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Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces

We consider a vector bundle with integrable connection (\cE,\na) on an analytic domain U in the generic fiber \cX_{\eta} of a smooth formal p-adic scheme \cX, in the sense of Berkovich. We define the \emph{diameter} \delta_{\cX}(\xi,U) of U at \xi\in U, the \emph{radius} \rho_{\cX}(\xi) of the point \xi\in\cX_{\eta}, the \emph{radius of convergence} of solutions of (\cE,\na) at \xi, R(\xi) = R_{\cX}(\xi, U,(\cE, \na)). We discuss (semi-) continuity of these functions with respect to the Berkovich topology. In particular, under we prove under certain assumptions that \delta_{\cX}(\xi,U), \rho_{\cX}(\xi) and R_{\xi}(U,\cE,\na) are upper semicontinuous functions of \xi; for Laurent domains in the affine space, \delta_{\cX}(-,U) is continuous. In the classical case of an affinoid domain U of the analytic affine line, R is a continuous function.


Introduction
The classical existence theorem of Cauchy [14, Chap.I] for local solutions of an analytic differential system at an ordinary point does not hold in general for differential equations on a smooth Berkovich analytic space X over a p-adic field k. We recall [3, 1.2.2] that to any point ξ ∈ X one associates a completely valued extension field H (ξ) of k, called the residue field at ξ; the point ξ is k-rational if H (ξ) = k. Any k-rational point ξ of X admits a neighborhood isomorphic to a polydisk centered at the origin O in an affine (analytic) k-space, the isomorphism sending ξ to O. However, the neighborhoods of a nonrigid point are in general too coarse. So, a differential equation does in general have no solutions analytic in a full neighborhood of a non-rigid point ξ ∈ X, even if the point is not a singularity of the equation. In the very inspiring paper [1, §3] Y. André concentrates on differential equations which after pull-back to a finiteétale covering admit a full set of multivalued analytic solutions. For such differential equations there is a notion of global monodromy group close to the one in the complex case. It would be interesting to pursue André's investigation into a description of integrable analytic connections locally for thé etale topology of [4]. But this is not our approach here: we use the natural topology on Berkovich analytic spaces and regard anétale covering f : Y → X as producing a highly non-trivial connection ( Moreover, the problem of the failure of Cauchy existence theorem would not be overcome in general by using someétale topology. On the other hand, it is possible and sometimes convenient to recover Cauchy's theorem at any given point ξ ∈ X, by performing the extension of scalars to X ⊗H (ξ), and passing to some canonical point ξ ′ of this space above ξ. This viewpoint has been systematically used by Dwork and Robba in their study of p-adic differential equations.
We actually assume that X comes with a local notion of distance, measured in terms of an embedding of X as an analytic domain in the generic fiber X η of a smooth formal scheme X over k • . This does not mean that we privilege formal schemes over k • ork-schemes, over k-analytic spaces. The formal model of X is here a technical tool for expressing "local" radii of convergence of solutions of differential equations in the above sense. We will show in a subsequent paper that certain expressions in these local radii are in fact absolute invariants of a connection on an analytic space.
In practice, we consider all over this paper the following Situation 1.1. The smooth formal scheme X = SpfA, is affine andétale over A d k • , the formal affine space over k • , with formal coordinates x = (x 1 , . . . , x d ). Then A := A ⊗ k, X = M(A ), and U is an analytic domain in X.
We are given an integrable system of partial differential equations of the form for y a column vector of unknown functions and G i a µ × µ matrix of analytic functions on U . Notice that, for any k-rational point ξ ∈ X, theétale map x : X → A d k to the affine k-analytic space of dimension d, admits a unique local section σ ξ : D d k (x(ξ), 1 − ) → X, sending x(ξ) to ξ, where The image of σ ξ will be denoted D X (ξ, 1 − ), and called the open disk of radius 1 centered at the k-rational point ξ ∈ X. Similarly, we define open and closed disks D X (ξ, r ± ), of radius r < 1 centered at ξ. Notice that we use the term "disk" to refer to "polydisk with equal radii". The diameter δ X (ξ, U ) of U at the k-rational point ξ, is the radius of the maximal open disk centered at ξ and contained in U , a notion obviously independent of the choice of the formal coordinates x on X . Then 0 < δ X (ξ, U ) ≤ 1 because, on the one hand, a k-rational point of U is necessarily an interior point of U in X; on the other hand, disks of radii ≥ 1 are not defined. Now, when ξ ∈ U is a k-rational point of U , the definition of the radius of convergence of the system (1.1.1) at ξ is completely natural. It is the radius r = R X (ξ, Σ) = R(ξ, Σ) of the maximal open disk D X (ξ, r − ) contained in U , where a fundamental solution matrix Y of (1.1.1) at ξ converges. Notice that Y is a matrix with and its convergence is really tested in D d k (x(ξ), 1 − ).
its radius of convergence is, as in the classical case, where |α| ∞ = α 1 + · · · + α d , and where the norm of a matrix is the maximum absolute value of its entries. Notice that the disk of radius R(ξ, Σ), centered at ξ ∈ A d k , is not necessarily contained in U , as the example of the trivial connection G i = 0, ∀i, on a small disk U ⊂ A d k shows. But we insist on defining The reason is that the determinant of the matrix Y may vanish at a point ζ ∈ D(x(ξ), R(ξ, Σ) − )\ D(x(ξ), δ X (ξ, U ) − ), while this cannot be the case in D(x(ξ), R(ξ, Σ) − ), otherwise the differential system for the wronskian w := det Y , namely would have a singularity in U , which is not the case. Notice that R(ξ, Σ) is then the maximum real number r ≤ 1 such that the system Σ admits a solution matrix Y ∈ GL(d, O(D X (ξ, r − ))). The advantage, and the intrinsic content, of this definition may be better appreciated if we consider the category MIC(U/k) of coherent O U -modules with integrable connection (E, ∇), ∇ : E → E ⊗ Ω 1 U/k , and the object (E := O µ X , ∇) associated to Σ. If e = (e 1 , . . . , e µ ) denotes the canonical basis of global sections of O µ X , then, by convention, so that Σ is the differential system satisfied by the horizonal sections of (E, ∇). The abelian sheaf E ∇ = Ker(∇ : E → E ⊗ Ω 1 U/k ) for the G-topology of U , is not in general locally constant. If, on some analytic domain V ⊂ U , E ∇ |V is locally constant, then it is necessarily a local system of k-vector spaces of rank µ on V and the canonical monomorphism is in fact an isomorphism: this is the intrinsic content of our previous statement on the wronskian equation. Taking into account the fact that a locally constant sheaf of finite dimensional k-vector spaces on a disk is necessarily constant, we see that is the maximal open disk centered at ξ ∈ U , and contained in U , where (E, ∇) is isomorphic to the trivial connection (O D , d D ) µ . We may then give the Definition 1.2 (Alternative form). Let (E, ∇) be an object of MIC(U/k), with E locally free of rank µ for the G-topology. For any k-rational point ξ ∈ U , we define the radius of convergence R X (ξ, (E, ∇)) of (E, ∇) at ξ as the maximal open disk D centered at ξ and contained in U , such that (E, ∇) |D is isomorphic to the trivial connection Coming back to the explicit situation (1.1.1), there is a nice compact formula for the solution matrix Y = Y ξ of (1.1.1) at ξ, such that Y ξ (ξ) = I µ . We write By iteration of the system (1.1.1) we obtain, for any α ∈ N d , the equations , µ× µ matrices of functions analytic in U . In particular, G 0 = I µ and G i is now written G [1 i ] = G 1 i , where 1 i = (0, . . . , 0, 1, 0, . . . , 0), with 1 only at the i-th place.
The G α satisfy the recursion relations The Taylor series of the fundamental solution matrix Y ξ of (1. (for the k-rational point ξ, H (ξ) = k, of course) with radius of convergence We now extend the previous definitions to all points ξ ∈ U . The function ξ → R(ξ, Σ) will be defined in general by formula (1.2.5). This amounts to the following consideration on Berkovich analytic spaces. As explained in [4, 1.4], we may consider the ground field extension of U to H (ξ), U H (ξ) = U ⊗ k H (ξ). It is a H (ξ)-analytic space equipped with a canonical compact projection map ψ ξ : U H (ξ) → U , and there is a canonical H (ξ)rational point ξ ′ above ξ. The system (1.1.1) may be viewed, with no change in notation, on , where the field of constants is now H (ξ), and formula (1.2.5) represents the radius of convergence of the fundamental solution matrix Y ξ ′ of (1.1.1), viewed on U H (ξ) at ξ ′ . We then define, for general ξ ∈ U , D X (ξ, r ± ) : , and R(ξ, Σ) = min( R(ξ, Σ), δ X (ξ, U )) (resp. R X (ξ, (E, ∇)) = R X H (ξ) • (ξ ′ , ψ * ξ (E, ∇))). We abusively call D X (ξ, r ± ) the open (resp. closed) disk of radius r centered at ξ ∈ X.
In the situation (1.1), under the further condition that U is a Laurent domain in X, we prove that the function ξ → R(ξ, Σ) is upper semicontinuous on U , for its natural Berkovich topology. A preliminary fact, and this is where we need U to be a Laurent domain in X, is that the function ξ → δ X (ξ, U ) is upper semicontinuous on U . Moreover, if U is the inverse image of a Laurent domain in D d k (0, 1 + ), the function ξ → δ X (ξ, U ) is continuous. If U = X then ξ → R(ξ, Σ) is continuous at the maximal point η X of X. If dim X = 1 and X = A 1 k • , we prove directly that ξ → R(ξ, Σ) is continuous on U . Combining the last two results, we deduce that ξ → R(ξ, Σ) is continuous if dim X = 1 and U is any affinoid neighborhood of η X .
Surprisingly enough, the simple statement above seems to be new even in the case when U is the closed unit disk D k (0, 1 + ) of dimension 1, a case extensively discussed in the literature (cf. [11] and [9] for reference). In the case of an ordinary differential system Σ as (1.1.1) on an annulus U = C(r 1 , r 2 ) = {ξ : r 1 < |x(ξ)| < r 2 } ⊂ A 1 k with 0 < r 1 , a simple convexity argument due to Christol and Dwork [5] shows that the function R is continuous when restricted to the segment of points (r 1 , r 2 ) → C(r 1 , r 2 ), r → t r , where t r = t 0,r is the "generic point at distance r from 0", i.e. the point at the boundary of the disk D k (0, r − ). They actually consider, precisely as we do, the more invariant function It is easy to show [5, 2.3] that the function log r → log R(t r , Σ) is concave (i.e. ∩-shaped), hence continuous, in (r 1 , r 2 ). The function r → R(t r , Σ) is therefore continuous on (r 1 , r 2 ). In this situation, the system is said to be solvable at r 2 if the lim r→r − 2 R(t r , Σ), which certainly exists, is = r 2 (and similarly for r 1 ). Systems solvable at r 2 (resp. r 1 ) are only understood on C(r 2 − ε, r 2 ) (resp. C(r 1 , r 1 + ε)), for small values of ε > 0, by the theory of factorization according to the slopes due to Christol and Mebkhout [7] [8]. In the special case of a Robba system [6, 3.1], i.e. of a system Σ on C(r 1 , r 2 ), such that R(t r , Σ) = r for every r ∈ (r 1 , r 2 ), it follows from Dwork transfer theorems [11,IV.5.2], that R(ξ, Σ) = |x(ξ)|, for every ξ ∈ C(r 1 , r 2 ). 1 This simplest case is of high interest, even (or maybe especially) when its features depend on the existence of a strong Frobenius structure. A notion of exponents is then available, and under an arithmetic condition on them (automatic in case of a strong Frobenius structure) the system admits a Fuchs-type decomposition over C(r 1 , r 2 ) [6] [10].

Our paper deals with the deviation of a system from being of Robba type.
We prove a far-reaching generalization of the Dwork-Robba theorem [11, IV.3.1] on effective bounds for the growth of local solutions (theorem (4.7) and its corollaries). The only difference from the version published by Gachet [13] is the formulation on Berkovich analytic spaces, which is however crucial in our proof of the upper semicontinuity of the radius of convergence (cf. §4.3).
It turns out that Berkovich analytic spaces represent an ideal framework for the study of p-adic differential equations. They contain the generic points in the sense of Dwork and Robba, as honest points. This gives great flexibility to the "rigid" geometry setting and permits in the end to generalize classical one-dimensional results of Dwork, Robba and Christol to analytic spaces.
Acknowledgement. We are indebted to Vladimir Berkovich for considerable help both in the formulation and in the proof of the theorems appearing in this paper. We are also grateful to Yves André and Kiran Kedlaya for showing so much interest in our results.

Generalities and notation
We refer to our assumptions (1.1). For any subset S ⊂ X, the notation || || S will refer to the supnorm on S. For example, for an analytic domain V ⊂ X, let A + V , denote the k-Banach algebra of bounded analytic functions on V , equipped with k-Banach norm || || V .
and the k-affinoid algebra A V will be viewed as a k-Banach algebra via || || V . We will denote by For a matrix G = (g ij ) of elements in a k-Banach algebra (B, || ||), we will set Then ||G · H|| ≤ ||G|| · ||H||, whenever multiplication of matrices makes sense, and the kalgebra M n×n (B) of n × n-matrices with entries in B, equipped with the norm || ||, is a Banach k-algebra.

Ground extension functor and continuity
We need a definition extracted from [4, 1.4]. Let L be any complete valued field extension of k; the ground extension functor associates to any k-analytic space Y an L-analytic space We will be dealing with a family F of functions defined on the analytic spaces over k in a class S = L S L , where S L is a class of L-analytic spaces and L varies over completely valued field extensionof k. We assume that taking values in a fixed topological space S. We will assume that the class S is stable by ground extensions, and that the family F is compatible with base change, in the sense The following general lemma shows that, to prove continuity of the functions in F k , no loss of generality is involved in assuming that the base field k is maximally complete and algebraically closed.
Proof. We first prove that the map ψ L is closed. Let C be a closed subset of Y L . Let y be a point of Y \ ψ L (C), and let D 2 be a compact neighborhood of y in Y . Then The conclusion follows from [12, 2.4].
It follows from the previous lemma that a function on Y is continuous if and only if its lift to Y L is continuous. In particular, This will allow us to assume in certain cases, without loss of generality, that the ground field k is maximally complete and algebraically closed.
We recall here for completeness that a function ϕ : T → R, where T is any topological space is upper semicontinuous or USC (resp. lower semicontinuous or LSC) if ∀t 0 ∈ T and ε > 0, there exists a neighborhood U t0,ε of t 0 in T such that is USC (resp. LSC).

(Semi-) continuity of formal invariants 3.1 Diameter
We work here under the assumptions of (1.1). We recall that we have defined open (resp. closed) disks D X (ξ, r ± ) centered at ξ ∈ X of radius r ∈ (0, 1] (resp. r ∈ (0, 1)). We say that a disk D X (ξ, r ± ) is k-rational if its center ξ may be chosen in X(k).
Proof. We follow the notation of [3, 2.5]. We may assume that ξ is a k-rational point of U , and that be the bounded character corresponding to ξ ∈ U (resp. ξ ∈ D). Notice that χ ξ may be viewed as a bounded k{r −1 X}-homomorphism A U → k. The reduced character χ ξ : , and r i , s j are positive real numbers.

Trivial estimate
We will assume here that the entries of the matrices G i in (1. For any ξ ∈ U we have: Proof. It follows from (1.2.3) that for any α ∈ N d , with α i > 0, we have Recursively we obtain Finally where S p (n) ≤ log p n is the sum of p-adic digits of the natural number n, from which we deduce the formula in the statement.

(Upper semi
, so that f i , g j ∈ A and r i , s j are positive real numbers. We will say that U = X( , with s > 0), and let ξ ∈ U . Then: (resp. In particular, for any Laurent domain U ⊂ X, the function ξ → δ X (ξ, U ) is upper semicontinuous on U . If f, g ∈ k{x 1 , . . . , x d }, the infima (3.3.1), (3.3.2) are realized on a finite set of α ∈ N d , depending only upon U . In particular, if U is a special Laurent domain in X, the function ξ → δ X (ξ, U ) is continuous on U .
If we now assume that f ∈ k{x 1 , . . . , x d }, then lim |α|∞→∞ ||f [α] || X = 0. Then there exists a natural number N such that ||f [α] || X < r, ∀x ∈ X, as soon as |α| ∞ ≥ N . The infimum in (3.3.5) is then really a minimum on the finite set |α| ∞ < N . The function ξ → δ X (ξ, U ) is continuous in this case. We now consider the case of U = X(sg −1 ), g ∈ A . As in the previous case, we extend our spaces to H (ξ), so that we have the canonical point ξ ′ over ξ with |g(ξ ′ )| = |g(ξ)| ≥ s. Suppose that there exists ω ∈ D d H (ξ) ((ξ 1 , . . . .ξ d ), ε + ), for some ε ∈ (0, 1), such that |g(ω)| < |g(ξ)|. We deduce from Corollary 5.6 in the appendix, that g has a zero in the disk In other words, we have proven that δ X (ξ, U ) is precisely the minimum distance of a zero of g from ξ ′ . We use Robba's theory of Newton polygons (cf. corollary 5.6 in the appendix) to obtain an explicit formula. The conclusion is that As in the previous case, the infimum is really a minimum, and if g ∈ k{x 1 , . . . , x d } it is a minimum in a finite set of α's. We conclude as in the previous case.

The global growth estimate
We set ourselves in the situation of (1.1.1). We will need the following estimate, a corollary of the generalized form of the theorem of Dwork and Robba [11, Chap. IV, Thm. 3.1] given below.
Corollary 4.4. For any ε > 0, there exists s ε ∈ N, such that for every α ∈ N d , with |α| ∞ ≥ s ε and every ξ ∈ U (4.4.1) We mention a variation of (4.1), which is often useful. Let cl X (U ) be the closure of U in X, and let A (U) be the localization of the algebra A with respect to the elements which do not vanish on cl X (U ). Let H (U ) ⊂ A + U denote the completion of A (U) in the supnorm || || U . The elements of H (U ) will be called analytic elements on U ; they define continuous real valued functions on cl X (U ). Namely, if h ∈ H (U ) is the uniform limit h = lim and S i does not vanish on cl X (U ), we may define for any limit point ξ ∈ cl X (U ), . Assume the entries of the matrices G i in (1.1.1) are analytic elements on the analytic domain U , and that the function ξ → δ X (ξ, U ) admits a continuous extension on cl X (U ). Then, |G [α] (ξ)| exists ∀ α and ∀ ξ ∈ cl X (U ), and R(ξ, Σ) is defined by formula (1.1.4) ∀ ξ ∈ cl X (U ). Let us define R(ξ, Σ), ∀ ξ ∈ cl X (U ), by formula (1.1.5).
Then we have

The generalized Dwork-Robba theorem
The following discussion, due to Dwork, has been previously made available by Gachet [13]. We rediscuss it here in the framework of Berkovich spaces. We set ourselves in a slightly more general situation, namely we assume that the matrices G i of the system Σ in (1.1.1), are meromorphic functions in a polydisk D(a, r − ), for a = (a 1 , . . . , a d ) ∈ k d , The field M(D(a, r − )) of meromorphic functions on D(a, r − ) is defined as the quotient field of the integral domain O (D(a, r − )). For any ρ = (ρ 1 , . . . , ρ d ), 0 < ρ i < r i , the maximal point t a,ρ of D(a, ρ + ) belongs to D(a, r − ) and defines a multiplicative map M(D(a, r − )) → H (t a,ρ ), f → f (t a,ρ ). For f ∈ M(D(a, r − )), the function ρ → f (t a,ρ ) is continuous, as shown in the appendix, but not necessarily bounded for 0 < ρ i < r i . We define the boundary seminorm || || a,r on O(D(a, r − )) as It is clear that ||f + g|| a,r ≤ sup(||f || a,r , ||g|| a,r ) , for all f, g ∈ M(D(a, r − )), and that ||f g|| a,r ≤ ||f || a,r ||g|| a,r , whenever the right side is defined (the only case excluded is ||f || a,r = 0, ||g|| a,r = ∞).
Notice that, in one variable X, ||1/ log(1 − X)|| 0,1 = 0. If f = α∈Z d a α (X − a) α , then a,R , || || a,R denotes the boundary seminorm on M(D d k (a, (R 1 , . . . , R d ) − )), and Proof. We may assume that a = 0, which will simplify notations. Let us consider the completion K b,R of the field k(b) = k(b 1 , . . . , b d ) (of rational functions in the variables b 1 , . . . , b d ) with respect to the absolute value | | b,R : f → |f (t 0,R )|, with respect to the variables b, so that |b i | b,R = R i , for any i, and |c| b,R = |c|, for any c ∈ k. We have an injective map of k-algebras .
For any α ∈ N d , we will shorten α! G [α] into G α , so that (1.2.2) becomes We denote by G α (Zb) the image of G α , via the injective morphism (4.7.2), and define for any l ∈ N (4.7.4) We reduce to the case of dimension 1 via a generic line argument: Lemma 4.8. Consider the system of ordinary differential equations where H 1 (Z) is the matrix of meromorphic functions on D K b,R (0, 1 − ) appearing in (4.7.4). Then, in the notation (4.7.4) Proof. It is enough to prove that the matrices H l (Z) verify the recursive relations induced by the Leibnitz formula, namely: We can now conclude the proof of the theorem. We denote by || || b,R the boundary seminorm on M(D K b,R (0, 1 − )), defined at the beginning of this section, relative to the complete field K b,R . We on the other hand keep denoting || || 0,R the boundary seminorm on M(D d k (0, (R 1 , . . . , R d ) − )). We have The classical theorem of Dwork-Robba in the one variable case (cf. [DGS, IV.3.2]) implies that for any l = |α| ∞ we obtain the estimate This ends the proof.
Then, for any α ∈ N d we have the following estimate Then, for any α ∈ N d we have the following estimate The proof of (4.1) now follows directly. We consider ξ ∈ U , and the canonical point ξ ′ ∈ U H (ξ) above it; let R = R(ξ, Σ) ≤ 1. The disk D X (ξ, R − ) ⊂ U H (ξ) is isomorphic via the coordinate functions to D H (ξ) (x(ξ), (R, . . . , R) − ). We apply (4.10) to the restriction of Σ to D X (ξ, R − ), taking r = R.

Upper semicontinuity of ξ → R(ξ, Σ)
We are now back to the assumptions in (1.1) and (4.1), so in particular the matrices G i are supposed to be bounded on U , and let us further insist that the function ξ → δ X (ξ, U ) be USC on U . For example, by (3.3), this happens when U be a Laurent domain in X. For s = 1, 2, . . . and for ξ ∈ U , let (4.10.2) where R X (ξ, Σ)) is the function introduced in (1.1.5). The corollary (4.4) of the Dwork-Robba theorem says that, So, Hence (4.10.6) because the sequence s → ϕ s is an increasing sequence of functions on U . Then, ∀ ε > 0, ∃ s ε such that Then ξ → R X (ξ, Σ) is a uniform limit of USC functions, and is therefore USC. We then state Similarly we have Theorem 4.12. Assume the matrices G i are analytic elements on the analytic domain U , and suppose the function ξ → δ X (ξ, U ) admits a continuous extension to cl X (U ). Let us define R X (ξ, Σ) on cl X (U ) as in theorem (4.5). Then ξ → R X (ξ, Σ) is USC on cl X (U ).

Continuity of ξ → R(ξ, Σ) at maximal points (Dwork's transfer theorem)
An immediate consequence of formula (4.3.1) is the following. Let U be an affinoid domain in X, and let Γ(U ) = {η 1 , . . . , η N } be the Shilov boundary of U . Then, for the constant C = C(Σ, U ) of (4.1), This shows that, for any ξ ∈ U , Proposition 4.13. Let us assume that U is a Laurent domain in X with a unique maximal point η U , and that the function ξ → δ X (ξ, U ) be continuous at η U . Then, ξ → R(ξ, Σ) is continuous at η U .
5 The one-dimensional case

The theorem of Christol-Dwork revisited
Christol and Dwork (cf. [5]) consider a differential system with G a µ × µ matrix of analytic elements on the annulus So, the entries of G are elements of the k-Banach algebra H (r 1 , r 2 ) of uniform limits on C(r 1 , r 2 ) of rational functions in k(x), having no pole in C(r 1 , r 2 ). Here cl X (U ) = C + (r 1 , r 2 ) = C(r 1 , r 2 ) ∪ {t 0,r1 , t 0,r2 } .
Christol and Dwork consider the function radius of convergence of (5.0.1), restricted to a segment of points in C + (r 1 , r 2 ), namely where t r = t 0,r is the point at the boundary of D(0, r − ). This coincides with our definitions (4.5) taking into account the fact that the function ξ → |x(ξ)| extends continuously ξ → δ X (ξ, C(r 1 , r 2 )) to C + (r 1 , r 2 ). So the problem is to describe . They use the well-known fact that, for any f ∈ H (r 1 , r 2 ), the function ρ → log |f (t e ρ )| is convex and continuous on the interval [r 1 , r 2 ]. It is an elementary fact that, if ∀i ∈ N, ϕ i : [r 1 , r 2 ] → R is a convex (resp. concave) function, then is convex (resp. concave). They conclude that the function ρ → log R(e ρ ) is concave (i.e. ∩shaped) in [log r 1 , log r 2 ]. So the function R is continuous in (log r 1 , log r 2 ) and LSC at log r 1 and log r 2 . Then the same is true for the function R. But we have proven in section (4.3), that the function R is USC in U , so, in the present case, R is continuous. The conclusion is that: , and assume the entries of G are analytic elements on C(r 1 , r 2 ). Then the function is continuous.
Remark 5.2. We do not claim that the function r → R(r, Σ) is continuous at r 1 , r 2 .
5.2 Continuity of ξ → R X (ξ, Σ) on an affinoid U ⊂ D(0, 1 + ) of dimension 1 In this section we consider a system Σ = Σ x,G,U of the form (5.0.1) on an affinoid domain U of D(0, 1 + ). So, this is the case of system (1.1.1) under the assumptions of (1.1), in dimension one, and with the further condition that U is affinoid and X = A 1 k • . We prove continuity of ξ → R(ξ, Σ). Since our definitions of diameter and radius of convergence of a system are invariant by base-field extension, we may apply the discussion of (2.1), and assume that the field k is maximally complete and algebrically closed. An affinoid U of D(0, 1 + ) is of the form where I is a finite set and a i is a k-rational point of D(0, 1 + ). We are left to prove LSC continuity of ξ → R(ξ, Σ) for this system. Notice that, because k is maximally complete and algebrically closed, the points of D(0, 1 + ) are either k-rational points or of the form t a,r = the boundary point of a disk D(a, r − ), centered at a k-rational point a and of radius r ∈ (0, 1]. Proof. We will have to restrict the system Σ to various affinoid subdomains V of U . We then write R(ξ, V ) for R(ξ, Σ), when Σ is restricted to the affinoid V ⊂ U . Let ξ ∈ U be a k-rational point. Then the function η → R(η, U ), which expresses the radius of the maximal open disk centered at η and contained in U on which Σ is the trivial differential system, is clearly constant in the neighborhood D k (ξ, R(ξ, U ) − ) of ξ. This neighborhood is non empty since R(ξ, U ) = min( R(ξ, Σ), δ X (ξ, U )), and we compute: and, by (3.2), We are left to prove continuity (in fact just LSC) of ξ → R(ξ, U ) at a point ξ ∈ U of the form ξ = t a,r ∈ U . Notice that we may (and will) assume R = R(t a,r , U ) ≤ r, otherwise on the disk D(a, R − ), which is an open neighborhood of t a,r , the function ξ → R(ξ, U ) would be constant of value R, and ξ → R(ξ, U ) would then be continuous at ξ = t a,r . On the other hand, δ X (t a,r , U ) ≥ r, so in particular we assume R(t a,r , Σ) ≤ δ X (t a,r , U ).
and let ε 0 := min i / ∈J |a − a i |. We further subdivide J into a disjoint union We want to construct a (not fundamental) system of affinoid neighborhoods {V ε } ε>ε0 of t a,r ∈ U , with the property that the Shilov boundary Γ(V ε ) of V ε consists of the points t ai,r−ε , for i ∈ J 1 , t ai,r = t a,r , for i ∈ J 2 and of the point t a,r+ε . Notice that t a,r+ε → t a,r , and t ai,r−ε → t a,r , as ε → 0, ∀ i ∈ J 1 . We simply take for ε > ε 0 (5.3.3) Notice that (5.3.4) δ X (t a,r+ε , V ε ) = r + ε ; δ X (t a,r , V ε ) = r ; δ X (t ai,r−ε , V ε ) = r − ε ∀i ∈ J .
Coming back to our differential system (5.0.1) and its iterates we have, by (4.12.2), ∀η ∈ V ε , t ai,r−ε , for i ∈ J 1 , t ai,r = t a,r , for i ∈ J 2 and of the point t a,r+ε .

Continuity of
We assume here that X is a smooth formal scheme of relative dimension 1 over Spf k • , and U is an affinoid neighborhood of the maximal point η X of X = X η (always satisfying the requirements in (1.1)). Notice that the special fiber X s of X is a smooth scheme overk, which we may assume to be connected. The reduction map π : X → X s , is such that the fiber at each closed point of X s is an open disk of radius one, called a residue class, while the inverse image of the generic point η Xs consists only of the maximal point η X of X. An affinoid U ⊂ X is a neighborhood of η X if and only if it contains almost all residue classes in X, and contains a non trivial annulus of outer radius one in each of the remaining residue classes. So, U is the disjoint union of the generic fiber Y = Y η of a smooth formal scheme Y (the union of full residue classes) and of a finite number {C 1 , . . . , C r } of analytic subdomains of the open disk of radius one, which are bound to contain some annulus of outer radius one. We are given the system (5.0.1) on U , and must prove continuity of ξ → R X (ξ, Σ) on U . Notice that, if we call Σ |Y , Σ |C1 , . . . , Σ |Cr , the restrictions of Σ to the various analytic subdomains of U , we have because δ X (ξ, Y ) = δ Y (ξ, Y ) = 1, ∀ ξ ∈ Y . Similarly, R Y (ξ, Σ |Ci ) = R X (ξ, Σ) , ∀ξ ∈ C i , ∀ i = 1, . . . , r .
Since δ X (ξ, U ) is continuous on U and takes the value 1 at η X , restricting to points ξ of the form t 0,r , as r → 1, it follows from (5.1) that (5.3.6) lim ξ→ηX R X (ξ, Σ |C1 ) = min(1, R(η X , Σ)) , which only depends upon η X , and is therefore independent of the residue class containing C 1 , chosen to approach η X . We conclude Corollary 5.4. If U is an affinoid neighborhood of η X , ξ → R(ξ, Σ) is continuous on U .

Appendix. Valuation polygon of an analytic function in several variables
For the reader's convenience we recall some facts from [15, §2] on the several variable Newton polygon theory, which has been applied in this paper. We set v(x) = − log |x| for any x in an extension of k. Let us assume that k is algebraically closed.  1. Let ξ ∈ k d and µ = (v(ξ 1 ), . . . , v(ξ d )) be in Conv(f ). If µ ∈ Reg(f ) then f (ξ) = 0 and v(f (ξ)) = v(f, µ).