previous notation, we can state our generalization of Christol’s theorem:

Theorem 4.9. Let

y(x) ∈ (A[[x]][\frac{1}{x}])^{n}

be a formal column solution of a standard system (4.7.0) in _{A,s}S_{A_{r}}. Let ρ be the type of (4.7.0). Then y(x) ∈ (A\{\{x\}\}_{rρ}[\frac{1}{x}])^{n} .

Corollary 4.9.1. Let

y(x) ∈ (A[[x]][\frac{1}{x}])^{n}

be a formal column solution of a standard system (4.7.0), in _{A,s}S_{A_{1}}. Suppose for each multiexponent (λ_{1}, …, λ_{s}) of (4.7.0) the number λ_{1} + … + λ_{s} ∈ Z_{p} is p-adically non-Liouville. Then y(x) ∈ (A\{\{x\}\}_{1}[\frac{1}{x}])^{n}.

§5 Proof of Theorem 4.9

5.1 We will assume, with no restriction (cf. (4.5.3)), that our system (4.7.0) is in fact in_{A,s}S_{B_{r}}. By (4.1.2) we observe that g = ∑_{α∈N^{s}} a_{α}x^{α} ∈ A[[x]] belongs to A{{x}}_{r} if and only if for every absolute value in (4.4.1) | − |_{j}, j = 1, …, l, and for every η ∈ R, 0 < η < r:

lim_{|α|_{R}→+∞} |a_{α}|_{j}η^{|α|_{R}} = 0.

On the other hand if g ∈ A{{x}}_{r}^{b} we have:

|g|(r) = sup_{α∈N^{s}} |a_{α}|_{A}r^{|α|_{R}} = max_{j} ( sup_{α∈N^{s}} |a_{α}|_{j}r^{|α|_{R}}).

It follows that a standard system

(5.1.1)  x_{i} \frac{∂}{∂x_{i}} y = G_{δ_{i}}(x)y  i = 1, …, s

with G_{δ_{i}}(x) ∈ M(A{{x}}_{r}^{b}), is in _{A,s}S_{B_{r}} if and only if the same system (5.1.1) belongs to _{K_{j},s}S_{B_{r}} for each j = 1, …, s when regarding each G_{δ_{i}}(x), i = 1, …, s, as an element M_{n}(K_{j}{{x}}_{r}^{b}).

5.2 In particular for each absolute value | − |_{j} in (4.1.1), the ordinary ( in the variable x_{i}) system associated to (5.1.1)

(5.2.1)  x_{i} \frac{d}{dx_{i}} y = G_{δ_{i}}(0)y

with G_{δ_{i}}(0) ∈ M(K_{j}), may be regarded as an element of _{K_{j},1}S_{B_{r}}. This is true because the recursion matrices G_{δ_{i}}(0)_{m}, m ∈ N, for (5.2.1), in the sense of (1.1.3), coincide with the recursion matrices G_{mδ_{i}}(0) = G(0)_{mδ_{i}}, m ∈ N, for the system (4.7.1) (cf. the symbolism of (4.4.6)).

It follows that for any i ∈ {1, …, s} the eigenvalues of G_{δ_{i}}(0) are in Z_{p} (cfr. §1) and this purely as a consequence of condition _{K_{j},1}S_{B_{r}} for one j in {1, …, l}.

5.3 Since each field (K_{j}, | − |_{j}) is an algebraic closed field of characteristic zero, complete under the absolute value | − |_{j}, in order to prove theorem 4.9 we need, by 5.1, only to prove its following special case:

Theorem 5.4. Suppose

y(x) ∈ (k[[x]][\frac{1}{x}])^{n}

is a formal column solution of a system (5.1.1) in _{k,s}S_{B_{r}} of type ρ. Then y(x) is an element of (k\{\{x\}\}_{rρ}[\frac{1}{x}])^{n}.

Proof. The idea is to reduce theorem 5.4 to corollary 1.7 by the method of the “generic line” ([S-S],[BA2, §3]). We consider a system of indeterminates γ_{1}, …, γ_{s} over k. We denote by k_{γ} the free Tate algebra [BGR]:

k_{γ} = k <γ_{1}, …, γ_{s} >

endowed with the Gauss norm, which is actually an absolute value. Hence we can extend such an absolute value to the quotient field and consider an algebraically closed extension field of k_{γ} complete with respect to an absolute value extending the absolute value in k_{γ}. We will denote this field by K_{γ} and its absolute value still by | − |.

We consider the substitution x_{i} = γ_{i}z, i = 1, …, s. Notice that for an element g(x) ∈ k{{x}}_{r}^{b} we have g(γ_{1}z, …, γ_{s}z) ∈ K_{γ}{{z}}_{r}^{b} and |g(x)|(r) = |g(γ_{1}z, …, γ_{s}z)|(r).

Suppose y(x) satisfies a system of the form (5.1.1) in _{k,s}S_{B_{r}}. We obtain a solution

y(z) = y(γ_{1}z, …, γ_{s}z) ∈ (K_{γ}[[z]][\frac{1}{z}])^{n},

of the system:

(5.4.1)  (z \frac{d}{dz} − G(z))y(z) = 0

where:

G(z) = ∑_{i=1}^{s} G_{δ_{i}}(γ_{1}z, …, γ_{s}z) ∈ M_{n}(B_{r})

(B_{r} = K_{γ}{{z}}_{r}^{b} is defined as in §0, with k replaced by K_{γ}). By [BA2, Lemma 3.3.2], for any ϵ > 0, y(z) converges for |z| < ϵ if and only if y(x) converges in the open polydisk {x ∈ K_{γ}^{s} | |x_{i}| < ϵ for i = 1, …, s}. We refer to (5.4.1) as the system induced on the generic line by (5.1.1).

We shall now prove that the system (5.4.1) is an element of S_{B_{r}} (cf. §1) and that its type is also ρ. If we denote by G_{m}(z), m ∈ N, the recursion matrices for the ordinary system (5.4.1) (according to (1.1.3)), and retain formulas (4.4.4) and (4.4.5) for the system (5.1.1), we have:

Lemma 5.4.2. For each m ∈ N:

(5.4.3)  \frac{G_{m}(z)}{m!} = ∑_{|α|_{R}=m} \frac{G_{α}(γ_{1}z, …, γ_{s}z)}{α!}.

Proof. We prove formula (5.4.3) by induction on m. We suppose (5.4.3) true for m. By applying (1.1.3) to (5.4.1) we obtain:

(5.4.4)  (m + 1)\frac{G_{m+1}(z)}{(m + 1)!} = z \frac{d}{dz} (\frac{G_{m}(z)}{m!}) + \frac{G_{m}(z)}{m!}(G(z) − m).

By induction, the matrix on the right hand side comes from the matrix

(5.4.5)  ∑_{i=1}^{s} x_{i} \frac{∂}{∂x_{i}} ( ∑_{|α|_{R}=m} \frac{G_{α}(x)}{α!}) + ( ∑_{|α|_{R}=m} \frac{G_{α}(x)}{α!})(∑_{i=1}^{s} G_{δ_{i}}(x) − m)

simply by the substitution x_{i} = γ_{i}z, i = 1, …, s. By rearranging the terms, (5.4.5) is seen to be equal to

(5.4.6)  ∑_{i=1}^{s} ∑_{|α|_{R}=m}[x_{i} \frac{∂}{∂x_{i}} (\frac{G_{α}(x)}{α!}) + \frac{G_{α}(x)}{α!}(G_{δ_{i}}(x) − α_{i})].

By (4.4.5) we have:

(5.4.7)  (α_{i} + 1)\frac{G_{α+δ_{i}}(x)}{(α + δ_{i})!} = x_{i} \frac{∂}{∂x_{i}} (\frac{G_{α}(x)}{α!}) + \frac{G_{α}(x)}{α!}(G_{δ_{i}}(x) − α_{i}).

Therefore (5.4.6) is equal to

∑_{i=1}^{s} ∑_{|α|_{R}=m}(α_{i} +1)\frac{G_{α+δ_{i}}(x)}{(α + δ_{i})!} = ∑_{i=1}^{s} ∑_{|β|_{R}=m+1} β_{i} \frac{G_{β}(x)}{β!} = (m+1) ∑_{|α|_{R}=m+1} \frac{G_{α}(x)}{α!}.

We conclude that (5.4.3) holds with m replaced by m + 1. Q.E.D.

Condition S_{B_{r}} may be expressed for the system (5.4.1) with G(z) ∈ M_{n}(B_{r}), by

lim_{m→∞} |\frac{G_{m}(z)}{m!}|(r)η^{m} = 0

for every 0 < η < 1. This holds as a consequence of lemma 5.4.2 , because:

|\frac{G_{m}(z)}{m!}|(r) = | ∑_{|α|_{R}=m}(\frac{G_{α}(γ_{1}z, …, γ_{s}z)}{α!})|(r) =

(5.4.8)  = sup_{|α|_{R}=m} |\frac{G_{α}(γ_{1}z, …, γ_{s}z)}{α!}|(r) = sup_{|α|_{R}=m} |\frac{G_{α}(x)}{α!}|(r).

As for the type of (5.4.1), we simply notice that its exponents are precisely the numbers μ = λ_{1} + … + λ_{s} each taken with multiplicity

e(μ) = ∑ ; ∑ λ_{i}=μ e(λ_{1}, …, λ_{s})

where e(λ_{1}, …, λ_{s}) is the multiplicity of (λ_{1}, …, λ_{s}) as multiexponent of (4.7.0). From the previous discussion it follows that y(z) is a formal solution of the system (5.4.1), which is in the set S_{B_{r}} and is of type ρ . We can then apply corollary 1.7 to conclude that y(z) is meromorphic in the disk of radius rρ with possible pole only at z = 0.The formal solution y(x) therefore converges in the open polydisk of radius rρ with possible poles at x_{1} … x_{s} = 0. This concludes the proof of theorem 5.4 and also of theorem 4.9. Q.E.D.

Remark 5.5 Consider a system (cf. (4.7.0))

(5.5.1)  x_{i} \frac{∂}{∂x_{i}} y = G_{δ_{i}}(x)y  i = 1, …, s

with G_{δ_{i}}(x) ∈ M_{n}(A{{x}}_{r}^{b}) (resp. G_{δ_{i}}(x) ∈ M_{n}(A{{x}}_{r})) in _{A,s}S_{B_{r}} (resp. in A,sS_{A_{r}}). Let us assume there exists a system (cf. [BA-CT2]):

(5.5.2)  x_{i} \frac{∂}{∂x_{i}} y = C_{δ_{i}}(G)y  i = 1, …, s

with C_{δ_{i}}(G) ∈ M_{n}(A) is in _{A,s}S_{B_{r}} (resp. in _{A,s}S_{A_{r}}) formally equivalent to (5.5.1). This means that there exists a matrix Y (x) ∈ GL_{n}(A[[x]][\frac{1}{x}]) satisfying:

(5.5.3)  x_{i} \frac{∂}{x_{i}∂} Y (x) = G_{δ_{i}}(x)Y (x) − Y (x)C_{δ_{i}}(G), i = 1, …, s.

Then Y (x) is a solution of a standard system of rank n^{2} (where Y (x) is written as a column vector in some definite order):

(5.5.4)  x_{i} \frac{∂}{∂x_{i}} Y (x) = G̃_{δ_{i}}(x)Y (x)  i = 1, …, s

with G̃_{δ_{i}}(x) ∈ M_{n^{2}}(A{{x}}_{r}^{b}) (resp.G̃_{δ_{i}}(x) ∈ M_{n^{2}}(A{{x}}_{r})). Under the previous assumptions the system (5.5.4) satisfies property _{A,s}S_{B_{r}} (resp. _{A,s}S_{A_{r}}). To show this, we may restrict ourselves to the case of systems in _{A,s}S_{B_{r}}. By use of the arguments explained in 5.1, we need only check the assertion in the case when the ring A is replaced by a valued field complete and algebraically closed. For any of the systems (5.5.j), j = 1, 2, 3, 4, where we assume A = k, let (5.5.j)_{γ} denote the system induced by (5.5.j) on the generic line x_{i} = γ_{i}z, i = 1, …, s. Lemma 5.4.2 shows that (5.5.4) satisfies _{k,s}S_{B_{r}} iff (5.5.4)_{γ} satisfies _{K_{γ},1}S_{B_{r}} . On the other hand (5.5.4)_{γ} is the system obtained from (5.5.1)_{γ} and (5.5.2)_{γ} as (1.5.1) was obtained from (δ − G)y = 0 and (δ − C(G))y = 0 . Remark 1.5 then shows that (5.5.4)_{γ} satisfies _{K_{γ},1}S_{B_{r}} . The theory developed in the present paper, shows then that, if (5.5.4) is of type ρ > 0, the systems (5.5.1) and (5.5.2) are equivalent over A\{\{x\}\}_{rρ}[\frac{1}{x}]. Actually, under the previous assumptions, the type of (5.5.4) may be foreseen from the types of the differences of the multiexponents of (5.5.1), exactly as in the 1-variable case.

Our next step will be to show that, under certain circumstances, there exists a system of the type of (5.5.2), equivalent to (5.5.1) over A[[x]][\frac{1}{x}]. We will show that this is the case when A = k, but also when A is a regular Tate algebra and the system (5.5.1) can be completed to an absolute (i.e. relative to k) integrable system by the action of all continuous derivations of A/k.

This will complete the analogy with Christol’s work [CH1] and will be the subject of a forthcoming paper ([BA-CT2]).

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Authors’ address: Dipartimento di Matematica Università di Padova Via Belzoni 7 I-35131 PADOVA - ITALY E-mail:

”BALDASSARRI@PDMAT1.UNIPD.IT” ”CHIARELLOTTO@PDMAT1.UNIPD.IT”

Fax: (39-49-)8758596