On André’s transfer theorem.

by Francesco Baldassarri and Bruno Chiarellotto

§0 Introduction

In this article we consider a system of linear differential equations of the form

(0.1)  (δ − G)y = 0

where δ = x\frac{d}{dx} and G is an n × n matrix of functions holomorphic in the p-adic open unit disk D(0, 1^{−}). We refer to system (0.1) as to ”the system δ − G”.

We are interested in both understanding and improving André’s results ([A1, Résultat 2], [A2, V.4.2], [CH-DW1, 8.4]), giving information on the radius of meromorphy of the uniform part of the solution matrix of (0.1) at 0, under some mild assumptions on convergence for the solution matrix at a generic point. As in our previous article [BA-CT0], we rather take the viewpoint of estimating the radius of p-adic meromorphy of formally meromorphic solutions of (0.1) at 0. We take advantage of arguments used by Christol and Dwork ([CH-DW1]) in the construction of a Frobenius transform of (0.1) when G(0) is nilpotent. We are able to extend those arguments to the general case. This generalization and the strong form of the so-called (cf.[CH2, 4.8.1]) Dwork-Frobenius theorem proved in [CH-DW2, 4.2], permit some improvement upon previous results. The reader should compare part (i) of Corollary 3.5 with [A1, loc.cit.], [A2, loc.cit.], [CH-DW1, loc. cit.], [CH-DW2, 4.4], noticing that we are not imposing any restriction on the Jordan form of G(0). Part (ii) of Corollary 3.5, however, gives sharper results in the semisimple case; the results of section 3 of [DW-VP] follow from that statement.

As in all the previous authors, the technique used here is that of iterating the Frobenius transformation in order to refine a first crude estimate of the radius of convergence of the solutions of (0.1) at 0.

We are indebted to Professor Dwork for his encouragement and for showing us remarkable simplifications on our original proofs. We acknowledge some useful discussions with G.Gerotto.

We hope our considerations will lead to improvements in estimates for the growth of solutions of (0.1) at 0 when approaching the boundary of their disk of convergence ([CH-DW1]).

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

1991 Mathematics Subject Classification. Primary 12H25.

§1 Notation and basic results ([CH-DW1],[CH-DW2], [BA-CT0]).

1.1 We recall the notation of [BA-CT0].

p= a prime number;

k= an algebraically closed field of characteristic zero, complete under a non-archimedean absolute value

| | : k ⟶ R_{≥0}

extending the p-adic absolute value of Q_{p} and normalized by |p| = p^{−1}; we also use the additive form:

ord(λ) = −\frac{log |λ|}{log p}

for λ ∈ k ( ord(0) = ∞); V= ring of integers of k. 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 (Ω, | − |) a universal domain for k and t a generic point (for k). For ρ ≥ 0 and c ∈ Ω we will denote by D(c, ρ^{−}) (resp. D(c, ρ)) the open (resp. closed) disk of radius ρ centered at c, in Ω:

D(c, ρ^{−}) = {x ∈ Ω | |x − c| < ρ}

(resp.

D(c, ρ) = {x ∈ Ω | |x − c| ≤ ρ}).

We also put:

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 (defined over k) on D(0, 1^{−}). The elements of E_{∘} are uniform limits on D(0, 1^{−}) of sequences of rational functions defined over k 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}.

The type of a number is invariant under translations by integers.

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

(1.2.2)  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 a k-linear transformation h : V → V of a finite dimensional k-vector space V, we will refer to the type of any matrix representation of h as the type of h. 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 then denote it by ρ(G).

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

(1.3.1)  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. We recall Lemma 1.10 of [BA-CT0]:

Lemma 1.4. Let G, H ∈ M_{n}(k[[x]]) and P ∈ GL(n, k((x))) be 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.

Corollary 1.5. Let G, H ∈ M_{n}(k[[x]]) be such that the systems δ − G and δ − H are equivalent over k((x)). Then ρ(G) = ρ(H).

We now have:

Lemma 1.6. If λ, λ′ ∈ V are such that p^{a}λ′ − λ ∈ Z, for some a = 0, 1,…, then ρ(λ′) = ρ(λ)^{p^{a}}.

Proof. We can assume a = 1; the proof of [BA-CT0] then applies to the present case. Q.E.D.

1.7 We recall that a point c ∈ Ω is said to be an apparent singularity for the system δ − G, where G is meromorphic in a neighborhood of c, if there exists a full solution matrix of δ − G meromorphic at c. Let H be the set of all matrices G ∈ M_{n}(E′_{∘}) satisfying the following conditions:

H1. G is holomorphic at 0.

H2. The system (δ−G)y = 0 has at most apparent singularities in D(0, 1^{−})\ {0}.

If G ∈ H , the entries of G are meromorphic in the region

D = D(0, 1^{−}) ∪ D(t, 1^{−}).

Let G be a matrix in H and c ∈ D. Let

S_{c}(G) = Sol(δ − G, Ω((x − c)))

denote the Ω-vector space of vector solutions y ∈ Ω((x − c))^{n} of (δ − G)y = 0. For G ∈ H and c ∈ D we then define:

R_{c}(G) = inf(1, maximum common radius of meromorphy of the elements of S_{c}(G))

We note that R_{c}(G) = R_{c}(G_{[H]}) if H ∈ GL(n, E′_{∘}). If G is holomorphic at c, we denote by U_{G,c}(x) ∈ M_{n}(Ω[[x − c]]) the solution matrix of the system δ − G normalized by U_{G,c}(c) = I_{n}. 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:

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

We then have, for G ∈ H:

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

Notice that for G ∈ H, we have:

(1.7.3)  G_{m}(0) = G(0)(G(0) − I_{n}) … (G(0) − (m − 1)I_{n}) .

Lemma 1.8. Let σ : k/Z ⟶ k be a section of the canonical projection k ⟶ k/Z. Let G ∈ M_{n}(k[[x]]) There exists S ∈ GL(n, V[x, x^{−1}]) such that G_{[S]} ∈ M_{n}(k[[x]]) and G_{[S]}(0) is an upper triangular matrix with diagonal entries in the image of σ.

Proof. Cf.[CH-DW1, 2.2 and 2.3]. Q.E.D.

We will refer to the transformation G ⟼ G_{[S]} of the previous lemma as a shearing transformation.

Lemma 1.9 (Elimination of apparent singularities). Let G be an element of H. Then there exists H ∈ Gl(n, k(x)), H unimodular, H and H^{−1} analytic at 0, such that G_{[H]} ∈ M_{n}(E_{∘}).

Proof. Cf. [CH-DW1,2.4], [CH2, 5.4.3].Q.E.D.

Lemma 1.10 (Dwork-Frobenius). Let G be an element of H. Then there exists H ∈ Gl(n, E′_{∘}), such that G_{[H]} ∈ H and

|G_{[H]}| ≤ sup(1, \frac{|π|}{R_{t}(G)}).

Proof. Cf. [CH-DW2,4.2].Q.E.D.

Lemma 1.11 (The trivial estimate). Let G be an element of H. Then

R_{0}(G) ≥ |π|^{n}ρ(G) max(1, |G|)^{−n}.

Proof. It follows from [BA-CT0,2.3], after elimination of apparent singularities. Q.E.D.

Corollary 1.12 (The basic estimate). Let G be an element of H. Then

R_{0}(G) ≥ |π|^{n}ρ(G) sup(1, \frac{|π|}{R_{t}(G)})^{−n}.

Proof. We reduce to the previous statement via Dwork-Frobenius.Q.E.D.

§2 Analytic calculations.

For a matrix α ∈ M_{n}(k), we define its spectral norm |α|_{sp} as:

|α|_{sp} = max{|λ| | det(λI_{n} − α) = 0}.

Let α ∈ M_{n}(k); we define a matrix of formal power series:

(2.1)  θ(α, x) = ∑_{s=0}^{∞} \binom{α}{s}x^{s} ∈ M_{n}(k[[x]]).

This applies in particular to the case n =1, for α = λ ∈ k, to give a formal power series θ(λ, x) ∈ k[[x]] . We recall that d(λ, Z_{p}) denotes the distance between λ and Z_{p}.

Proposition 2.2. Let l ∈ N, l = 1, 2, 3 …, and λ ∈ k. Suppose that the series θ(λ, x) converges in the closed disk D(0, |π|^{\frac{1}{p^{l−1}}} ) i.e. that:

(2.2.1)  lim_{m→+∞} |\binom{λ}{m}| |π|^{\frac{m}{p^{l−1}}} = 0

Then d(λ, Z_{p}) < p^{−l+1}.

Proof. (Cf. [CH2, 6.2.9], [CL]). Our hyphothesis implies |λ| ≤ 1. In fact otherwise:

|\binom{λ}{p^{m}}| > \frac{1}{|p^{m}!|} = p^{\frac{p^{m}−1}{p−1}} .

But

p^{\frac{p^{m}−1}{p−1}} |π|^{\frac{p^{m}}{p^{l−1}}} = p^{\frac{p^{m}−p^{m−l+1}−1}{p−1}} ,

which does not converge to 0 for m → +∞.

Suppose on the other hand that λ ∉ Z_{p}. Let

d(λ, Z_{p}) = p^{−r}

where r is a positive real number and r = σ + ϵ, where σ = [r] is the integral part of r and ϵ ∈ [0, 1).

Consider m ∈ N such that

|λ + m| = p^{−r}.

For s ∈ N and s > m we have:

(λ)_{s} = (λ)_{m}(λ + m)_{s−m}.

It follows that for each s ∈ N with s > m:

|(λ)_{s}| = c|(λ + m)_{s−m}|,

where c ∈ R_{>0} is a constant independent of s. Let α = λ + m, so that ord(α) = σ + ϵ ≥ ord(α + i), ∀i = 0, 1, 2,. . .. An elementary calculation shows that :

ord(α)_{s} = ord(α) + [\frac{s − 1}{p}] + [\frac{s − 1}{p^{2}}] … + [\frac{s − 1}{p^{σ}}] + ϵ[\frac{s − 1}{p^{σ+1}}]

Therefore

ord(λ)_{s} = s(\frac{1}{p}+\frac{1}{p^{2}} +…+\frac{1}{p^{σ}})+\frac{sϵ}{p^{σ+1}} +O(1) = s(\frac{1}{p − 1} − \frac{1}{p^{σ}} \frac{1}{p − 1} + \frac{ϵ}{p^{σ+1}} )+O(1),

where O(1) denotes a constant depending upon d(λ, Z_{p}) but not upon λ,s. For fixed λ and s → +∞ we then get the asymptotic estimate:

ord\binom{λ}{s} = s(−\frac{1}{p^{σ}} \frac{1}{p − 1} + \frac{ϵ}{p^{σ+1}} ) + O(log(s)) = \frac{s}{p^{σ}} (\frac{ϵ}{p} − \frac{1}{p − 1}) + O(log(s)).

So, the series θ(λ, x) converges exactly for:

|x| < p^{\frac{1}{p^{σ}} ( \frac{ϵ}{p} − \frac{1}{p−1} )}.

Our hypotheses then say that :

−\frac{1}{p^{l−1}(p − 1)} < \frac{1}{p^{σ}} (\frac{ϵ}{p} − \frac{1}{p − 1}).

Then, if ϵ = 0 we have:

\frac{1}{p^{l−1}(p − 1)} > \frac{1}{p^{σ}(p − 1)}

which implies l − 1 < σ. If on the other hand ϵ ≠ 0, we have:

\frac{1}{p^{l−1}(p − 1)} > \frac{1}{p^{σ}} (\frac{1}{p − 1} − \frac{1}{p})

i.e.

\frac{1}{p^{l−1}(p − 1)} > \frac{1}{p^{σ+1}(p − 1)},

so l − 1 < σ + 1. But σ, l ∈ N, so that

l − 1 < σ + ϵ.

Q.E.D.

We may generalize the previous result:

Proposition 2.3. Suppose that for a matrix α ∈ M_{n}(k) the series θ(α, x) converges in the closed disk

D(0, |π|^{\frac{1}{p^{l−1}}} )

for some positive integer l. Then for each eigenvalue λ of α,

d(λ, Z_{p}) < p^{1−l}.

Proof. This is a consequence of 2.2. Q.E.D.

2.4 For α ∈ M_{n}(k) , we have the formal identity (cf. [CH-DW1, 2.8]):

(2.4.1)  ∑_{s=0}^{+∞} \binom{α}{s}x^{s} = ∑_{j=0}^{+∞} \frac{α^{j}}{j!} L(x)^{j}

where

L(x) = log(1 + x) = ∑_{i=1}^{+∞}(−1)^{i−1} \frac{x^{i}}{i} .

Lemma 2.5. Let l be a positive integer and assume λ ∈ k is such that |λ| < p^{1−l}. Then the series of functions

∑_{j=0}^{+∞} \frac{λ^{j}}{j!} L(x)^{j}

converges uniformly in D(0, |π|^{\frac{1}{p^{l−1}}} ).

Proof. Consideration of the Newton polygon of

L(x) = ∑_{i=1}^{+∞}(−1)^{i−1} \frac{x^{i}}{i}

shows that if |x| ≤ |π|^{\frac{1}{p^{l−1}}} = p^{− \frac{1}{p^{l−1}(p−1)}} then

|L(x)| ≤ p^{l−1− \frac{1}{p−1}} = |π|p^{l−1}.

The statement now follows from our hyphotheses about λ. Q.E.D. Remark 2.5.1. One may avoid the use of Newton polygons by the following argument. For j = 1, 2,…, let

h(j) = \frac{j}{(p − 1)p^{l}} − ord(j) .

It is enough to show that

inf_{j≥1} h(j) ≥ −l + \frac{1}{p − 1} .

Since p |̸m implies h(mj) ≥ h(j), we may restrict our attention to j = 1, p, p^{2},. . .. Let h_{1}(s) = h(p^{s}). A trivial calculation shows that h_{1}(s) ≤ h_{1}(s+1) if and only if s ≥ l. Thus the maximum of h is assumed at j = p^{l}, which completes the proof.

As a consequence of the previous lemma we can state:

Corollary 2.6. Let l be a positive integer and let λ ∈ k be such that |λ| < p^{1−l}. Then:

(i) θ(λ, x) converges in the closed disk D(0, |π|^{\frac{1}{p^{l−1}}} )

(ii) If ζ ∈ k satisfies ζ^{p^{l}} = 1, we have:

θ(λ, ζ − 1) = 1.

Proof. For the (i) part we simply notice that formula (2.4.1) presents θ(λ, x) as the Taylor expansion at 0 of the analytic function

∑_{j=0}^{+∞} \frac{λ^{j}}{j!} L(x)^{j};

we then apply lemma 2.5.

For (ii), we remark that if ζ ∈ k satisfies ζ^{p^{l}} = 1 then |ζ − 1| = |π|^{\frac{1}{p^{l−1}}} . Q.E.D.

Finally we can state:

Theorem 2.7. Let l a positive integer and α ∈ M_{n}(k) be such that

|α|_{sp} < p^{1−l}.

Then if ζ ∈ k satisfies ζ^{p^{l}} = 1,

θ(α, ζ − 1) = I_{n}.

Proof. We decompose α ∈ M_{n}(k) as

α = α_{ss} + α_{nilp}

where α_{ss}, α_{nilp} ∈ M_{n}(k), α_{ss} is semisimple, α_{nilp} is nilpotent and

[α_{nilp}, α_{ss}] = 0.

We then obtain the formal identity:

(2.7.2)  θ(α, x) = θ(α_{nilp}, x)θ(α_{ss}, x).

In [CH-DW1, 2.8] it is proved that:

a) θ(α_{nilp}, x) converges in D(0, 1^{−}),

b) θ(α_{nilp}, ζ−1) = I_{n} for any ζ ∈ k such that ζ^{p^{N}} = 1, for some N = 1, 2,. . .. We may then suppose α semisimple and finally α = λ ∈ k (so, λ is an

eigenvalue of the original α) .

The hypotheses of the theorem imply that |λ| < p^{1−l}: we then apply part (ii) of corollary 2.6. Q.E.D.

§3 Frobenius action and main theorem.

For a positive integer l, q = p^{l}, and u ∈ Ω((x)) we define by u^{ϕ_{q}} to be the composition x → u(x^{q}). We sometimes informally refer to the map x → x^{q} as the Frobenius transformation of order l. We consider, for l = 1, 2, 3 … and G ∈ H, the condition :

H_{l} : The formal matrix solution U_{G,t} converges in the closed disk D(t, |π|^{\frac{1}{p^{l−1}}} ).

We will also denote by H_{l} the set of matrices in H satisfying condition H_{l}.

Proposition 3.1. If G ∈ H_{l} there exists a unimodular H ∈ GL(n, k(x)) such that the following two conditions hold:

(i) G_{[H]} ∈ H_{l} ∩ M_{n}(E_{∘}),

(ii) for each eigenvalue λ of G_{[H]}(0)

d(λ, Z_{p}) < p^{1−l}.

Proof. We use the matrix H of lemma 1.9 to eliminate apparent singularities. The fact that H is holomorphic in the open disk D(t, 1^{−}), while det H does not vanish on that disk, shows that G_{[H]} ∈ H_{l}. According to formulas (1.7.2) and (1.7.3), our hypotheses imply that

θ(G_{[H]}(0), x)

converges in D(0, |π|^{\frac{1}{p^{l−1}}} ). We then apply proposition 2.3. Q.E.D.

Corollary 3.2. Let G ∈ H_{l}. There exists a unimodular U ∈ GL(n, k(x)) such that G_{[U]} ∈ H_{l} ∩ M_{n}(E_{∘}) and

|G_{[U]}(0)|_{sp} < p^{1−l}.

Proof. We write, using the matrix H of the previous lemma 3.1,

U = T H

for a suitable “ shearing transformation” (cf.(1.8)) T ∈ GL(n, V[x, x^{−1}]) .Q.E.D.

In the sequel we will refer to the matrix G_{[U]} of corollary 3.2 as a normal form of G ∈ H_{l}. A matrix G ∈ H_{l} will be said to be in normal form , if G ∈ H_{l} ∩ M_{n}(E_{∘}) and |G(0)|_{sp} < p^{1−l}.

We can finally state :

Theorem 3.3. Let G ∈ H_{l} be in normal form. Then there exists H ∈ M_{n}(E_{∘})∩ GL(n, E′_{∘}), H(0) = I_{n} and F ∈ H such that for q = p^{l}

(3.3.1)  qF^{ϕ_{q}} = G_{[H]}.

We have

(3.3.2).  ρ(F (0)) = ρ(G(0))^{q}

For each c ∈ D the map:

S_{c^{q}}(F) ⟶ S_{c}(G)
y ⟶ H^{−1}y^{ϕ_{q}}

is an isomorphism of Ω-vector spaces. In particular, for c = 0:

(3.3.3)  R_{0}(F) = R_{0}(G)^{q} ,

and, for c = t:

(3.3.4)  R_{t}(F) = R_{t^{q}}(F) = R_{t}(G)^{q} .

Proof. We follow the proof of Theorem 8.2 of [CH-DW1].

For G ∈ H_{l} we define a series of functions of (x, z) ∈ (D \ {0}) × (D \ {0}) (cf. (1.7.1))

Y(x, z) = ∑_{s=0}^{+∞} \frac{G_{s}(x)}{s!}(\frac{z − x}{x})^{s}.

By our hypotheses this is a convergent series in the region

(x, z) ∈ (D \ \{0\}) × (D \ \{0\}) , |\frac{z − x}{x}| ≤ |π|^{\frac{1}{p^{l−1}}} .

For c, x, z ∈ D \ {0} and for

|\frac{z − x}{x}| ≤ |π|^{\frac{1}{p^{l−1}}} , |\frac{x − c}{c}| ≤ |π|^{\frac{1}{p^{l−1}}}

we have:

U_{G,c}(z) = Y(x, z)U_{G,c}(x).
For q = p^{l}, we define:

H = \frac{1}{q} ∑_{ζ^{q}=1} Y(x, ζx) = ∑_{m=0}^{+∞}( ∑_{ζ^{q}=1} \frac{(ζ − 1)^{m}}{q})\frac{G_{m}(x)}{m!} .

Since for ζ^{q} = 1

|ζ − 1| ≤ |π|^{\frac{1}{p^{l−1}}} ,

H(x) ∈ M_{n}(E_{∘}). We now calculate H(0). We have:

∑_{m=0}^{+∞} \frac{G_{m}(0)}{m!}(ζ − 1)^{m} = θ(G(0), ζ − 1) = I_{n}

by theorem 2.7 and by our hypothesis that G be in normal form; therefore

H(0) = \frac{1}{q} ∑_{ζ^{q}=1} θ(G(0), ζ − 1) = I_{n}.

We conclude that H(x) ∈ Gl(n, E′_{∘}). For the remaining part of the proof we refer to [CH-DW1, 8.2]. In particular we observe that (8.2.2) of loc. cit. implies that F ∈ M_{n}(E′_{∘}) by Lemma 3.2 of [BA-CT0]. Notice that, while (3.3.3) is obvious, (3.3.4) follows from [CH-DW1, Lemma 6.3] and the hypothesis H_{l}. Q.E.D.

Finally our main theorem:

Theorem 3.4. Let G ∈ H. Then:

R_{0}(G) ≥ ρ(G(0))R_{t}(G)^{n}.

Proof. If G does not belong to H_{1} we deduce the result from the basic estimate (1.12). Otherwise, let l be maximal such that G ∈ H_{l}; we may assume G to be in normal form. In the notation of theorem 3.3, we have R_{0}(G) = R_{0}(F)^{\frac{1}{q}}, ρ(F (0)) = ρ(G(0))^{q}, R_{t}(F) = R_{t^{q}}(F) = R_{t}(G)^{q} and

R_{0}(F) ≥ |π|^{n}ρ(F (0)) sup(1, \frac{|π|}{R_{t}(F)})^{−n}

by the basic estimate (1.12). We then deduce

(3.4.1)  R_{0}(G) ≥ |π|^{\frac{n}{q}} ρ(G(0)) sup(1, \frac{|π|}{R_{t}(G)^{q}})^{− \frac{n}{q}} .

By the maximality of l

|π|^{\frac{1}{p^{l−1}}} ≤ R_{t}(G) ≤ |π|^{\frac{1}{p^{l}}} ,

so that, for q = p^{l}

|π|^{p} ≤ R_{t}(G)^{q} ≤ |π|,

now (3.4.1) gives

R_{0}(G) ≥ |π|^{\frac{n}{q}} ρ(G(0))(\frac{R_{t}(G)^{q}}{|π|})^{\frac{n}{q}} ≥ ρ(G(0))R_{t}(G)^{n} .

Q.E.D.

Corollary 3.5. Let G ∈ H and let ρ be the type of the linear transformation:

M_{n}(k) ⟶ M_{n}(k)
X ⟼ G(0)X − XG(0).

(i) Let Y_{G}(x)x^{G(0)} (cf. [CH-DW1, Introduction], [A2, III.1.1]) be a formal solution matrix of (0.1) at 0, with Y_{G}(x) ∈ M_{n}(k((x))). Then Y_{G}(x) is meromorphic in the open disk

D(0, R_{t}(G)^{n^{2}}ρ^{−}) .

(ii) Let λ ∈ k be an eigenvalue of G(0), and let y(x)x^{λ}, with y(x) ∈ k((x))^{n}, be a formal column solution of (0.1) at 0. Then y(x) is meromorphic in the open disk

D(0, R_{t}(G)^{n}ρ(G(0) − λI_{n})^{−}) .

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[CH-DW 2] Christol G., Dwork B. : “ Differential modules of bounded spectral norm ”, in this volume.

[CL] Clark D. : “A note on the p-adic convergence of solution of linear differential equations”, Proc. Am. Math. Soc. 17 (1966), 262-269.

[DW-VP] Dwork B., van der Poorten A. : “The Eisenstein constant”, Duke Math. Journal 65 (1992), 23-43.

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