É Delaygue

T Rivoal

On Abel's problem and Gauss congruences

A classical problem due to Abel is to determine if a differential equation y = ηy admits a non-trivial solution y algebraic over C(x) when η is a given algebraic function over C(x). Risch designed an algorithm that, given η, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when η admits a Puiseux expansion with rational coefficients at some point in C∪{∞}, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of y = ηy if and only if the coefficients of the Puiseux expansion of xη(x) at 0 satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations y = ηy with an algebraic solution when xη(x) is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks.

Introduction

Given an algebraic function η over C(x), Abel's problem, as mentionned by Boulanger in [8, p. 93], consists in determining if the differential equation

y = ηy (1.1)
admits a non-trivial solution y algebraic over C(x). This problem naturally occurs in procedures to decide in a finite number of steps if all solutions of a linear differential equation with polynomial coefficients are algebraic over C(x), see [START_REF] Baldassarri | On second order linear differential equations with algebraic solutions[END_REF][START_REF] Singer | Algebraic solutions of n-th order linear differential equations[END_REF]. A decision procedure to solve Abel's problem was first given by Risch [START_REF] Risch | The problem of integration in finite terms[END_REF] and later independently by Baldassarri and Dwork [2, Section 6].

In this paper, we are interested in the restriction of Abel's problem to the functions η algebraic over C(x) that admit a Puiseux expansion with rational coefficients at some point δ in C ∪ {∞}.

Obviously the above mentioned decision procedures also apply in this context but we present here an arithmetic criterion based on Gauss congruences for xη(x) that proves effective in applications. It is stated as Theorem 1 in Section 1.1, where we use in particular a consequence of Grothendieck's conjecture proved in rank one by Chudnovsky and Chudnovsky [START_REF] Chudnovsky | Applications of Padé approximations to the Grothendieck conjecture on linear differential equations, Number theory[END_REF].

Secondly, we apply our criterion to give in Section 1.3 a complete resolution -Theorem 2 -of Abel's problem when xη(x) is an algebraic hypergeometric series with rational parameters. We obtain a criterion -Theorem 4 -for a globally bounded hypergeometric series with rational parameters to satisfy Gauss congruences. This enables us to confirm a prediction attributed to Golyshev by Zagier in [31, p. 757].

Finally in Sections 2.1 and 2.2, we briefly study the case when η is a rational function and we apply our criterion to various examples issued from diagonals of multivariate rational fractions and random walks in the quarter plane.

An arithmetic criterion via Gauss congruences

Given a prime number p, we set Z (p) := {r ∈ Q : v p (r) ≥ 0}, where v p (r) is the p-adic valuation of r. In other words, Z (p) is the ring of rational numbers the denominator of which is not divisible by p (recall that v p (0) = +∞ by convention). We consider the following congruences. Definition 1. Let p be a prime number. We say that a sequence (a n ) n∈Z of rational numbers satisfies Gauss congruences for the prime p if a np -a n ∈ npZ (p) for all integers n. This is equivalent to a mp s+1 -a mp s ∈ p s+1 Z (p) for all integers m ∈ Z and s ≥ 0, a property that often appears in this form in the literature. These congruences hold for an integer sequence (a n ) n≥0 and for all prime numbers if and only if, for all n ≥ 0, we have

d|n µ(n/d)a d ≡ 0 mod n, (1.2)
where µ is the Möbius function. The congruence (1.2) was first proved by Gauss when a n = r n and r is a prime number, and was later generalized to all integers r ∈ Z. We refer to [START_REF] Zarelua | On congruences for the traces of powers of some matrices[END_REF] for a survey of these congruences and to [START_REF] Beukers | Gauss congruences for rational functions in several variables[END_REF][START_REF] Beukers | Dwork crystals I[END_REF][START_REF] Minton | Linear recurrence sequences satisfying congruence conditions[END_REF] for recent results on Gauss congruences for multivariate rational fractions. Following [START_REF] Beukers | Gauss congruences for rational functions in several variables[END_REF], when a sequence of rational numbers (a n ) n∈Z satisfies Gauss congruences for almost all prime numbers p (1 ), we say that it has the Gauss property. In this case, we also say that its bilateral generating series has the Gauss property. In the particular situation that a n = 0 for all n < r for some r ∈ Z, i.e. the generating series is a Laurent series in Q((x)), we shall say that the sequence (a n ) n≥r satisfies Gauss congruences. However, it is easily shown (see Section 3.2) that in this case necessarily a n = 0 for negative n, or in other words that if f ∈ Q((x)) has the Gauss property, then

f ∈ Q[[x]].
Our arithmetic criterion, which is proved in Section 3, is the following.

Theorem 1. Let η ∈ Q((x)) be algebraic over Q(x). Then the differential equation y = ηy admits a non-trivial solution y algebraic over Q(x) if and only if xη(x) has the Gauss property.

Let us explain how Theorem 1 yields an arithmetic characterization of Abel's problem for algebraic series that admit a rational Puiseux expansion at some point δ ∈ C ∪ {∞}. Let η be an algebraic function over C(x). Let δ ∈ C, respectively δ = ∞, and consider the Puiseux expansion of η at δ given respectively by

∞ n=r p n (x -δ) n/d and ∞ n=r p n x -n/d , (1.3)
where r and d ≥ 1 are both integers. We call (p n ) n∈Z the sequence of the coefficients of the Puiseux expansion of η at δ, where p n = 0 for n < r. We say that the Puiseux expansion is rational if p n ∈ Q for all n. In this case, we say that η has the Gauss property at δ if (p n ) n∈Z has the Gauss property (by the remark preceding Theorem 1, the latter property implies that p n = 0 for n < 0). This is a generalization of the above definition for Laurent series in Q((x)), when d = 1 and δ = 0. A direct consequence of Theorem 1 is the following criterion. Proof. We first assume that η has a Puiseux expansion at δ ∈ C of the form (1.3) with p n ∈ Q for all n. We make the change of variables x = t d + δ which yields the differential equation y (t) = η(t) y(t), with y(t) := y(t d + δ) and η(t

) := dt d-1 η(t d + δ) ∈ Q((t)).
Applying Theorem 1 to η(t), we obtain that the differential equation y = ηy admits a non-trivial solution y algebraic over C(x) if and only if t η(t) = ∞ n=r dp n t n+d has the Gauss property (at 0). This is equivalent to saying that the sequence (dp n-d ) n∈Z has the Gauss property, or equivalently that (x -δ)η(x) has the Gauss property at δ.

Similarly, when δ = ∞, the change of variables x = 1/t d shows that y = ηy has a non-trivial algebraic solution over C(x) if and only if (-dp n+d ) n∈Z has the Gauss property, which is equivalent to saying that xη(x) has the Gauss property at ∞. It follows from Corollary 1 that the Gauss property is somehow independent of the point where η admits a rational Puiseux expansion. If η is an algebraic function and δ ∈ C, then we write η δ (x) := (x -δ)η(x) and η ∞ (x) = xη(x).

Corollary 2. Let η be an algebraic function over C(x) and δ 1 and δ 2 two points in C∪{∞} at which η admits rational Puiseux expansions. Then η δ 1 has the Gauss property at δ 1 if and only if η δ 2 has the Gauss property at δ 2 .

We shall from now on focus on the case when η admits a rational Puiseux expansion at δ = 0. For y = ηy to have a non-trivial algebraic solution, η must admit a Puiseux expansion at 0 of the form (see Section 3.2 for details)

η(x) = ∞ n=0 a n x n/d-1 ∈ C((x 1/d )).
(1.4)

Then the sequence of coefficients of the Puiseux expansion of xη(x) at 0 reads (a n ) n≥0 and a non-trivial solution of (1.1) is given by

y(x) := x a 0 exp d ∞ n=1 a n n x n/d =: x a 0 g(x) ∈ x a 0 C((x 1/d )). (1.5)
We shall also use the less precise notation y := exp η with the meaning of (1.5). Note that an algebraic function over C(x) which admits a Puiseux expansion at a rational point with rational coefficients is necessarily algebraic over Q(x). In particular, η as given in (1.4) with a n ∈ Q is algebraic over Q(x), and thus satisfies a linear differential equation over Q(x) (this is based on the fact that η ∈ Q(x, η), see [START_REF] Comtet | Calcul pratique des coefficients de Taylor d'une fonction algébrique[END_REF]). Similarily, when y given by (1.5) is algebraic over C(x), it is automatically algebraic over Q(x); observe that y is algebraic over Q(x) if and only if g is too.

Remarks. Let us make a few remarks on Theorem 1 and its corollaries.

• When y is algebraic over Q(x), Corollary 1 shows that, for all but finitely many prime numbers p, the sequence (a n ) n≥0 defined by (1.4) satisfies Gauss congruences for p. This bound on p will be made more precise in the proof when d = 1: Gauss congruences hold at least for any p that does not divide the smallest positive integer λ such that g(λx) ∈ Z[[x]] (where g is defined in (1.5) with d = 1).

• When d = 1 in (1.5), the assertion g ∈ Z[[x]] has a well-known interpretation in formal group theory, see [START_REF] Beukers | Some congruences for the Apéry numbers[END_REF][START_REF] Honda | Formal groups obtained from generalized hypergeometric functions[END_REF].

• When y is algebraic over Q(x), we have η ∈ Q(x, y) because y ∈ Q(x, y) (see above) and η = y /y. It is also easy to prove that y is algebraic over Q(x) if and only if there exists a positive integer m such that mη is the logarithmic derivative of an element of Q(x, η), which yields y m ∈ Q(x, η). A procedure to determine in a finite number of steps whether mη is the logarithmic derivative of an element of Q(x, η) for some m is given in [2, Section 6]. It would be interesting to bound explicitely m and so the degree of y in terms of η. The case η ∈ Q(x) already shows that it is not always true that m = 1 works (see Section 1.2).

When η(x) is a rational fraction

If η belongs to Q(x), then the non-trivial algebraic solutions of y = ηy are arithmetic Nilsson-Gevrey series of order 0 (see [START_REF] Fischler | A note on G-operators of order 2[END_REF] for the definition which is not essential here). By [START_REF] Fischler | A note on G-operators of order 2[END_REF]Proposition 3], we obtain that

y(x) = c i∈I (x -λ i ) s i and η(x) = i∈I s i x -λ i , (1.6)
where c ∈ Q * , I is a finite set, λ i ∈ Q and s i ∈ Q for all i ∈ I.

In our context, we are interested in the particular case η ∈ Q(x). Applying (1.6) in this case yields the equivalence: there is a non-trivial algebraic solution to y = ηy with η ∈ Q(x) if and only if we have

y(x) = c i∈I u i (x) s i and η(x) = i∈I s i u i (x) u i (x) , where c ∈ Q * , I is a finite set, u i ∈ Z[x]
and s i ∈ Q for all i ∈ I. Combining this equivalence with our criterion Theorem 1, we retrieve the following recent result of Minton.

Theorem A (Minton [START_REF] Minton | Linear recurrence sequences satisfying congruence conditions[END_REF]). A rational fraction f ∈ Q(x) has the Gauss property (at 0) if and only

if f (x) is a Q-linear combination of terms xu (x)/u(x) with u ∈ Z[x].
Here we implicitly say that f ∈ Q(x) has the Gauss property at 0 when the coefficients of its Laurent expansion at the origin f (x) = n∈Z a n x n , with a n = 0 for large negative integers n, defines a sequence (a n ) n∈Z with the Gauss property.

The hypergeometric case and Golyshev's predictions

In this section, we state our results which give a complete resolution of the case when xη(x) is an algebraic hypergeometric series with rational parameters. In particular, such a hypergeometric series is algebraic over Q(x).

Let α := (α 1 , . . . , α r ) and β := (β 1 , . . . , β s ) be tuples of rational numbers in Q \ Z ≤0 . The generalized hypergeometric series with rational parameters [START_REF] Slater | Generalized Hypergeometric Functions[END_REF] is defined by

r F s α 1 , . . . , α r β 1 , . . . , β s ; x := ∞ n=0 (α 1 ) n • • • (α r ) n (β 1 ) n • • • (β s ) n x n n! ,
where (α) n is the Pochhammer symbol defined by (α

) n := α(α + 1) • • • (α + n -1)
for n ≥ 1 and 1 if n = 0. We discard the case when xη(x) is a polynomial, already considered in Section 1.2, by assuming α i / ∈ Z ≤0 for all i. We set

Q α,β (n) := (α 1 ) n • • • (α r ) n (β 1 ) n • • • (β s ) n and F α,β (x) := ∞ n=0 Q α,β (n)x n , so that F α,β (x) = r+1 F s α 1 , . . . , α r , 1 β 1 , . . . , β s ; x ,
which has a finite positive radius of convergence if and only if r = s. Hence, in the rest of this section, we assume that r = s because this is a necessary condition for F α,β (x) to be a non-polynomial algebraic function.

We say that F α,β is factorial if there exists a non-zero rational constant C and tuples of positive integers e = (e 1 , . . . , e u ) and f = (f 1 , . . . , f v ) such that

F α,β (Cx) = ∞ n=0 (e 1 n)!(e 2 n)! • • • (e u n)! (f 1 n)!(f 2 n)! • • • (f v n)! x n . (1.7)
Because r = s, we necessarily have j e j = j f j ; this property will be implicit below.

We shall denote the series on the right hand side of (1.7) by F e,f (x) and we also set

Q e,f (n) := (e 1 n)!(e 2 n)! • • • (e u n)! (f 1 n)!(f 2 n)! • • • (f v n)! .
The series F α,β is factorial if and only if α and β are R-partitioned in the sense of [14, §7], see Section 4 for more details.

According to Zagier [31, p. 757], Golyshev has predicted, based on an argument about extensions of motives, that

y e,f (x) := exp F e,f (x) x dx = x exp ∞ n=1 Q e,f (n) n x n is always algebraic over Q(x) when F e,f is itself algebraic over Q(x).
By ad hoc explicit computations, Zagier has proved in [31, pp. 757-759] that Golyshev's predictions (2 ) hold when Q e,f (n) is one of

en f n , (6n)!n! (3n)!(2n)! 2 or (10n)!n! (5n)!(4n)!(2n)!
, with e ≥ f ≥ 1. Zagier has also mentionned that Bloch had sketched to him a proof when the algebraic curve defined by F e,f (x) is rational, but that the general case seemed to be open. In passing, let us mention that the algebraicity of the function y e,f associated to en f n naturally appears in subjects pertaining to physics; see for instance [13, §1, Eq. (1.

2)] and [22, §7.5].

By characterizing the globally bounded hypergeometric series with rational parameters having the Gauss property, we prove that the first part of Golyshev's predictions is true and optimal within the class of algebraic hypergeometric series with rational parameters.

Theorem 2. Let α and β be tuples of parameters in

Q \ Z ≤0 such that F α,β is algebraic over Q(x). Then the function exp F α,β (x) x dx is algebraic over Q(x) if and only if F α,β is factorial.
Remarks. Let us make two comments on the other predictions made by Golyshev.

• In the algebraic and factorial case, by the remarks that follow Corollary 2, F e,f ∈ Q(x, y e,f ) and there exists a positive integer m such that y m e,f ∈ Q(x, F e,f ). According to Zagier, Golyshev has predicted that m = 1 and that y e,f is a unit over Z[1/x]. In particular, it would follow that y e,f and F e,f are algebraic functions of the same degree over Q(x).

• We observe that in very similar but not exactly hypergeometric situations, none would be true. Consider for instance f

(x) = 1 + x 2(1-x) : we have exp f (x) x dx = x √ 1 -x / ∈ Q(x, f ) = Q(x)
and it is not an integer over Z[1/x] because its minimal polynomial over

Z[1/x] is (1/x 2 - 1/x)T 2 -1.
This example shows in particular that if the remaining parts of Golyshev's predictions are true, their proof could not be based only on the fact that the analytic continuations of algebraic hypergeometric series have exactly one finite singularity.

When F α,β is algebraic over Q(x) but not factorial, Theorem 2 shows that

y α,β := exp F α,β (x)
x dx is transcendental over Q(x). But it turns out that the interlacing criterion of Beukers and Heckman [START_REF] Beukers | Monodromy for the hypergeometric function n F n-1[END_REF] naturally produces algebraic hypergeometric series associated with F α,β whose product of the corresponding series y α,β is always algebraic. Let us be more precise. Let {•} stand for the fractional part function and consider the slight modification of {•} given, for all real numbers x, by x = {x} if x is not an integer and 1 otherwise (instead of 0). We define • on tuples component-wise, that is (α 1 , . . . , α r ) := ( α 1 , . . . , α r ).

Theorem 3. Let α and β = (β 1 , . . . , β r ), with β r = 1, be disjoint tuples of rational numbers in (0, 1] such that F α,β is algebraic over Q(x). Let d ≥ 1 be the least common multiple of the exact denominators of the α i 's and β j 's. Then the function

d k=1 gcd(k,d)=1 exp F kα , kβ (x) x dx (1.8) is algebraic over Q(x).
Remarks. Let us make few remarks on Theorem 3.

• When F α,β is factorial, then for every k ∈ {1, . . . , d} coprime to d, we have kα = α and kβ = β so that each term of the product (1.8) is equal to y α,β . It follows that when F α,β is algebraic and factorial, then Theorem 3 also gives the conclusion of Theorem 2: y α,β is algebraic over Q(x).

• A contrario, when F α , β is not factorial, then none of the F kα , kβ 's is factorial. But as explained in Section 4.2, the interlacing criterion of Beukers and Heckman gives that, for every k ∈ {1, . . . , d} coprime to d, f k (x) := F kα , kβ (x)/x is algebraic over Q(x). By Theorem 2, each y k := exp f k is transcendental over Q(x). It turns out, by Theorem 3, that the product of the y k 's is algebraic over Q(x).

Theorems 2 and 3 relie on our criterion Theorem 1 and the following characterization of globally bounded hypergeometric series with rational parameters which have the Gauss property. We recall that a series f (x) ∈ Q [[x]] is said to be globally bounded if it has a non-zero radius of convergence and if there exist

C, D ∈ Q\{0} such that D•f (Cx) ∈ Z[[x]].
Theorem 4. Let α and β be tuples of parameters in Q \ Z ≤0 such that F α,β is globally bounded. Then F α,β has the Gauss property if and only if it is factorial. Furthermore, in the latter case, if F α,β (Cx) = F e,f (x) for some non-zero C ∈ Q, then F e,f satisfies Gauss congruences for all prime numbers p.

Let us discuss two examples to illustrate the above theorems.

• The sequence of coefficients

Q e,f (n) = (30n)!n! (15n)!(10n)!(6n)! ∈ Z
was used by Chebyshev in his work on the distribution of prime numbers. According to Rodriguez-Villegas [START_REF] Rodriguez-Villegas | Integral ratios of factorials and algebraic hypergeometric functions[END_REF], they define an algebraic hypergeometric series F e,f with a surprisingly large degree over Q(x): 483, 840. By Theorem 2, the function exp F e,f (x)

x dx is also algebraic over Q(x). We do not know its degree over Q(x) but according to Golyshev's prediction, it is also 483, 840.

• Let α and β be such that

F α,β (x) = ∞ n=0 (1/4) n (11/12) n (1/2) n n! x n .
The interlacing criterion of Beukers and Heckman amounts to comparing the elements of kα and kβ for every k ∈ {1, . . . 12} coprime to 12, that is k = 1, 5, 7 or 11. The interlacing condition is easily verified and we obtain four algebraic hypergeometric series: f 1 , f 5 , f 7 and f 11 which are respectively defined by their Taylor coefficients:

(1/4) n (11/12) n (1/2) n n! , (1/4) n (7/12) n (1/2) n n! , (5/12) n (3/4) n (1/2) n n! and (1/12) n (3/4) n (1/2) n n! .
Since none of those series is factorial, Theorem 2 implies that, for all i ∈ {1, 5, 7, 11}, the function

y i (x) = exp f i (x)
x dx is transcendental over Q(x). But Theorem 3 shows that the product y 1 y 5 y 7 y 11 is algebraic over Q(x).

Acknowledgements.

Applications of Theorem 1

We give two further applications of Theorem 1 in two different directions. First, we show that diagonals of rational fractions provide a wide variety of examples of algebraic series η with the Gauss property, then leading, by Theorem 1, to the algebraicity of y = exp η.

Secondly, we use Theorem 1 in combination with the work of Banderier and Flajolet [START_REF] Banderier | Basic analytic combinatorics of directed lattice paths[END_REF] to prove that the numbers of directed two-dimensional bridges satisfy Gauss congruences for all prime p.

Diagonals of rational fractions

We present a way to produce examples of algebraic functions η in Q((x)) such that y is also algebraic: when η is the diagonal of a bivariate rational fraction. To that end, for every positive integers r and s, we consider the diagonal operator ∆ r,s defined for every bivariate series

f (x, z) = n,m≥-1 a n,m x n z m ∈ (xz) -1 C[[x, z]] by ∆ r,s (f )(x) := ∞ n=-1 a rn,sn x n ,
where a rn,sn = 0 if either rn < -1 or sn < -1. When r = s = 1, the operator preserves only the coefficients on the main diagonal and we simply write ∆ for ∆ 1,1 .

By a result of Furstenberg [START_REF] Furstenberg | Algebraic functions over finite fields[END_REF], diagonals of bivariate rational fractions are algebraic.

In particular, if f (x, z) ∈ Q(x, z) admits an expansion in (xz) -1 Q[[x, z]], then its corre- sponding diagonal ∆(f ) is algebraic over Q(x).
Another related way to produce algebraic series was found earlier by Pólya [START_REF] Pólya | Sur les séries entières dont la somme est une fonction algébrique[END_REF]: if ϕ and ψ are algebraic functions regular at the origin, then

f (x, z) := ψ(z) 1 -xϕ(z) (2.1)
admits an expansion as a bivariate power series and, for every positive integers r and s, ∆ r,s (f ) is algebraic. We will use this result in the particular case when both ϕ and ψ are rational functions.

Let f (x, z) be a bivariate rational fraction in (xz) -1 Q[[x, z]], and r, s two positive integers such that

η(x) := ∆ r,s (f )(x) = ∞ n=0 a n x n-1
is algebraic. By Theorem 1, y := exp η is also algebraic if and only if (a n ) n≥0 has the Gauss property. To study when does the latter occur, we follow [START_REF] Beukers | Gauss congruences for rational functions in several variables[END_REF] and extend Definition 1 of Gauss congruences to several variables. Definition 2. Let p be a prime number. We say that a family (a n ) n∈Z k of rational numbers satisfies Gauss congruences for the prime p if a mp s+1 -a mp s ∈ p s+1 Z (p) for all m ∈ Z k and all s ∈ Z ≥0 .

We retrieve Definition 1 by setting k = 1. When a family satisfies Gauss congruences for almost all prime p, we also say that it has the Gauss property.

We shall consider Gauss congruences for rational fractions below and they have to be understood as follows. As explained in [6, Section 2], a rational fraction f = P/Q has Laurent series associated with each vertex of the Newton polytope (3 ) of Q. It is proved in [START_REF] Beukers | Gauss congruences for rational functions in several variables[END_REF]Proposition 3.4] that one of these Laurent series has the Gauss property if and only if all Laurent series have the Gauss property. In this case, we also say that f has the Gauss property.

Gauss congruences are stable by any diagonalization ∆ r,s . That is, if (a n,m ) (n,m)∈Z 2 has the Gauss property then it is also the case for (a rn,sn ) n∈Z . Hence we are interested in Gauss congruences for multivariate rational fractions. The following two recent results produce examples of rational fractions with the Gauss property.

Theorem B (Beukers-Houben-Straub [START_REF] Beukers | Gauss congruences for rational functions in several variables[END_REF]). Let m ≤ n and let f 1 , . . . , f m ∈ Q(x) be non-zero. Then the rational fraction

x 1 • • • x m f 1 • • • f m det ∂f j ∂x i 1≤i,j≤m
has the Gauss property.

Theorem C (Beukers-Vlasenko [START_REF] Beukers | Dwork crystals I[END_REF]). Let f be a Laurent polynomial with integer coefficients such that lattice points in its Newton polytope ∆ ⊂ R n are vertices. Let g be a Laurent polynomial with integer coefficients and support in ∆. Then g/f has the Gauss property.

Applying Theorems B and C with dimension n = 2 produces bivariate rational fractions with the Gauss property. Then taking their principal diagonal and applying the result of Furstenberg together with Theorem 1, we obtain algebraic functions η such that y is also algebraic. For example, the central Delannoy numbers

D(n) = n k=0 n k n + k k ,
are the numbers of S-walks from (0, 0) to (n, n) with steps S = {(1, 0), (0, 1), (1, 1)} (see Section 2.2 for a definition). They are also given by the algebraic diagonal

∞ n=0 D(n)x n = ∆ 1 1 -x -y -xy = 1 √ 1 -6x + x 2 .
By Theorem C, the sequence (D(n)) n≥0 has the Gauss property, so that

y(x) = x exp x 0 1 √ 1 -6t + t 2 -1 dt t
is also algebraic. Of course, this could be checked directly without first proving that (D(n)) n≥0 has the Gauss property, because an antiderivative of the integrand is

-arctanh (1 -3x)/ √ 1 -6x + x 2 -log(x).
But this example illustrates the possibilities and is reminiscent of that of directed twodimensional walks, that we briefly describe below.

Directed two-dimensional walks

Fix a finite set of vectors S = {(a 1 , b 1 ), . . . , (a m , b m )}. A S-walk is a sequence v = (v 1 , . . . , v n ) such that each v j ∈ S. The geometric realization of a walk (v 1 , . . . , v n ) is the sequence of points (P 0 , P 1 , . . . , P n ) such that P 0 = (0, 0) and P j = P j-1 + v j . The integer n is referred to as the size of the walk. In the rest of this section, we assume that a 1 = • • • = a m = 1. In this case, S is said simple and a S-walk is a directed two-dimensional lattice path.

A bridge is a walk whose end-point P n lies on the x-axis. An excursion is a bridge that lies in the quarter plane Z ≥0 × Z ≥0 . We set B(x), respectively E(x), the generating function of the number of bridges, respectively of excursions. That is

B(x) := ∞ n=0 B n x n and E(x) := ∞ n=0 E n x n ,
where B n and E n are respectively the numbers of bridges and excursions of size n (in this simple case the size is equal to the length of a walk).

Banderier and Flajolet [START_REF] Banderier | Basic analytic combinatorics of directed lattice paths[END_REF] proved that B and E are algebraic series over Q(x) satisfying

E (x) = B(x) -1 x E(x).
By a direct application of Theorem 1, we obtain that the sequence of the number of bridges satisfies Gauss congruences for all prime p.

Proposition 1. For all prime numbers p and every non-negative integers m and s, we have B mp s+1 ≡ B mp s mod p s+1 .

Banderier and Flajolet also gave in [START_REF] Banderier | Basic analytic combinatorics of directed lattice paths[END_REF] an expression for B as the diagonal of a rational fraction. To state this result, we introduce the characteristic polynomial of S which is the Laurent polynomial P (z) := m i=1 z b i . We write c for the integer such that z c P (z) is a polynomial with constant term 1. We assume that c ≥ 1, otherwise there is no bridge nor excursion. Then B is the diagonal ∆ 1,c of the rational fraction

1 1 -xz c P (z) ∈ Q[[x, z]]. (2.
2)

The algebraicity of B follows by the above result of Pólya. But Theorem C of Beukers and Vlasenko does not apply when 3 or more steps are allowed in S. In fact, in this case, the Newton polytope of 1 -xz c P (z) contains at least one lattice point which is not a vertex.

Proof of the arithmetic criterion

This section is devoted to the proof of Theorem 1.

The Dieudonné-Dwork Lemma

A fundamental tool for studying arithmetic properties of exponentials is the following.

Lemma 1 (Dieudonné-Dwork). Let F (x) ∈ 1 + xQ[[x]
] and p a prime number. Then

F (x) ∈ 1 + xZ (p) [[x]] if and only if F (x p )/F (x) p ∈ 1 + pxZ (p) [[x]].
In our context, we aim at studying g(x) = exp(s(x)) where s(x) ∈ xQ [[x]] and this lemma yields

Corollary 3. Let s(x) ∈ xQ[[x]] and p a prime number. Then exp(s(x)) ∈ 1 + xZ (p) [[x]] if and only if s(x p ) -ps(x) ∈ pxZ (p) [[x]].
This lemma is a particular case of [16, Lemma 1], another proof of which can be found in [START_REF] Beukers | Some congruences for the Apéry numbers[END_REF].

Proof of Theorem 1

Let η ∈ Q((x)) be algebraic over Q(x). We must prove that the differential equation y = ηy admits a non-trivial solution y algebraic over Q(x) if and only if xη(x) has the Gauss property. First we show that, without loss of generality, we can assume that xη(x) ∈ Q [[x]].

On the one hand, if xη(x) has the Gauss property, then set

xη(x) = ∞ n=r a n x n ,
with r ∈ Z. For all primes p large enough and at least such that p > |r|, and all negative integers n, we have both a np -a n ∈ npZ (p) and a np = 0 so that a n ∈ pZ (p) . This shows that a n = 0 for every negative integer n, and thus that xη(x

) ∈ Q[[x]].
On the other hand, assume the differential equation y = ηy admits a non-trivial solution y algebraic over Q(x). By the Newton-Puiseux theorem [START_REF] Serre | Local Fields[END_REF]p. 68,Proposition 8], y admits a Puiseux expansion:

y(x) = ∞ n=r b n x n/d = x r/d ∞ n=0 b n+r x n/d ∈ C((x 1/d )),
for some integers r and d ≥ 1 and with b r non-zero. We also have

y (x) = 1 d ∞ n=r nb n x n/d-1 = x r/d-1 d ∞ n=0 (n + r)b n+r x n/d .
It follows that η = y /y admits a Puiseux expansion of the form

η(x) = ∞ n=0 a n x n/d-1 ∈ C((x 1/d )). (3.1)
Since by assumption η ∈ Q((x)), we can take d = 1 so that xη(x

) ∈ Q[[x]].
In the rest of the proof of Theorem 1, we thus assume without loss of generality that

xη(x) = ∞ n=0 a n x n ∈ Q[[x]].
It remains to prove the arithmetic criterion for the sequence (a n ) n≥0 . We write

y(x) := x a 0 exp ∞ n=1 a n n x n =: x a 0 g(x), which is in x a 0 Q[[x]].
Obviously, y is algebraic over Q(x) if and only if g is too. We write

f (x) := ∞ n=1 a n x n-1 and F (x) := ∞ n=1 a n n x n , so that g = exp(F ) and g = f g. Since η is algebraic over Q(x), f (x) = η(x) -a 0 /x is too.
We prove Theorem 1 in two steps. First we prove the theorem when f belongs to Z [[x]]. Then we use this result to prove the general case when

f ∈ Q[[x]].

When f ∈ Z[[x]]

In this section we assume that the sequence (a n ) n≥1 is integer-valued and we prove Theorem 1 in this particular case, that is: y is algebraic over Q(x) if and only if (a n ) n≥0 has the Gauss property.

Proof of the if part. We assume that (a n ) n≥0 has the Gauss property and show that y is algebraic over Q(x). We prove this implication in two steps. First we show that there exists a positive integer λ such that g(λx) ∈ Z [[x]]. Then we apply a result of Chudnovsky and Chudnovsky to conclude.

The sequence (a n ) n≥0 satisfies Gauss congruences for all prime numbers p ≥ N , for some integer N . By Corollary 3, the assertion g ∈ 1 + xZ (p) [[x]] is equivalent to the assertion F (x p ) -pF (x) ∈ pxZ (p) [[x]]. This in turn is equivalent to the two following assertions together:

∀n ∈ N * : p n ⇒ pa n n ∈ pZ (p) , (3.2)
∀n ∈ N * : a n -a np n ∈ pZ (p) . (3.3)
In our situation, (3.2) holds because 1/n ∈ Z (p) when p n and a n ∈ Z, and (3.3) holds as well for p ≥ N by assumption. Therefore, g ∈ Z (p) [[x]] for all p ≥ N . Moreover, we have

g(x) = exp F (x) = ∞ k=0 1 k! ∞ n=1 a n n x n k = 1 + ∞ n=1 σ n x n ,
where

σ n := n k=1 1 k! m 1 +•••+m k =n m j ≥1 a m 1 a m 2 • • • a m k m 1 m 2 • • • m k . We have m 1 m 2 • • • m k | n! for all integers m 1 , . . . , m k ∈ {1, . . . , n} such that k j=1 m j = n because n! m 1 m 2 • • • m k = (m 1 + • • • + m k )! m 1 ! • • • m k ! • (m 1 -1)! • • • (m k -1)! ∈ Z.
Consequently, for all n ≥ 1, n! 2 σ n ∈ Z because a m ∈ Z for all m ≥ 1. For all primes p and all n ≥ 1, we have

v p n! 2 = 2 ∞ k=1 n p k ≤ 2n p -1 ≤ 2n,
and we deduce that p 2n σ n ∈ Z (p) . Letting λ := p<N p 2 , it follows that g(λx) ∈ Z (p) [[x]] for all primes p < N . Since g(x) ∈ Z (p) [[x]] for all primes p ≥ N , we conclude that

g(λx) ∈ Z[[x]].
We are now in position to apply a fundamental result due to Chudnovsky and Chudnovsky in [START_REF] Chudnovsky | Applications of Padé approximations to the Grothendieck conjecture on linear differential equations, Number theory[END_REF]

: if h ∈ Z[[x/µ]] (for some µ ∈ Q * ) is such that h /h is algebraic over Q(x),
then h is itself algebraic over Q(x) (see also [9, §3], [START_REF] Kassel | Algebraicity of the zeta function associated to a matrix over a free group algebra[END_REF]Theorem 4.4] as well as a generalization in [1, pp. 123-124] where h is only assumed to be a G-function). We can apply this with h := g and µ := λ defined above, and deduce that g is algebraic over Q(x).

Proof of the only if part. We assume that y is algebraic over Q(x) and we prove that (a n ) n≥0 has the Gauss property. Since

g ∈ 1 + xQ[[x]] is algebraic over Q(x), there exists a positive integer λ such that g(λx) ∈ Z[[x]] by Eisenstein's theorem. Hence, g ∈ 1 + xZ (p) [[x]
] for any p that does not divide λ. By Corollary 3, this implies that F (x p ) -pF (x) ∈ pxZ (p) [[x]] for any p that does not divide λ. Hence, by (3.3), we have a np -a n ∈ npZ (p) for all n ≥ 1 and any p that does not divide λ. Then (a n ) n≥0 has the Gauss property. This completes the proof of Theorem 1 in the particular case

f ∈ Z[[x]]. 3.2.2 When f ∈ Q[[x]]
In this section we assume that a n ∈ Q for n ≥ 0. We complete the proof of Theorem 1, that is: y is algebraic over Q(x) if and only if (a n ) n≥0 has the Gauss property.

We first remark that for any integer r = 0, f (x

) := r/(1 -rx) ∈ Z[[x]] is such that g(x) = 1/(1 -rx) ∈ Z[[x]
], so that by the proven case of Theorem 1, the sequence (r n ) n≥0 satisfies Gauss congruences for all prime p; this well-known generalization of Fermat's little theorem will be used below.

Proof of Theorem 1. By Eisenstein's theorem, there exists an integer λ ≥ 1 such that

λf (λx) = ∞ n=1 λ n a n x n-1 ∈ Z[[x]]
and remains algebraic over Q(x). We can thus apply the proven case of Theorem 1 to η(x) := a 0 x + λf (λx) and y(x

) := x a 0 exp ∞ n=1 λ n a n n x n .
It follows that y is algebraic over Q(x) if and only if, for all n ≥ 0 and large enough p, we have λ np a np -λ n a n ∈ npZ (p) .

Now, the algebraicity of y is equivalent to that of x a 0 exp ∞ n=1 an n x n . Moreover for any n ∈ N and any p that does not divide λ, we have

λ np a np -λ n a n = λ np (a np -a n ) + (λ np -λ n )a n ≡ λ np (a np -a n ) mod npZ (p) ,
because λ np -λ n ∈ npZ (p) by the remark above with r = λ and a n ∈ Z (p) as λ n a n ∈ Z. Since λ is invertible in Z (p) , we obtain the following equivalence.

λ np a np -λ n a n ∈ npZ (p) ⇐⇒ a np -a n ∈ npZ (p) .
This completes the proof of Theorem 1.

The hypergeometric case

This section is devoted to the proofs of Theorems 2 and 3. By Theorem 1, this amounts to studying Gauss congruences for hypergeometric series.

Throughout those proofs, we will make an intensive use of the notations of Section 1.3. Furthermore, it is well-known that F α,β is factorial if and only if α and β are tuples of parameters in Q ∩ (0, 1] satisfying

(x -e 2iπα 1 ) • • • (x -e 2iπαr ) (x -e 2iπβ 1 ) • • • (x -e 2iπβs ) ∈ Q(x),
which is equivalent to saying that α and β are R-partitioned in the sense of [14, §7]. In this case, there exist tuples of positive integers e = (e 1 , . . . , e u ) and

f = (f 1 , . . . , f v ) such that (x -e 2iπα 1 ) • • • (x -e 2iπαr ) (x -e 2iπβ 1 ) • • • (x -e 2iπβs ) = (x e 1 -1) • • • (x eu -1) (x f 1 -1) • • • (x fv -1)
.

We write |e| := u i=1 e i and |f | := v j=1 f j . Then we have r -s = |e| -|f | and F α,β (Cx) = F e,f (x) with

C := e e 1 1 • • • e eu u f f 1 1 • • • f fv v . (4.1)
By Theorem 4 and Section 4.2.2 in [START_REF] Delaygue | On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps[END_REF], if r = s, then C is also the smallest positive rational number such that

F α,β (Cx) ∈ Z[[x]].

Hypergeometric series and Gauss congruences

This section is devoted to the proof of Theorem 4. To that purpose, we need two lemmas on congruences for hypergeometric terms. If α and β are tuples of non-zero rational numbers, then we write d α,β for the least common multiple of the exact denominators of the α i 's and β j 's.

Lemma 2. Let α and β be tuples of parameters in Q \ Z ≤0 such that F α,β is globally bounded. There exists a constant c α,β such that the following holds. Let p > c α,β be a prime number and k ∈ {1, . . . , d α,β } be such that kp ≡ 1 mod d α,β . Then, for all nonnegative integers m and s, we have

Q α,β (mp s+1 ) -Q kα , kβ (mp s ) ∈ p s+1 Z (p) .
Proof. We write d for d α,β . Let p > d be fixed. For every α ∈ Z (p) , there is a unique element D p (α) in Z (p) such that pD p (α) -α ∈ {0, . . . , p -1}.

The map α → D p (α) was used by Dwork in [START_REF] Dwork | p-adic cycles[END_REF] (denoted there as α → α ) to study the p-adic valuation of Pochhammer symbols. By a result of Dwork [17, Lemma 1] applied with a = µ = 0, if α ∈ Z (p) , then for all non-negative integers m and s, we have

(α) mp s+1 (D p (α)) mp s ∈ (-p) p s ε p s m 1 + p s+1 Z (p) , (4.2)
where

ε k = -1 if k = 2
Q α,β (mp s+1 ) Q kα , kβ (mp s ) ∈ 1 + p s+1 Z (p) , (4.3)
where the term with ε p s disappeared because α and β are tuples of the same length. To finish the proof of this lemma, it suffices to show that, for m ∈ Z ≥0 and p > c α,β , we have

Q kα , kβ (mp s ) ∈ Z (p) . (4.4)
Let us consider the total order on R defined by

x y ⇐⇒ x < y or ( x = y and x ≥ y) .

Christol has proved in [START_REF] Christol | Fonctions hypergéométriques bornées[END_REF] that F α,β is globally bounded if and only if α and β have the same length and if, for every a ∈ {1, . . . , d} coprime to d and every x ∈ R, we have

ξ α,β (a, x) := #{1 ≤ i ≤ r : aα i x} -#{1 ≤ j ≤ r : aβ j x} ≥ 0.
Let k ∈ {1, . . . , d} coprime to d be fixed. By [START_REF] Delaygue | On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps[END_REF]Proposition 16], we also have d kα , kβ = d and, for every a ∈ {1, . . . , d} coprime to d and every x ∈ R, we have ξ kα , kβ (a, x) ≥ 0. It follows by Christol's criterion that F kα , kβ is globally bounded.

Since kα and kβ are tuples of rational numbers in (0, 1], [START_REF] Delaygue | On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps[END_REF]Theorem 4] shows that F kα , kβ is a power series with coefficients in Z (p) for every prime p > d. This yields (4.4) which, together with (4.3), gives

Q α,β (mp s+1 ) -Q kα , kβ (mp s ) ∈ p s+1 Z (p) ,
as expected. Lemma 2 will be used to prove that a hypergeometric series has to be factorial to have the Gauss property. But we need the following lemma, proved in [14, Lemme 10] (4 ), to prove that those congruences remain valid for p ≤ c α,β .

Lemma 3 (Lemma 10 of [START_REF] Delaygue | Critère pour l'intégralité des coefficients de Taylor des applications miroir[END_REF]). Let e and f be tuples of positive integers such that |e| = |f |. Then, for all prime numbers p, all s ∈ N, all c ∈ {0, 1, . . . , p s -1} and all m ∈ N, we have

Q e,f (c) Q e,f (cp) Q e,f (cp + mp s+1 ) Q e,f (c + mp s ) ∈ 1 + p s+1 Z (p) .
Remark. The proof of this lemma does not require that Q e,f (n) ∈ Z for all n ≥ 0.

Proof of Theorem 4. Let α and β be tuples in Q \ Z ≤0 such that F α,β is globally bounded. We can assume without loss of generality that the tuples α and β are disjoint. We write d for d α,β and we recall that α and β have the same length.

• First, we assume that F α,β is factorial. Let e and f be tuples of positive integers such that F α,β (Cx) = F e,f (x) with C given by (4.1). By [15, Section 4.2.2] and [15, Theorem 4], we have F α,β (Cx) ∈ Z[[x]] so, for every n ∈ Z ≥0 , we have Q e,f (n) ∈ Z. We also have |e| = |f | because α and β have the same length so we can apply Lemma 3 with c = 0. Let m and s be non-negative integers and p a prime number. We write Q as a shorthand for Q e,f . We obtain that

Q(mp s+1 ) Q(mp s ) ∈ 1 + p s+1 Z (p) , which yields Q(mp s+1 ) -Q(mp s ) ∈ p s+1 Q(mp s )Z (p) ⊂ p s+1 Z (p) ,
because Q(n) is an integer for all n ≥ 0. It follows that F e,f satisfy Gauss congruences for all prime p.

• Conversely, assume that F α,β has the Gauss property. By Lemma 2, there exists a constant c α,β such that every prime p > c α,β and all non-negative integers m and s, we have

Q α,β (mp s+1 ) -Q kα , kβ (mp s ) ∈ p s+1 Z (p) , (4.5)
where k ∈ {1, . . . , d} satisfies kp ≡ 1 mod d. By definition of the Gauss property, there exists a constant κ > c α,β such that, for every prime p > κ, we also have

Q α,β (mp s+1 ) -Q α,β (mp s ) ∈ p s+1 Z (p) . (4.6)
Let a and k in {1, . . . , d} be such that ka ≡ 1 mod d. By subtracting Congruences (4.6) and (4.5) with s = 0, we obtain that, for every prime p > κ satisfying p ≡ a mod d and all non-negative integers m, we have

Q α,β (m) -Q kα , kβ (m) ∈ pZ (p) .
By Dirichlet's theorem, there are infinitely many prime numbers p > κ satisfying p ≡ a mod d so, for every m ∈ Z ≥0 , we have

Q α,β (m) = Q kα , kβ (m), (4.7)
this equation being valid for every k ∈ {1, . . . , d} coprime to d. Since α and β are disjoint, by [14, Proposition 1] and Equation (4.7) we obtain that, for every k ∈ {1, . . . , d} coprime to d, we have α = kα and β = kβ up to a permutation within the tuples. In particular, α and β are tuples of elements in (0, 1] and the polynomials r j=1

X -e 2πiα j and r j=1

X -e 2πiβ j are left invariant by the action of every Galois automorphism σ ∈ Gal(Q/Q). Hence those polynomials have integer coefficients and F α,β is factorial.

Abel's problem for hypergeometric series

In this section, we prove Theorems 2 and 3. The combination of Theorems 1 and 4 easily gives Theorem 2 as follows.

Proof of Theorem 2. Let α and β be tuples of parameters in Q \ Z ≥0 such that F α,β is algebraic over Q(x). By Theorem 1, the function exp F α,β (x)

x dx is algebraic over Q(x) if and only if F α,β has the Gauss property. Since F α,β is algebraic, it is globally bounded by Eisenstein's theorem. Now Theorem 4 shows that F α,β has the Gauss property if and only if it is factorial, which ends the proof of Theorem 2.

Before proving Theorem 3, we recall the interlacing criterion of Beukers and Heckman. We say that the sets {a 1 , . . . , a r } and {b 1 , . . . , b r } interlace on the unit circle if and only if either

λ 1 < µ 1 < λ 2 < µ 2 < • • • < λ r < µ r or µ 1 < λ 1 < µ 2 < λ 2 < • • • < µ r < λ r .
Let α = (α 1 , . . . , α r ) and β = (β 1 , . . . , β r ), with β r = 1, be tuples of rational numbers in (0, 1] such that α i = β j for all i and j. Let d be the common denominator of the α i 's and β j 's and write a j := exp(2πiα j ) and b j := exp(2πiβ j ) for all j. By the criterion of Beukers and Heckman [START_REF] Beukers | Monodromy for the hypergeometric function n F n-1[END_REF]Theorem 4.8], the hypergeometric series F α,β is algebraic over Q(x) if and only if, for every k ∈ {1, . . . , d} coprime to d, the sets {a k 1 , . . . , a k r } and {b k 1 , . . . , b k r } interlace on the unit circle.

Proof of Theorem 3. Let α and β = (β 1 , . . . , β r ), with β r = 1, be disjoint tuples of rational parameters in (0, 1] such that F α,β is algebraic over Q(x). We write d for d α,β . For every k ∈ {1, . . . , d} coprime to d, the tuples kα and kβ are disjoint and, by the above interlacing criterion, F kα , kβ is algebraic over Q(x) and de facto globally bounded. Write It follows that f is algebraic over Q(x). By Lemma 2 applied with kα and kβ instead of α and β respectively, we obtain that, for every large enough prime p and all non-negative integers m and s, we have Together with (4.8), we obtain, that for every large enough prime p and all non-negative integers m and s, we have

Q(mp s+1 ) -Q(mp s ) ∈ p s+1 Z (p) .
Hence f has the Gauss property. By Theorem 1, it follows that exp f (x)

x dx is algebraic over Q(x) and Theorem 3 is proved.

Corollary 1 .

 1 Let η be an algebraic function over C(x) which has a rational Puiseux expansion at δ ∈ C, respectively at δ = ∞. Then the differential equation y = ηy admits a non-trivial solution y algebraic over C(x) if and only if (x -δ)η(x), respectively xη(x), has the Gauss property at δ.

Definition 3 .

 3 Let a j = exp(2πiλ j ) and b j = exp(2πiµ j ), 1 ≤ j ≤ r, be two sets of numbers on the unit circle inC. Suppose 0 ≤ λ 1 ≤ • • • ≤ λ r < 1 and 0 ≤ µ 1 ≤ • • • ≤ µ r < 1.

F

  kα , kβ and Q(n) := d k=1 gcd(k,d)=1 Q kα , kβ (n).

Q

  kα , kβ (mp s+1 ) -Q a kα , a kβ (mp s ) ∈ p s+1 Z (p) ,(4.8)where a ∈ {1, . . . , d} is such that ap ≡ 1 mod d. Since a kα = akα = bα for b ∈ {1, . . . , d} satisfying ak ≡ b mod d, we haved k=1 gcd(k,d)=1 Q a kα , a kβ (n) = d b=1 gcd(b,d)=1 Q bα , bβ (n) = Q(n).

  Both authors have partially been funded by the ANR project De Rerum Natura (ANR-19-CE40-0018) for this research. The first author has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under the Grant Agreement No 648132.

  , and ε k = 1 otherwise. By[START_REF] Delaygue | On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps[END_REF] Lemma 23] applied with = 1, there exists a constant c α such that, for p > c α , we have D p (α) = kα where k ∈ {1, . . . , d} satisfies kp ≡ 1 mod d. Together with Equation (4.2), it follows that there exists a constant c α,β > d such that, for all p > c α,β , we have

We say that a property holds for almost all prime numbers p when it holds for all but finitely many prime numbers p.

Zagier has not explicitely proved that y e,f ∈ Q(x, F e,f ), but the indications he has given on the degrees of y e,f and F e,f prove this fact in the cases he considered, using the third remark made after Corollary 2.

If f (x) = k i=1 f i x αi is a Laurent polynomial in x 1 , . . . , x n with f i ∈ Q for all i,where we use the vector notation x e = x e1 1 • • • x en n , then the Newton polytope ∆ ⊂ R n of f is the convex hull of its support {α i : f i = 0}.

As essentially all p-adic congruences of this type proved till 2015 (say) in the context of "integrality of mirror maps", the proof of Lemma 3 uses in particular p-adic congruences for Morita p-adic Gamma functions proved by Lang [23, Chapter 14, Section 1, Lemma 1.1]. As it appeared later, Lang's congruences do not hold in one case (p = 2 and s = 2) and a corrected version is given in[15, §4.4]: a minus sign must be introduced. It turns out that Lemma 3 is still correct when p = 2 and s = 2 because this minus sign only contributes a harmless factor (-1) ej -fj = 1 in its proof.


Mariam Mnatsakanyan

Joel J Thevarajah

Rozet Sallom Roi

Antonio Lauto

Marianne Gaborieau
email: m.gaborieau@uws.edu.au

Patrice Castignolles

Acetylation using Simple Free Solution Capillary Electrophoresis

Keywords: capillary electrophoresis, chitosan, degree of acetylation, critical conditions, NMR spectroscopy

Chitosan is a biopolymer of increasing significance, as well as a renewable and sustainable material. Its main molecular characteristics are molar mass and degree of acetylation (composition). Precise average degrees of acetylation were measured by quantitative 1 H solutionstate NMR spectroscopy. While number-average degree of acetylation had already been determined by 1 H NMR spectroscopy, weight-average degree of acetylation are also determined and may be more relevant for some properties, such as mechanical properties. We report the first separation of chitosan according to its degree of acetylation using free solution capillary electrophoresis. Capillary electrophoresis separates chitosan in the 'critical conditions': the molar mass plays little role and the separation is by the degree of acetylation. It characterizes the heterogeneity of chitosan samples in terms of composition (dispersity of the distribution of degrees of acetylation). This heterogeneity (broad distribution of degrees of acetylation) cannot be neglected contrary to a common assumption found in the literature. This fast and easy separation will allow establishing structure-property relations.

Introduction

Chitin is the second most abundant polymer in the world [1]. This waste of the seafood industry is a renewable but also sustainable biopolymer source, whose partial deacetylation yields a copolymer: chitosan (Fig. S-1) including N-acetyl-D-glucosamine as well D-glucosamine units [1]. The copolymer composition is also named the degree of acetylation (DA) in the case of chitosan. Chitosan is biocompatible, biodegradable, antimicrobial and antifungal. The great variety of chitosan properties is determined by the polysaccharide's chemical structure [1,2], and this chemical structure changes with the source of origin and the extraction process. The natural variability of chitosan makes the characterization of its chemical structure very important. The DA, the degree of polymerization (DP, or the molar mass MM) and their distributions are among the most critical parameters that govern the properties of chitosan [1,2] Different methods are used to estimate or quantify an average DA. Some are based on indirect measurements such as chemical / enzymatic analysis or liquid chromatography [2]. Further methods are based on direct measurements with techniques such as 1 H, 13 C, and 15 N NMR spectroscopy in solution and in the solid state. 1 H solution-state NMR spectroscopy provides the most precise and accurate measurement of the average DA for chitosan [2], as it takes advantage of the high sensitivity of the 1 H nucleus in NMR. The measurement is quantitative if parameters are set up to allow full relaxation between scans (see Electronic Supplementary Material). Quantitative 1 H solution-state NMR was used in this work to quantify the average DA of the investigated chitosan samples. The literature never mentions whether number-average or weight-average DA are measured, although this distinction is nowadays the norm for molar masses in the literature.

Very little has been done to characterize the distribution of DAs. The distribution of DAs affects properties important for biomedical applications, such as solubility and biodegradability: samples with the same average DA (and the same distribution of molecular weights) are expected to exhibit different solubility and biodegradation behaviours. It has been shown recently for the first time using fractionation by preparative size-exclusion chromatography (SEC/GPC) coupled offline to 1 H NMR spectroscopy that chitosan had a significant distribution of DAs [3]. While the results of this SEC×NMR characterization are invaluable, the method is expensive and time-consuming. Furthermore, SEC does not separate by the DA (or molar mass) but by the hydrodynamic volume [4]. The most common alternative is liquid chromatography at the critical adsorption point (LC-CC). LC-CC does not separate the compound of interest by its molar mass and can thus separate complex polymers by their structure, including separation of copolymers by their composition [5]. The critical conditions in LC are however tedious to determine and recovery is poor (see Electronic Supplementary Material). We are proposing free-solution capillary electrophoresis (CE) as an alternative.

Free solution CE is used to separate DNA and proteins. Applications of free solution CE to other polymers have been successful [6], but applications to chitosan have been very limited (see Electronic Supplementary Material). No separation of chitosan by DA has ever been published (by CE or by any other separation method). In the case of polymers, no separation by molar mass is observed in free solution CE of single-stranded or double-stranded DNA, poly(styrene sulfonate) and a few other polyelectrolytes [6,7]. This effectively corresponds to the "critical conditions" sought in LC-CC. Free solution CE technique can be used as a method to separate polyelectrolytes by their microstructure, such as composition or branching. In analogy with LC-CC, we named the method CE in the critical conditions (CE-CC), as the absence of separation by molar mass allows separation by the microstructure [8,9]. Critical conditions in CE do not correspond to free-draining, but rather quasi-parallel increase of the nominal charge and the friction with the degree of polymerization (see Electronic Supplementary Material). The aim of this work is to explore the potential of NMR spectroscopy and CE-CC for the characterization of the heterogeneity of the DA in chitosan (dispersity of its distribution of degrees of acetylation) and for the fast separation of chitosan by its DA. In the case of molecular weight, the number-average is widely used to relate to the kinetics of polymerization, while the weight-average is widely used for different purposes, for examples to relate to the polymer properties. The relationship between DAn and DAw can easily be derived (see Electronic Supplementary Material, Equations S-4 to S-12):

Results and discussion

Details of the materials,

𝐷𝐴 w = 𝐷𝐴 n •𝑀 A 𝐷𝐴 n •𝑀 A +(1-𝐷𝐴 n )•𝑀 D (1)
Equation ( 1) is valid for any copolymer (in the particular case of chitosan, see Equation (S-12). DAw has never been used in the literature, but its correlation with different properties can be easily tested. Equation (S-12) was used to calculate DAw for the chitosan samples analysed in this work (Table S-1). It is to be noted that the weight-average degrees of acetylation are systematically higher than the number-average degrees of acetylation, and do not exactly correspond to the DA value given by the supplier. Like in the case of DAn NMR mentioned above, the trend in DAw values is the same as the trend in DA supp , for each supplier, but it is not the case between suppliers. The value of the dispersity from these averages correlates with DAn:

𝐷 = 𝐷𝐴 𝑤 𝐷𝐴 𝑛 = 1 𝑀 𝐷 𝑀 𝐴 ⁄ +(1- 𝑀 𝐷 𝑀 𝐴 ⁄ )𝐷𝐴 𝑛 (2)
This dispersity thus does not bring any information on the heterogeneity (in terms of degree of acetylation) of the chitosan polymer chains. This is due to the fact that NMR is a local method determining the acetyl groups at the monomer unit's level. In order to get information about the heterogeneity of the polymer chains, separation by DA is required.

Separation by the degree of acetylation using CE in the critical conditions

Fused-silica capillaries were used since they lead to no significant chitosan adsorption onto their surface at low pH (see Electronic Supplementary Material).

Capillaries used had a 50 m inner diameter, a 360 m outer diameter, a total length of 64, 61 or 40 cm, corresponding to a length to the detector of 55.5 cm, 51.5 cm or 31.5 cm, respectively; a fused silica capillary with extended light-path detection window was also used, with a 64 cm total length. Phosphate buffer at pH 2 was used previously to determine the amount of chitosan (it is transparent in UV contrary to acetic acid). In this work, we do not only want to determine chitosan but also to separate it by its DA. Electrophoretic mobilities of chitosan are the same at pH 3 and 3.5 indicating complete ionization at pH 3, which was thus preferred to avoid any degradation. In order to minimize or prevent chitosan adsorption and interaction on the capillary wall, a high concentration phosphate buffer is required. The separations are fast, taking less than 10 min (Fig. 1a), are observed [7]. The molar masses of the samples are not known, but they are high enough to be in the critical conditions. This is also consistent with the random order of migration in terms of viscosity: 55; 55; 570; 590; 320 and then 32 cps with increasing migration time. This CE-CC method is currently using the free solution CE technique, although most free solution CE separations currently published rely on different separations mechanisms than critical conditions. Note that CE-CC does not have to be restricted to the free solution CE technique.

Electrophoretic mobility according to the degree of acetylation

The lower the DA is, the higher the number of free amino groups is, which subsequently results in a higher cationic charge of the molecule and thus in a higher electrophoretic mobility. An equivalent behavior has already been observed for pectins and carboxymethyl cellulose. The electrophoretic mobility of pectins increases with the pectin's charge. Furthermore in an electrolyte solution the net charge of polyelectrolytes with a high charge density is not proportional to their charge density due to the condensation of counter ions onto the polymer backbone [10]. Counter ion condensation is a critical factor responsible for a nonlinear relationship between the degree of substitution (DS) and mobility of the carboxymethyl cellulose with a high charge density (DS) [10]. and DA may thus also best described by a nonlinear behavior. A lower selectivity for degrees of acetylation lower than about 17 % is consistent with Manning's condensation of the counter-ions (see Electronic Supplementary Material). The non-linear behavior of the mobility with DA was thus expected. The electrophoretic mobility has been empirically observed to be inversely proportional to the degree of substitution of pectins (the migration time has been observed to linearly correlate with the degree of substitution). In the case of chitosan, the correlation of the inverse electrophoretic mobility with either DAn or DAw leads to correlation coefficients poorer than the correlation of the mobility with the DAs (see Fig. 2b). We do not observe with our chitosan samples the same correlation as observed in the literature with pectins. This may be due to a broader distribution of substitution / acetylation in the case of chitosan or to solubility issues, which will be the subject of future investigations. The calibration of the electrophoretic mobility by the DAw or the DAn will allow the determination of the distribution of DAs, similarly to the distribution of molecular weights. Fig. 1 shows that a distribution of DAs exists for each sample: none seems to be monodisperse and they have different dispersities: as a crude estimate the full width at half maximum (fhwm) can be used. The samples with larger average DA values exhibit a higher dispersity of their distribution of DAs (see Fig. S-9 in Electronic Supplementary Material).

Conclusions

Chitosan's exceptional biological properties will enable it to achieve its potential Chitosan, Veraval, India and was deacetylated using a previously published method [3].

Quantitative 1 H solution-state NMR measurements

Background:

A high measurement temperature (70 °C to 90 °C) yields a higher resolution [4], while room temperature yields shorter measurement time and could be less damaging to the sample [5].

Experimental conditions

The samples were dissolved at 1 to 1.5 gL -1 in 50 mM DCl in D 2 O (same concentrations as in Table S-1) and heated at 65 C for 45 min. Solution-state 1 H NMR spectra were recorded on a Bruker Avance 400 spectrometer operating at a Larmor frequency of 400 MHz for 1 H and equipped with a 5 mm BBO probe, using a 23 s 90° pulse at 90 °C and a 21 s 90° pulse at 25 °C. Quantitative 1 H NMR spectra were recorded with a repetition delay of 10 s at 90 °C and of 6 s at 25 °C (see below for the determination of these repetition delays). At least 64 transients were accumulated. The chemical shift scale was calibrated with the HOD signal at 4.72 ppm at 25 °C and at 4.10 ppm at 90 °C [6]. Spectra were processed with Topspin 3.1 (Bruker Biospin).

Care was taken to ensure that the magnetization was fully recovered between pulses for all signals of interest, in order to determine true relative signal integrals [7]. To ensure full magnetization recovery between pulses, 1 H NMR spectra must be recorded with a delay between pulses longer than 5 times the longitudinal relaxation time (T 1 ) for each signal of interest. One-dimensional inversion-recovery T 1 relaxation measurements (1D T 1 ) were conducted, each with a single delay in the 'indirect dimension', and corresponding delays between pulses of 5  1.44 times that value. Phasing the spectra the same way as a conventional 1 H spectrum recorded in the same conditions, each signal is negative for short delay values, positive for long delay values, with a zero crossing occurring at T 1 /1.44.

At 90 °C, delays ranged from 1.042 to 3.47 s. for a delay of 1.389 s, the chitosan signals of interest changed from negative to positive, and thus their T 1 relaxation time is less than 1.389  1.44 = 2 s.

Measuring all subsequent 1 H solution-state NMR spectra at room temperature with a repetition delay of than 5  2 = 10 s thus ensured that full magnetization recovery took place and the 1 H spectra were quantitative.

S-4

At room temperature, delays ranged from 0.695 to 1.042 s. For a delay of 0.833 s, the chitosan signals of interest changed from negative to positive, and thus their T 1 relaxation time is less than 0.833  1.44 = 1.2 s. Measuring all subsequent 1 H solution-state NMR spectra at room temperature with a repetition delay of than 5  1.2 = 6 s thus ensured that full magnetization recovery took place and the 1 H spectra were quantitative.

Determination of average DA from 1 H solution-state NMR measurements

For the quantitative 1 H solution-state NMR spectra recorded at 90 °C, the number-average DA was calculated from Equation (S-1):

  3 / ) (CH ) 1 (H 3 / ) (CH 100 % 3 d 3 I I I DA NMR n    (S-1)
where I(CH 3 ) is the integral of the signal of the methyl group of the acetyl group at 1. For the quantitative 1 H solution-state NMR spectra recorded at 25 °C, the average DA was calculated from Equation (S-2):

  6 / 6) H 5 H 4 H 3 H 2 (H 3 / ) (CH 100 % 3 ,       I I DA RT NMR n (S-2)
where I(CH 3 ) is the integral of the signal of the methyl group of the acetyl group at 1.

Number-average versus weight-average degree of acetylation

Usual methods used to quantify the DA in chitosan, namely FT-IR and NMR spectroscopy, rely on the detection of individual monomer units via their relative numbers. Therefore they measure a number-average degree of acetylation DA n on the whole sample (not related to single polymer chains, but to the monomer units):

(S-4)

where n A is the number of acetylated monomer units, n D is the number of deacetylated monomer units.

The weight-average degree of acetylation DA w of the whole sample can be defined as:

• • • (S-5)
where M A is the molar mass of an acetylated monomer unit, M D is that of a deacetylated monomer unit.

(S-4) can be rearranged into 1 Combining (S-8) with (S-7):

• D A (S-9)

Capillary electrophoresis: experimental part and determination of mobilities

CE experiments were carried out using an HP 3D (Hewlett Packard, now Agilent Technologies, Australia) and an Agilent 7100 CE systems (Agilent Technologies, Waldbronn, Germany) equipped with a diode array detector (DAD). Polyimide-coated fused silica capillaries (50 m inner diameter, 360 m outer diameter, from Polymicro, USA) were used. Fused silica capillaries thermostated at 25 °C had a total length of 64 cm (55.5 cm effective length) or 60 cm (51.5 cm effective length) or 40 cm (31.5 cm effective length).

A fused silica capillary with extended light-path detection window (Agilent) was also used (64 cm total length). The capillary was pretreated by flushing for 10 min with 1 M NaOH, for 5 min with 0. Equation (S-13) was used to calculate the apparent electrophoretic mobility m i app (in m 2 •V -1 s -1 ) [START_REF] Deschamps | Size-Based Characterization of an Ionic Polydiacetylene by Taylor Dispersion Analysis and Capillary Electrophoresis[END_REF].

v • • (S-13)
where  i is the velocity, E is the electric field, L d is the length of capillary to the detector (m), L t is the total length of the capillary (m), t m is the migration time (s), and V is the electric field (V).

Equation (S-14) was used to calculate the electroosmotic mobility m eof :

• • (S-14)
where t eof is the migration time of the electroosmotic flow. t eof was measured to be 40 min.

Electrophoretic mobilities are presented rather than migration times for two reasons. First the separation is according to the electrophoretic mobility, and electrophoretic mobility can be used to understand and calibrate the separation [START_REF] Guillotin | Determination of the degree of substitution, degree of amidation and degree of blockiness of commercial pectins by using capillary electrophoresis[END_REF]. Second, the electrophoretic mobility is a molecular property of the samples while migration time is not [START_REF] Starkweather | Polyelectrolyte electrophoresis in a dilute solution of neutral polymers: Model studies[END_REF].

The electroosmotic flow and the electrophoretic mobility are in the same direction. Comparing the different samples, it is clear that higher degrees of acetylation correspond to lower apparent mobilities. The electroosmotic mobility was measured around 5.10 -9 m 2 .V -1 .s -1 (See Equation S-14) which is one order of magnitude lower than the apparent mobility. It is constant and it is negligible in a first approximation compared to the electrophoretic mobility. The apparent mobility is thus considered in this work to be equal to the electrophoretic mobility of chitosan in a first approximation (an internal standard could be added to improve the precision of the electrophoretic mobility values).

A shoulder is observed on the high mobility side of the electropherogram of the chitosan sample with an average DA 16.5 %. This might be due to some limited aggregation. Chitosan is known to be prone to aggregate in solution. [START_REF] Yanagisawa | SEC-MALS study on aggregates of chitosan molecules in aqueous solvents: Influence of residual N-acetyl groups[END_REF] Guillotin and coworkers [START_REF] Guillotin | Determination of the degree of substitution, degree of amidation and degree of blockiness of commercial pectins by using capillary electrophoresis[END_REF] characterized high methyl-esterified pectins by their charge density and concluded that the electrophoretic mobility increases with the pectin charge. The electrophoretic mobility has been empirically observed to be inversely proportional to the degree of substitution of pectins (the migration time has been observed to linearly correlate with the degree of substitution) [START_REF] Jiang | Determination of the degree of esterification of alkaline de-esterified pectins by capillary zone electrophoresis[END_REF]. In the case of chitosan, the correlation of the inverse electrophoretic mobility with either DA n or DA w (Fig. 2a) leads to similar correlation coefficients, respectively 0.84 and 0.83. These correlation coefficients are poorer than the ones for correlation of the mobility with the DAs (Fig. 2bS-5).

Fig. S-7

Reproducibility of the separation of chitosan chitAL (DA w NMR = 4.0 %). Samples were previously prepared by two analysts, then injected at 0.98 g L -1 using HP 3D CE with conventional capillary and buffers (red lines) or using 7100 CE with high sensitivity capillary (blue lines)

  (see Fig. S-1 for a description of DP and DA).

  Material, including Fig. S-2 and S-3), and the resulting DAn NMR was used in this work. The trend in DAn NMR values is the same as the trend in DA values provided by the supplier, DA supp , for each supplier. However, it is not the case between suppliers, presumably due to different experimental methods being used by different suppliers to determine DA. Note that the suppliers do not specify what method they use to determine DA, or if they determine a number-average (Equation S-4) or a weight-average (Equation S-5) DA.

  repeatable and reproducible (Fig.S-5to S-7). Seven different chitosan samples were successfully separated by their DA (Fig.1) and the separation reproduced by another operator (Fig.S-4). The shoulder observed at low migration times of Fig.1afor the chitosan with 16.5 % average DA might be due to some aggregation (see Electronic Supplementary Material).Derivatization or indirect detection was not necessary in this work to obtain sufficient sensitivity since polymers and not oligomers are studied. The sensitivity of the UV detection (at 200 nm) is high enough especially using a modern instrument. A high-sensitivity capillary can further increase the sensitivity (Fig.S-7). At 200 nm, the acetyl groups of the chitosan chains are detected, therefore the samples with higher DAs appear with larger peak areas when injected in similar concentrations (see Fig.S-8in Electronic Supplementary Material). The separations do not reveal the presence of oligomers: no sharp contiguous peaks

  Fig. 2a represents the relation between the DAw or the DAn and the electrophoretic mobility. Both give a relatively poor linear correlation (regression coefficient of 0.86 with both DAw and DAn) in the case of chitosan in this work. The relation between mobility

  as biomaterial with proper separation and characterization methods. We have shown that the existence of a distribution of degrees of acetylation (DA) is of importance (as is the distribution of molar masses). NMR spectroscopy is usually only used to determine number-average DAs (DAn), but weight-average DAs (DAw) can also been determined without any additional experimental work. In analogy to molar mass, the weight-average DA may be the most relevant parameter to correlate for example with rheological properties. Separation according to the distribution of DA is possible with simple free solution CE: CE in the critical conditions, CE-CC. CE-CC does not separate by molar mass but only by DA. The method is thus complementary to size-exclusion chromatography (SEC or GPC), the most popular separation method for polysaccharides. SEC separates by hydrodynamic volume that depends on both on DA and molar mass. By separating solely by DA, CE-CC can shed a light on the distribution of DA but also on the accuracy of the distribution of molar masses determined by SEC.

Fig. 1

 1 Fig. 1 Separation of chitosan samples by their DA with CE: electropherogram shown (a) as a function of migration time, and (b) as a function of apparent electrophoretic mobility. Samples AKbioD1, AKbioV2, AKbioV3, MedMW, AKbioD2, AKbioD3 and chitAL are shown. Conditions: 64 cm capillary total length (55.5. cm effective length), 30 kV voltage, sodium phosphate buffer (pH 3, 100 mM) as background electrolyte

Fig. 2 a

 2 Fig. 2 a Electrophoretic mobility against DAn (red circles, linear fit not shown, R 2 =0.861; the red dotted lines are a guide to the eye) and DAw (black triangles, linear fit not shown, R 2 =0.856; the black dashed lines are a guide to the eye); b Inverse electrophoretic mobility against DAn (linear fit shown as red dotted line: R 2 =0.837) and DAw (linear fit shown as black dashed-dotted line: R 2 =0.831); mobility and DA were measured respectively by CE and NMR for chitosan samples AKbioD1, AKbioV2, AKbioV3, MedMW, AKbioD2, AKbioD3 and

Fig. S- 1

 1 Fig. S-1 Chemical structure of a chitosan chain with a degree of polymerization DP and a degree of acetylation DA. Chitin corresponds to DA=1. Note that a chitosan sample contains a wide distribution of chains with different DP and different DA values. The glucosamine and N-acetyl glucosamine units are not arranged in a block copolymer structure as could be inferred from the figure, but have a more complex arrangement along the polymer chain (see introduction of the manuscript). The numbers from 1 to 6 are used for the signal assignment of the NMR spectra shown in Fig. S-2 and S-3

  7 ppm (corresponding to 3 protons of the acetylated units only), and I(H1 d ) is the integral of the signal of the anomeric proton of the deacetylated unit at 4.6 ppm (corresponding to one proton of the deacetylated units only). A typical spectrum is shown on Fig. S-2.

Fig. S- 2

 2 Fig. S-2 Typical quantitative 1 H NMR spectrum of chitosan at 90 °C (sample AKbioV3 is shown)

  8 ppm (corresponding to 3 protons of acetylated units only), and I(H2+H3+H4+H5+H6) is the integral of the signals of the protons H2 to H6 between 2.8 and 4.2 ppm (corresponding to 6 protons of all units, see Fig. S-1 for assignment). A typical spectrum is shown on Fig. S-3.

Fig. S- 3

 3 Fig. S-3 Typical quantitative 1 H NMR spectrum of chitosan at 25 °C (sample AKbioV2 is shown)

Average degrees of acetylation measured by quantitative NMR spectroscopy

  1 H solution-state NMR spectroscopy and capillary electrophoresis experimental conditions are given in Electronic Supplementary Materials.

	While measurements of average DAs at 25 °C and 90 °C are both precise, the
	high-temperature measurement is more accurate (see Electronic Supplementary

Table S - 1

 S1 Characteristics of investigated chitosan samples Average degree of acetylation (DA) as given by the supplier, analytical method not specified. b Viscosity as given by the supplier. Number-average DA measured in this work with1 H NMR at 90 °C. d Number-average DA measured in this work with 1 H NMR at 25 °C. e Weight-average DA determined in this work from DA n NMR and Equation (1). f Dispersity calculated as the ratio of DA w to DA n NMR . g Injection concentration used in the CE experiments. h Viscosity of 1% chitosan solution in a 1% acetic acid solution.

	Sample	DA supp (%) a	Visc supp (cps) b	DA n (%) c NMR	DA n (%) d NMR, RT	DA w (%) e	D f	C inj (mg.mL -1 ) g
	MedMW	20	590 h	15.5	16.7	18.7 1.21	1.5
	AKbioD1	13.9	35 i	19.8		23.6 1.19	1.4
	AKbioD2	9.8	55 i	13.6		16.5 1.21	1.2
	AKbioD3	4.4	55 i	3.5		4.3 1.24	1.3
	AKbioV2	14	320 i	18.7	16.9	22.4 1.20	1.4
	AKbioV3	14	570 i	16.5		19.8 1.20	1.5
	chitAL	2.5		3.2	2.7	4.0 1.24	1.3

a c i Viscosity measured at 20 °C.

  1 M NaOH, MQ water and 200 mM borate buffer. Three different running buffer concentrations corresponding to 50, 100 and 500 mM were tested. A 500 mM concentration results in too high currents around 290 µA, leading to unstable baselines and currents. Concentrations of 50 mM and 100 mM resulted in high but reasonable currents around 100 µA for capillaries total length (L t ) of 40 cm and 60 cm, respectively. For capillaries with L t of 64 cm total length the current was around 103 µA. Data analysis was carried out using the software

	Origin 8.

Acknowledgments

We thank Catherine Lefay and Denis Bertin (Marseilles, France) for discussions and providing chitosanand UWS for an infrastructure grant for the capillary electrophoresis and an equipment grant. MG thanks UWS for the research support funding associated with her Research Lectureship.

Electronic Supplementary Material by

Mariam Mnatsakanyan, 1,2,3 Joel J. Thevarajah, 1,2 Rozet Sallom Roi, 1 Antonio Lauto, 3 Marianne Gaborieau, 2 * Patrice Castignolles 1 for Separation of Chitosan by Degree of Acetylation using Simple Free Solution Capillary Electrophoresis

University of Western Sydney, Bioelectronics and Neuroscience (BENS) Research Group, The Marcs Institute, Campbelltown, Australia

This file contains the description of chitosan and of chitosan samples, the detailed experimental conditions for quantitative 1 H solution-state NMR measurements, the determination of average DA from 1 H solutionstate NMR measurements, the derivation of the relationship between number-average and weight-average degrees of acetylation, a description of the critical conditions for polymer separation in liquid chromatography and in capillary electrophoresis, the detailed experimental conditions for capillary electrophoresis, the calculation of the apparent electrophoretic mobility and the electroosmotic mobility, the separation of chitosan according to the degree of acetylation, repeatability and reproducibility studies, as well as the calculation of the Bjerrum length.

From which follows:

And finally:

The molar mass of the monomer units are M A = 203 g•mol -1 for the acetylated monomer unit and M D = 162 g•mol -1 for the (positively charged) deacetylated monomer unit. Thus (S-11) can be re-written as:

. (S-12)

Critical conditions: Liquid chromatography at the critical point of absorption (LC-CC) and capillary electrophoresis in the critical conditions (CE-CC)

Liquid chromatography at the critical point of adsorption (LC-CC) [9] is a powerful method to separate polymers according to their structure. It is mainly applied to copolymers (a polymer derived from more than one species of monomer), for example block copolymers [10], but LC-CC also applies to branched polymers [START_REF] Samman | Retention Behavior of Linear, Branched, and Hyperbranched Polyesters in Interaction Liquid Chromatography[END_REF][START_REF] Im | Retention Behavior of Star-Shaped Polystyrene near the Chromatographic Critical Condition[END_REF]. The critical conditions are a separation mechanism and refer to the absence of separation of a homopolymer by its molar mass. For example, critical conditions for a copolymer such as chitosan implies that chitosan homopolymers (DA=0) with different molar masses all elute at the same time. The molar mass thus becomes effectively "invisible". If copolymers based on chitosan were injected, the separation would then take place only according to the other comonomer. These critical conditions thus allow decomposing a complex problem in a series of simpler ones. The principle of the method is excellent and has attracted a lot of enthusiasm. LC-CC has however been typically plagued by low sample recovery and critical conditions that are very tedious to establish and maintain. [START_REF] Favier | Liquid chromatography at the critical adsorption point (LC-CAP) of high molecular weight polystyrene: pushing back the limits of reduced sample recovery[END_REF]14] On the contrary, we have shown that critical conditions in free-solution CE are very robust and that capillary electrophoresis in the critical conditions (CE-CC) can be used to investigate colloidal systems in which molecules are aggregating in solution, as is the case of the polysaccharide gellan gum. [START_REF] Taylor | Characterization of Gellan Gum by Capillary Electrophoresis[END_REF] CE-CC allows to determine the electrophoretic mobility, which is the proportionality constant between the velocity of the molecule under an electric field and this electric field. The electrophoretic mobility of a polyelectrolyte is independent of its molar mass [START_REF] Stellwagen | The free solution mobility of DNA[END_REF][START_REF] Cottet | From small charged molecules to oligomers: A semiempirical approach to the modeling of actual mobility in free solution[END_REF][START_REF] Meagher | End-labeled free-solution electrophoresis of DNA[END_REF] ("critical conditions"), but changes with the structure of the polyelectrolyte (such as copolymer composition, or branched structure). Critical conditions in CE do not correspond to free-draining as defined by Flory [START_REF] Flory | Configurational and frictional properties of the polymer molecule in dilute solution[END_REF], but rather to a quasi-parallel increase of the nominal charge and the friction with the degree of polymerization [START_REF] Muthukumar | Theory of electrophoretic mobility of a polyelectrolyte in semidilute solutions of neutral polymers[END_REF][START_REF] Barrat | Theory of polyelectrolyte solutions[END_REF]. It applies not only to chitosan but also to polyelectrolyte architectures in general. [START_REF] Taylor | Characterization of Gellan Gum by Capillary Electrophoresis[END_REF][START_REF] Maniego | Separation of poly(acrylic acid) salts according to the topology using capillary electrophoresis in the critical conditions[END_REF]

Separation by degree of acetylation by capillary electrophoresis

Free-solution CE proved successful without derivatization to determine chitosan [START_REF] Ban | Capillary electrophoresis of high-molecular chitosan: The natural carbohydrate biopolymer[END_REF], to separate chitosan from conjugates [START_REF] Fu | Analysis of natural carbohydrate biopolymer-high molecular chitosan and carboxymethyl chitosan by capillary zone electrophoresis[END_REF], to demonstrate successful chemical modification [28], or separate oligomers according to their DP [START_REF] Blanes | Direct detection of underivatized chitooligosaccharides produced through chitinase action using capillary zone electrophoresis[END_REF] (oligomers were also separated using indirect detection [START_REF] Hattori | Capillary electrophoresis of chitooligosaccharides in acidic solution: Simple determination using a quaternary-ammonium-modified column and indirect photometric detection with Crystal Violet[END_REF] or after derivatization [START_REF] Wang | Analysis of chitin oligosaccharides by capillary electrophoresis with laserinduced fluorescence[END_REF][START_REF] Beaudoin | Capillary electrophoresis separation of a mixture of chitin and chitosan oligosaccharides derivatized using a modified fluorophore conjugation procedure[END_REF] in CE or earlier by polyacrylamide gel electrophoresis [START_REF] Audy | Gel-electrophoretic analysis of chitosan hydrolysis products[END_REF]). Fused-silica capillaries were used since they proved to lead to no significant chitosan adsorption onto their surface at low pH [START_REF] Ban | Capillary electrophoresis of high-molecular chitosan: The natural carbohydrate biopolymer[END_REF] (although adsorption has been reported for oligomers [START_REF] Hattori | Capillary electrophoresis of chitooligosaccharides in acidic solution: Simple determination using a quaternary-ammonium-modified column and indirect photometric detection with Crystal Violet[END_REF]). Phosphate buffer at pH 2 was used previously to determine the amount of chitosan [START_REF] Fu | Analysis of natural carbohydrate biopolymer-high molecular chitosan and carboxymethyl chitosan by capillary zone electrophoresis[END_REF]. In this work, we do not only want to determine chitosan but also to separate it by its DA. A pH of 2 was not used to avoid degradation. A phosphate buffer at pH 3.01 was chosen for separations, as it protonates the amine groups of chitosan molecules and it is transparent in UV contrary to acetic acid, which is widely used as a mobile phase in LC [START_REF] Nguyen | Fractionation and characterization of chitosan by analytical SEC and H-1 NMR after semi-preparative SEC[END_REF]. Electrophoretic mobilities are the same at pH 3 and 3.5 indicating complete ionization of the chitosan at pH 3 [START_REF] Hattori | Capillary electrophoresis of chitooligosaccharides in acidic solution: Simple determination using a quaternary-ammonium-modified column and indirect photometric detection with Crystal Violet[END_REF].

The separation by DA presented in Fig.

Repeatability of the capillary electrophoresis separations

A good repeatability was obtained when using a 5 min wash with 10 mM phosphoric acid before each injection (See Fig

Fig. S-6

Reproducibility of the CE separation method studied by different analysts on HP 3D (black and red solid lines) and Agilent 7100 CE (blue and grey dotted lines) systems using capillaries with L t of 64 and 60 cm. All injections were hydrodynamic (30 mbar), injection time was 10 s

Influence of the degree of acetylation on the detection sensitivity

The samples were all injected with the same injection volume, in similar concentrations ranging from 1.2 to 1.5 mgmL -1 , but they strongly differ by their DAs, ranging from 4 to 23.6 % (see Table S-1). The observed peak areas for the different samples were plotted as normalized with the sample concentration, and showed no constant behaviour or systematic variation (see Figure S-8). The UV detection is at 200 nm. At this wavelength, it is expected that the acetyl group of the acetylated monomer units are predominantly detected. Therefore, the detection is expected to exhibit a higher sensitivity for the chains with a higher DA.

The observed peak areas for the different samples were plotted as normalized with the DA (and with the sample concentration), and a smaller scatter of the data was observed (see Figure S-8). This is in agreement with a higher detection sensitivity at 200 nm for the polymer chains with a higher DA.

Counter-ion condensation and DA

The Bjerrum length, l B , is calculated as [START_REF] Manning | Limiting laws and counterion condensation in poly-electrolyte solutions .7. Electrophoretic mobility and conductance[END_REF]:

where e is the elementary charge, taken as 1.60×10 -19 C,  0 is the vacuum permittivity, taken as 8.85×10 -12

Fm -1 ,  r is the dielectric constant taken as 80 (water), k B is the Boltzmann constant taken as 1.38×10 -23 m 2 kg -1 s -2 , and T is the temperature taken as 298 K.

It leads to:

If ones assumes a distance between monomer units of 8.5 Å on average for glucosamine and N-acetyl glucosamine, and a typical DA of 17 % (charge density of 83 %), then the distance between charges is 7.1 Å. This is thus consistent with counter-ion condensation for DA below 17 %.

Estimation of the dispersity of the distribution of DAs