Ricardo P Érez-Marco
email: ricardo.perez-marco@imj-prg.fr

E ÑE PRODUCT IN THE TRANSALGEBRAIC CLASS

Keywords: 2000 Mathematics Subject Classification. 08A02, 30D99, 30F99 Eñe product, exponential singularities, transalgebraic class

come

Introduction and Euler miscellanea

2. The transalgebraic class on a compact Riemann surface.

3. Eñe product structure on the transalgebraic class of P 1 (C).

Figure 1. From Évariste Galois' manuscripts, Ms 2108, Institut de France, [START_REF] Galois | [END_REF].

Introduction and Euler miscellanea

One of Euler's major discoveries is the transalgebraic nature of the exponential function as the unique "polynomial", normalized to take the value 1 at 0 as well as its derivative, with only one zero of infinite order at ∞ (see [START_REF] Euler | Introductio in Analysin, Facsimil edition RSME[END_REF]). Euler writes

e -z = 1 - z ∞ +∞ .
Hence, the exponential appears as a "transalgebraic polynomial". Observe that in this heuristic formula both infinite symbols are of a different nature: One is an infinite point (∞ in the Riemann sphere) and the other an infinite number (+∞ is the infinite order of the zero). Obviously, the proper justification of this heuristic comes from Euler formula, n can be found in Galois manuscripts [START_REF] Galois | [END_REF] (these may be course notes from his professor P.-L. É. Richard at Lycée Louis-le-Grand, so the authorship is unclear),

e z = lim
lim n→+∞ 1 + z n n = lim n→+∞ 1 + z nz nz = lim n→+∞ 1 + 1 n n z = e z
Note, that we do not want to use the logarithm function in the proof since the proper order to develop the theory is to define the exponential first. We also have a purely geometric proof that is the proper geometrization of Euler's transalgebraic heuristics, by using Carathéodory's convergence Theorem of the uniformizations for the Carathéodory convergence of the log-Riemann surfaces of n √ z to the log-Riemann surface of log z when n → +∞ (see [START_REF] Biswas | Tube-log Riemann surfaces[END_REF] for details, and [START_REF] Biswas | Caratheodory convergence of log-Riemann surfaces and Euler's formula, Geometry, groups and dynamics[END_REF] for background on log-Riemann surfaces). The construction of the log-Riemann surface of log z does not require the previous definition of the logarithm (on the contrary, we can define the logarithm function from it).

The main property of the exponential can be derived "à la Euler" as follows (using ∞ -2 << ∞ -1 ):

e z1 .e z2 = 1 + z 1 ∞ +∞ . 1 + z 2 ∞ +∞ = +∞ 1 + z 1 ∞ . 1 + z 2 ∞ = +∞ 1 + z 1 + z 2 ∞ + z 1 .z 2 ∞ 2 = +∞ 1 + z 1 + z 2 ∞ = 1 + z 1 + z 2 ∞ +∞ = e z1+z2
The exponential function is the link between the additive and the multiplicative structure on C. It serves also as the link between the multiplicative and the eñe product structure.

The eñe product.

We briefly recall the definition of the eñe product (see [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF]). Given two polynomials P, Q ∈ C[z], normalized such that P (0) = Q(0) = 1 say

P (z) = 1 + a 1 z + a 2 z 2 + . . . = α 1 - z α Q(z) = 1 + b 1 z + b 2 z 2 + . . . = β 1 - z β
where (α) and (β) are the respective zeros counted with multiplicity, then we define the eñe product ( [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF]) by

P Q(z) = α,β 1 - z αβ
Therefore, the divisor of P Q is the multiplicative convolution of the divisors of P and Q. If we write,

P Q(z) = 1 + c 1 z + c 2 z 2 + . . . then for n ≥ 1, c n = C n (a 1 , a 2 , . . . , a n , b 1 , b 2 , . . . , b n ) = -na n b n + . . . where C n ∈ Z[X 1 , . . . , X n , Y 1 , . . . , Y n ]
is a universal polynomial with integer coefficients and the dots in the left side of the formula represents a polynomial on a 1 , . . . , a n-1 , b 1 , . . . , b n-1 (see [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF]). This allows to define the eñe product f g of two formal power series

f (z) = 1 + a 1 z + a 2 z 2 + . . . g(z) = 1 + b 1 z + b 2 z 2 + . . . with coefficients a n , b n ∈ A, in a general commutative ring A by f g(z) = 1 + c 1 z + c 2 z 2 + . . . with c n = C n (a 1 , a 2 , . . . , a n , b 1 , b 2 , . . . , b n ) (
so without any reference to the zeros that we don't have in this general setting). We refer to [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF] for the rich algebraic and analytic properties of the eñe product. From the previous Eulerian heuristic of the exponential of a polynomial with an infinite order zero at ∞, it is natural to expect that the eñe product with an exponential must make sense and be an exponential. We do have a much more precise result: The exponential linearizes the eñe product. More precisely, we have a linear the exponential form of the eñe product (Theorem 4.1 from [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF]) in the following sense: If we write the power series in exponential form

f (z) = e F (z) g(z) = e G(z)
with formal power series (that have a finite non-zero radius of convergence when f and g are polynomials),

F (z) = F 1 z + F 2 z 2 + . . . G(z) = G 1 z + G 2 z 2 + . . . then we have f g(z) = e H(z)
with

H(z) = -F 1 G 1 z -2F 2 G 2 z 2 -3F 3 G 3 z 3 + . . . = - +∞ k=0 kF k G k z k
We denote e the linearized exponential form of the eñe product

F e G(z) = - +∞ k=0 kF k G k z k
Note that we could have defined the eñe product for polynomials in this formal way, but we would miss the original interpretation with the convolution of zeros (in particular because the disk of convergence of the exponential form never contains zeros!).

With similar heuristic ideas as before we can derive à la Euler the exponential form of the eñe product (for a rigorous proof see Theorem 4.1 from [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF]). Write as before

f (z) = e F (z) = 1 + F (z) ∞ +∞ g(z) = e G(z) = 1 + G(z) ∞ +∞
Then, note the double ∞ on the products and ∞ -3 << ∞ -2 , and compute

f g(z) = 1 + F (z) ∞ +∞ 1 + G(z) ∞ +∞ = +∞,+∞   1 + k≥1 F k ∞ z k     1 + k≥1 G k ∞ z k   = +∞,+∞ 1 + k -k F k ∞ . G k ∞ + O 1 ∞ 3 z k = +∞,+∞   1 + 1 ∞.∞ k≥1 -kF k G k z k   = 1 + (F e G)(z) ∞.∞ (+∞).(+∞) = e (F e G)(z)
Motivated by these transalgebraic heuristic considerations, the purpose of this article is to extend the eñe product to the class of transalgebraic functions, a class of functions with exponential singularities that we define precisely in next section. This is a good demonstration of the dual analytic-algebraic character of the eñe product.

The transalgebraic class on a compact Riemann surface.

We first define exponential singularities. The exponential singularities of finite order play the role of zeros of infinite order following Euler's heuristics. We are mostly interested in this section in the case of the Riemann sphere, or 1-dimensional projective space over C, X = C = P 1 (C) (genus g = 0), but there is little extra effort to define the transalgebraic class of functions T (X) for a general compact Riemann surface X. We denote M(X) the space of meromorphic functions on X, and M(X) * the non-zero meromorphic functions. Definition 2.1. A point z 0 ∈ C is an exponential singularity of f if for some neighborhood U of z 0 , f is a holomorphic function f : U -{z 0 } → C, f has no zeros nor poles on U and f does not extend meromorphically to U .

The exponential singularity z

0 ∈ C of f is of finite order 1 ≤ d = d(f, z 0 ) < +∞ if d is the minimal integer such that lim sup z→z0 |z -z 0 | d log |f (z)| < +∞
If no such finite order d exists, the exponential singularity is of infinite order and d = d(f, z 0 ) = +∞.

Let X be a Riemann surface. A point z 0 ∈ X is an exponential singularity for f if it is an exponential singularity in a local chart. The order d ≥ 1 has the same definition and is independent of the local chart.

Observe that we cannot have d = 0 because f would be bounded in a pointed neighborhood of z 0 and by Riemann's removability Theorem f will have an holomorphic extension at z 0 . Note also that the definition means that f has no monodromy around z 0 (i.e. f is holomorphic in a pointed neighborhood of z 0 ), and that z 0 is not a regular point nor a pole for f . Also z 0 is an exponential singularity for f if and only if it is also one for f -1 . We have a more precise result.

Proposition 2.2. The point z 0 ∈ X is an exponential singularity for f if and only if there is a local chart z where z 0 = 0 and f can be written in a neighborhood U of 0 as

f (z) = z n e h(z)
where n ∈ Z and h : U -{0} → C is holomorphic. The integer n ∈ Z if the residue of the logarithmic differential d log f = f /f dz at z = 0. The order d is finite if and only if h is meromorphic with a pole of order d at z 0 . The order of the pole of the logarithmic differential of f at z

0 is d + 1 ≥ 2.
Proof. The problem is local and we can assume z 0 = 0 and take U a simply connected neighborhood of 0 in a local chart with local variable z where f has no zeros nor poles on U . Then, the logarithmic derivative f /f is holomorphic on U -{0} and has a Laurent expansion

f (z) f (z) = n∈Z a n z n
The residue a -1 is an integer n ∈ Z since f has no monodromy around 0. We take for h the holomorphic function on U -{z 0 } defined by

h(z) = c 0 + n∈Z * a n-1 n z n
where c 0 ∈ C is an arbitrary constant to be chosen later. Then we have

f (z) f (z) = n z + h (z)
and, choosing the constant c 0 properly we have

f (z) = z n e h(z)
as desired. Conversely, such an expression has clearly an exponential singularity at z 0 = 0. From this expression, the order d is finite if and only if h has finite polar part of order d which is equivalent to have a pole of order d + 1 for the logarithmic derivative of f . Proposition 2.3. If f has an exponential singularity at z 0 then in any pointed neighborhood of z 0 the function f takes any value c ∈ C * infinitely often.

Proof. This is a direct application of Picard's Theorem since f does not take the values 0, ∞ ∈ C in a small pointed neighborhood of z 0 .

Definition 2.4. For a compact Riemann surface X and a finite number of punctures S ⊂ X, we define the space F(X, S) as the set of non-zero meromorphic functions f on X -S such that S is a set of zeros, poles, or exponential essential singularities of f or of its meromorphic extension. We define also

F(X) = S⊂X;S finite F(X, S)
Examples.

• For X = C and S = ∅ we have that F(C, ∅) = C(z) * is the space of non-zero rational functions.

• For X = C and S = {∞}, F(C, {∞}) is the set of functions f (z) = R(z)e h(z) where R ∈ C(z) * and h : C → C is an entire function, in the chart z ∈ C = C -{∞}. The exponential singularity at infinite is of finite order d ≥ 1 if and only if h ∈ C[z] is a polynomial of degree d ≥ 1. For d = 0, h must be constant and f is a non-zero rational function.

Proposition 2.5. The spaces F(X, S) and F(X) endowed with the multiplication of functions are multiplicative abelian groups. If f ∈ F(X) then f has a finite number of zeros, poles and exponential singularities.

Proof. These spaces are multiplicative groups from the remark made previously that z 0 ∈ X is an exponential singularity for f if and only if it is one for f -1 . The finiteness of exponential singularities follows from the finiteness of S. By compactness, an infinite sequence of zeros must have an accumulation point on X. It cannot accumulate a point of X -S or f would be identically zero. It can neither accumulate a point of S since they all have small zero free pointed neighborhoods. Hence we must have a finite number of zeros. The same argument for poles (or applied to f -1 ) gives a finite number of them. Definition 2.6. Let X be a compact Riemann surface, S ⊂ X a finite subset, and n ≥ 0. We define F n (X) ⊂ F(X) (resp. F n (X, S) ⊂ F(X, S)) as the subset of functions having at most n zeros and poles (both not counted with multiplicity), and exponential singularities (resp. with the exponential singularities located at S). We define T n (X) ⊂ F n (X) ⊂ F(X), resp. T n (X, S) ⊂ F n (X, S) ⊂ F(X, S), as the subset of functions with finite order exponential singularities.

We have the filtrations

F(X) = n≥0 F n (X) F(X, S) = n≥0 F n (X, S)
We define the class of transalgebraic functions T (X) and T (X, S) by

T (X) = n≥0 T n (X) T (X, S) = n≥0 T n (X, S)
We define also M n (X) ⊂ F n (X) to be the subset of meromorphic functions. We have, for n ≥ 0,

M n (X) ⊂ T n (X) ⊂ F n (X) M n (X) ⊂ T n (X, S) ⊂ F n (X, S) and M(X) ⊂ T (X) ⊂ F(X) M(X) ⊂ T (X, S) ⊂ F(X, S)
Remarks.

• The space F 0 (X) = C * is the set of non-zero constant functions.

• For X = C and n = 1, F 1 (C) is the set of functions which are Moebius conjugated to some e h(z) where h is an entire function. To see this, observe first that the function cannot be a constant nor a rational non constant function since for such a function the number of zeros and poles would be at least 2. For f ∈ F 1 (C) We can send the unique singularity to ∞ by a Moebius map, and f would be of the form e h(z) with h holomorphic in C. It follows that T 1 (C) are those functions Moebius conjugated to e h(z) where h(z) ∈ C[z] is a polynomial. Proposition 2.7. The spaces T (X) and T (X, S) endowed with the multiplication are groups.

Proof. From previous remarks they are invariant by taking inverses and their are clearly multiplicative invariant.

We define transalgebraic divisors. Definition 2.8. For f ∈ F(X), S(f ) denotes the set of exponential singularities of f . We define the transalgebraic divisor of f as the formal sum

Div(f ) = ρ∈X n ρ .(ρ) + ρ∈S(f ) d ρ .(ρ) ∞
where n ρ ∈ Z is the positive, resp. negative, order of the zero, resp. pole, at ρ or n ρ is the residue of the logarithmic derivative of d log f at ρ, or n ρ = 0 if ρ is neither a zero nor pole nor an exponential singularity, and 1 ≤ d ρ ≤ +∞ is the order of the exponential singularity at ρ when ρ is an exponential singularity, i.e. d ρ + 1 is the order of the polar part of d log f at ρ. The integer d ρ = d ρ (f ) is also called the transalgebraic degree of f at ρ.

The algebraic part of the divisor is

Div 0 (f ) = ρ∈X n ρ .(ρ)
and the transcendental part of the divisor is

Div ∞ (f ) = ρ∈S(f ) d ρ .(ρ) ∞ so that Div(f ) = Div 0 (f ) + Div ∞ (f ) .
The support of the transcendental part of the divisor is supp(Div ∞ (f )) = S(f ). The support of the algebraic part supp(Div 0 (f )) is the classical support of Div 0 (f ).

The support of the transalgebraic divisor of f is the finite subset of X supp(Div(f )) = supp(Div 0 (f )) ∪ supp(Div ∞ (f )) .

With these notations, we have

f ∈ F(X, S(f )) ∩ F | supp(f )| .
We recall that the set of compact subsets of a compact metric space is endowed with a natural Hausdorff distance and a Hausdorff topology. Any distance on X defining its compact surface topology defines the same Haussdorf topology on compact subsets. We fix one such distance d H and we define a topology on F(X) that is independent of the choice of d H . Definition 2.9. We define the topology on F(X) of uniform convergence out of the support of the divisor by defining a sequence

(f k ) convergent to f ∈ F if supp(Div(f k )) → supp(Div(f ))
in Hausdorff topology, and f k → f uniformly on compact sets out of supp(Div(f )).

We can construct a bases of neighborhoods (U (f )) >0 of an element f ∈ F(X) for this topology by taking for > 0, the -Hausdorff neighborhood of supp(Div(f )), V (supp(Div(f ))) and defining U (f ) to be the subset of g ∈ F(X) such that supp(Div(g)) ∈ V (supp(Div(f ))), i.e.

d H (supp(Div(g)), supp(Div(f ))) < and ||g -f || C 0 (X-W (supp(Div(f ))) <
where W (supp(Div(f ))) denotes the -neighborhood of supp(Div(f )) in X.

Proposition 2.10. The groups M(X) * , M(X, S) * , T (X), T (X, S), F(X) and F(X, S) are topological groups.

Proof. For the groups of fuctions with singularities in S, the multiplication and inverse are continuous since the set of zeros, poles and singularities are restricted to S. The larger groups are unions of those according to the filtration from Definition 2.4).

Proposition 2.11. The subgroup F(X, S) and the subspace F n (X) are closed in F(X).

Proof. We prove first that

F n (X) is closed in F(X). Consider a sequence (f k ) ⊂ F n (X) such that f k → f ∈ F(X).
The cardinal of finite sets is upper semi continuous for the Hausdorff topology, hence | supp(Div(f ))| ≤ n. Moreover, by Hurwitz Theorem the limit function f has no singularities and cannot have zeros nor poles in X -supp(Div(f )) (note that f cannot be the constant function 0 or ∞ since 0, ∞ / ∈ F). Hence f ∈ F n (X). Now the subspace F n (X, S) is closed in F(X) with the same proof.

The space M n (X) * is not closed as we can see using the Euler example where X = C with

f k (z) = 1 + z k k ∈ M 2 (C) * but f k (z) → e z / ∈ C(z) = M(C).
We can define a transalgebraic degree:

Definition 2.12. Let f ∈ T (X). The total transalgebraic degree of f is

d ∞ (f ) = ρ∈S(f ) (d ρ (f ) + 1)
We also define d 0 (f ) = | supp(Div 0 (f )) -S(f )|. We define the space T d0,d∞ (X) ⊂ T (X) as the subspace of those f ∈ T (X) with

d ∞ (f ) = d ∞ d 0 (f ) = d 0 .
For a finite set S ⊂ X we define T d0,d∞ (X, S) = T d0,d∞ (X) ∩ T (X, S)

Observe that for d ∞ = 0, T d0,0 (X) = M d0 (X) and for d ∞ ≥ 1, T d0,d∞ (X) ⊂ T d0+d∞-1 (X) since for any f ∈ T d0,d∞ (X) we have

| supp(Div(f ))| ≤ d 0 (f ) + d ∞ (f ) -1. Also we have |S(f )| ≤ d ∞ (f ).
The main Theorem in this section is that the closure of M n (X) is in the transalgebraic class. More precisely, Theorem 2.13. The closure of

M n (X) in F(X) is M n (X) ⊂ d0,d∞≥0 d0+d∞≤n T d0,d∞ (X) ⊂ M n (X) ∪ T n-1 (X) ⊂ T n (X) ⊂ T (X) For n ≤ |S|, the closure of M n (X, S) is M n (X, S) ⊂ d0,d∞≥0 d0+d∞≤n T d0,d∞ (X, S) ⊂ M n (X, S) ∪ T n-1 (X, S) ⊂ T n (X, S) ⊂ T (X, S)
The following two Lemmas are clear from the local analysis.

Lemma 2.14. Let f ∈ M(X) be a meromorphic function, f : X → C. The poles and zeros of f correspond bijectively to simple poles of the logarithmic derivative form f /f dz = d log f . The residue at these simple poles is the positive, resp. negative, multiplicity of the zero, resp. pole, of f . Lemma 2.15. Let f ∈ T (X). The logarithmic derivative d log f is a meromorphic differential d log f ∈ Ω 1 (X) with integer residues. Conversely, if f ∈ F(X) is such that d log f ∈ MΩ 1 (X) has integer residues at poles, then f ∈ T (X).

The following Lemma shows that the only way to have a creation of an exponential singularity ρ of transalgebraic degree d ρ (f ) at the limit for a converging sequence f k → f of rational functions is to have d ρ (f ) + 1 distinct sequences of poles and zeros of the f k converging to ρ (there must be both, poles and zeros). Lemma 2.16. Let (f k ) ⊂ M n (X) converging to f ∈ F(X) and let ρ ∈ S(f ) ⊂ X. Then there exists at least d ρ (f ) + 1 distinct sequences of poles and zeros of the (f k ) converging to ρ.

Proof. Consider a local chart at ρ and the logarithmic derivative f k /f k in this chart that has simple poles corresponding to zeros and poles of f k . We have f k /f k → f /f and at the limit we have at ρ a pole of order d ρ (f ) + 1 for f /f . The result follows from Rouché's Theorem. Note that poles and zeros can also annihilate each other is the have the same multiplicity.

Proof of Theorem 2.13. We consider a sequence of meromorphic functions (f k ) ⊂ M n (X) converging to f ∈ F(X). The associated sequence of meromorphic logarithmic derivatives d log f k have the support of their divisor Hausdorff converging to supp(Div(f )), and uniformly on compact sets outside X -supp(f ) we have d log f k → d log f . If m poles of the d log f k converge to a pole of d log f then the order of the pole is less or equal to m -1 and the residue is an integer as the sum of integer residues of d log f k of the converging points. Hence using the second Lemma 2.15 and counting poles we have that f ∈ T d0,d∞ (X). We have the same proof for the closure of M n (X, S).

Observe that from the proof we have that if the residues of the poles of the d log f k (or the order of the zeros or poles of f k ) that collapse into a pole of d log f are bounded, then the pole of d log f must be simple. Hence, the only way to have higher order poles for d log f is when poles and zeros of f k collapse into a point, the orders are unbounded, but their sum is asymptotically constant.

We have a converse, and any f ∈ T (x) can be approximated by a sequence (f k ) ⊂ M(X), and each exponential singularity can be realized as a limit of poles and zeros, but these more precise results will be studied elsewhere, since we are interested in this article on the simpler case X = C. We specialize the above results to X = C.

Observe that M n = M n (C) (we drop the dependence on X since X = C from now on) is the group of non-zero rational functions R : C → C with support of cardinal bounded by n, i.e. such that

|supp(R)| = |R -1 (0) ∪ R -1 (∞)| ≤ n .
We abuse the notation by writting R -1 (0), resp. R -1 (∞) to denote the set of zeros, resp. poles, of R, including the possible one at ∞ ∈ C. Note that we do not count multiplicities. So we have

M n = C(z) ∩ F n ⊂ F n .
Theorem 2.17. The group of transalgebraic functions T n are the functions f ∈ F n of the form

f = R 0 e R1
where R 0 = 0 and R 1 are meromorphic functions with R 1 = 0 or

(1) |R -1 0 (0) ∪ R -1 0 (∞)| + deg R 1 ≤ n .
In particular, the group of transalgebraic functions T is

T = {f = R 0 e R1 ; R 0 , R 1 ∈ C(z); R 0 = 0}
Proof. For n = 0 the result is clear, so we assume n ≥ 1. Such a function f = R 0 e R1 is clearly in T n . Conversely, given a function f ∈ T n , then the degree of the divisor Div 0 (f ) is 0 and we can choose a rational function K 0 such that Div K 0 = Div 0 (f ) . We can also choose a rational function K 1 with polar part matching the polar part of d log(f /K 0 ), in particular with the same integral residues. We consider the primitive exp

K 1 which is of the form exp K 1 = L 0 exp(R 1 )
where L 0 is a rational function coming from the integration of the order 1 polar part of K 1 , and R 1 from the higher order polar part and both are rational functions since K 1 has no monodromy at the support of Div ∞ (f ), and

R -1 1 (∞) = Div ∞ (f ) . Now, for R 0 = L 0 K 0 we have that R 0 e R1 /
f is a meromorphic function with no zeros nor poles, thus it is a constant and we can multiply R 0 by a non-zero constant so that f = R 0 e R1 .

It is instructive to understand how there transalgebraic functions arise as a limit of rational functions. When f = R 0 e R1 we have when k → +∞,

f k = R 0 1 + R 1 k k → f and f k ∈ M n (C) because of the inequality (1). Conversely, let f k → f ∈ F n (C) with f k ∈ M n (C).
The zeros and poles collapse into the divisor of f . Only when zeros and poles cohalesce with the sum of residues becoming asymptotically constant that we can have the emergence of an exponential singularity of finite order for f . We note a particular case of the previous theorem in the next Corollary.

Corollary 2.18. For a positive integer n ≥ 0, consider the space of non-zero polynomials P n ⊂ C[z] * normalized having exactly n zeros, not counted with multiplicity. We endow this space with the topology of uniform convergence on compact sets off the zeros as before. Then we have

P n = P n ∪ T P n ,
where T P n is the space of functions of the form f = P 0 e P1 where P 0 ∈ C[z] * is non-zero and

P 1 ∈ C(z), with |P -1 0 (0)| + deg P 1 + 1 ≤ n .
Proof. From the previous Theorem we get that all limits are of the form f = R 0 e R1 but the limit is holomorphic on C hence R 0 = P 0 and R 1 = P 1 are polynomials, and f = P 0 e P1 and P 0 is not identically 0. Each polynomial in P ∈ P n is a rational function with

|P -1 (0) ∪ P -1 (∞)| = |P -1 (0)| ∪ {∞} = n + 1 .
We apply the general theorem getting

|P -1 0 (0) ∪ P -1 0 (∞)| + deg P 1 + 1 = |P -1 0 (0)| + deg P 1 + 2 ≤ n + 1 and the result follows, |P -1 0 (0)| + deg P 1 + 1 ≤ n A particular case.
Let (P k ) be a sequence of polynomials with exactly n zeros, all escaping to ∞. If they are normalized conveniently in order to have a limit (for example, such that P k (0) = 1, P k (0) = 1), then the limit has no finite zero, nor pole, thus the limit must be of the form exp(P 1 ) where P 1 is a polynomial of degree at most n.

Historical comments.

Fields generated by functions with exponential singularities of finite order on a compact Riemann surfaces X of genus g > 0 have been studied by P. Cutillas Ripoll in a series of remarkable papers starting with [START_REF] Cutillas Ripoll | Construction of certain function fields associated with a compact Riemann surface[END_REF] (see also [START_REF] Cutillas Ripoll | isomorphisms between certains function fields over compact Riemann surfaces[END_REF] and [START_REF] Cutillas Ripoll | On ramification divisors of functions in a punctured compact Riemann surface[END_REF] and more recent articles). Cutillas proves the existence of minimal field of functions associated to the compact Riemann surface containing the space of meromorphic functions and realizing any divisor on X -S, S finite and non-empty. These minimal fields are all isomorphic and independent of S, so there is an abstract Cutillas field C(X) associated to any compact Riemann surface X. Moreover, he generates these function fields using functions in T (X, S), i.e. with exponential singularities located at S. This fantastic result seems to not be well known 1 . In [START_REF] Cutillas Ripoll | Construction of certain function fields associated with a compact Riemann surface[END_REF] it is pointed out that these functions have been considered by Clebsch and Gordan, and also by Weierstrass. They appear in chapter VII of the treatise by Baker [START_REF] Baker | Abel's theorem and the allied theory including the theory of theta functions[END_REF] where in footnote remarks there is some not very precise reference to Weierstrass work (see also [START_REF] Baker | An introduction to the theory of multiply periodic functions[END_REF]). Continuing the work of Baker, that was somewhat forgotten, functions with exponential singularities on hyperelliptic curves have been used, under the name of Baker-Akhiezer functions, by the russian school to construct explicit solutions of KP and KdV equations, see the surveys [START_REF] Dubrovin | Theta functions and non-linear equations[END_REF] and [START_REF] Buchstaber | Multi-Dimensional Sigma-Functions functions[END_REF].

Transalgebraic functions on the Riemann sphere were studied by Nevanlinna ([19], [START_REF] Nevanlinna | Analytic functions[END_REF]), and Taniguchi ( [START_REF] Taniguchi | Explicit representation of structurally finite entire functions[END_REF], [START_REF] Taniguchi | Synthetic deformation space of an entire function[END_REF]). They also appear naturally as uniformizations of log-Riemann surfaces defined by K. Biswas and the author ( [START_REF] Biswas | Tube-log Riemann surfaces[END_REF], [START_REF] Biswas | Caratheodory convergence of log-Riemann surfaces and Euler's formula, Geometry, groups and dynamics[END_REF], [START_REF] Biswas | Uniformization of simply connected finite type log-Riemann surfaces, Geometry, groups and dynamics[END_REF]) which roughly speaking are Riemann surfaces with canonical flat charts. In particular, in [START_REF] Biswas | Caratheodory convergence of log-Riemann surfaces and Euler's formula, Geometry, groups and dynamics[END_REF] the Caratheodory convergence of log-Riemann surfaces is defined, and a generalization to this setting of Caratheodory's Kernel Convergence Theorem is proved (Theorems 1.1 and 1.2 in [START_REF] Biswas | Caratheodory convergence of log-Riemann surfaces and Euler's formula, Geometry, groups and dynamics[END_REF]). As Corollary (Corollary 1.3 in [START_REF] Biswas | Caratheodory convergence of log-Riemann surfaces and Euler's formula, Geometry, groups and dynamics[END_REF])) a purely geometric proof of Euler's limit (which is central to this article) is obtained,

e z = lim n→+∞ 1 + z n n
Transalgebraic curves are defined in [START_REF] Biswas | Uniformization of simply connected finite type log-Riemann surfaces, Geometry, groups and dynamics[END_REF]. They generalize classical algebraic curves allowing infinite ramification points. More precisely, they are defined as log-Riemann surfaces having a finite number of finite (algebraic) and infinite (transcendental) ramification points. The Caratheodory closure of algebraic curves with uniformly bounded number of ramification points are proved to be transalgebraic curves (Theorem 2.11 in [START_REF] Biswas | Uniformization of simply connected finite type log-Riemann surfaces, Geometry, groups and dynamics[END_REF]). This result is the geometric counterpart of the previous Corollary 2. [START_REF] Hadamard | Théorème sur les séries entières[END_REF]. The more general closeness Theorem 2.13 is related to uniformizations of higher genus log-Riemann surfaces and more precisely to the main Theorem in [START_REF] Biswas | Uniformization of higher genus finite type log-Riemann surfaces[END_REF]. More precisely, it is proved in [START_REF] Biswas | Uniformization of higher genus finite type log-Riemann surfaces[END_REF] that any log-Riemann surface of finite topology (finitely generated fundamental group), is biholomorphic to a pointed compact Riemann surface X -S equipped with a transalgebraic differential form ω ∈ T Ω 1 (X), i.e. a differential form that is locally of the form ω = f (z)dz with f with exponential singularities, holomorphic out of S. Some of the transalgebraic properties of periods of transalgebraic curves are discussed in [START_REF] Biswas | The ramificant determinant[END_REF].

1 Cutillas article [START_REF] Cutillas Ripoll | Construction of certain function fields associated with a compact Riemann surface[END_REF] from 1984 has no citations according to Math Reviews, which shows how misguided is modern research.

3. Eñe product structure on the transalgebraic class of P 1 (C).

In this section we extend the eñe product to the transalgebraic class of the Riemann sphere T (C). The starting observation is the remarkable Convolution formula from [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF] (Theorem 5.3)

e z 1-z f (z) = exp α z α -z = α e z α-z
where (α) is the sequence of zeros of f . Observe that f (z) = e z 1-z ∈ T 0,1 ⊂ T 1 is a non-meromorphic transalgebraic function with a single point support for its divisor Div(f ) = (1) ∞

The function f (z) = e z 1-z is just the exponential function pre-composed by the Moebius transform mapping ∞ to z = 1 and tangent to the identity at 0. The convolution formula is what to expect if we consider the exponential singularity at 1 of e z 1-z as a zero of infinite order. We define a sequence of rational functions (R k ) ≥1 appearing in Euler's computations in [START_REF] Euler | Remarques sur un beau rapport entre les séries de puissances tant directes que réciproques[END_REF] p.85 (it is indeed -z -1 R k (-z) that Euler considers) for the summation of integer powers (see also [START_REF] Cartier | Fonctions polylogarithmesm nombres polyzêtas et groupes pro-unipotents[END_REF] where the rational functions Φ k are considered, and

R k = -Φ k-1 ), R 1 (z) = z z -1 = - +∞ n=1 z n and for k ≥ 0, R k = R 1 e . . . e R 1 = R ek 1 Therefore, for k ≥ 1, R k (z) = - +∞ n=1 n k-1 z n
and we can also define these rational functions by R k (0) = 0 and

(2) R k+1 = z dR k dz There follow the first seven rational function listed by Euler in [START_REF] Euler | Remarques sur un beau rapport entre les séries de puissances tant directes que réciproques[END_REF]

R 1 (z) = - z 1 -z R 2 (z) = - z (1 -z) 2 R 3 (z) = - z(1 + z) (1 -z) 3 R 4 (z) = - z(1 + 4z + z 2 ) (1 -z) 4 R 5 (z) = - z(1 + 11z + 11z 2 + z 3 ) (1 -z) 5 R 6 (z) = - z(1 + 26z + 66z 2 + 26z 3 + z 4 ) (1 -z) 6 R 7 (z) = - z(1 + 57z + 302z 2 + 302z 3 + 57z 4 + z 5 ) (1 -z) 7
In general we have, Proof. Using Theorem 2.17 any function f ∈ T (C) is of the form f = R 0 e R1 with R 0 , R 1 ∈ C, R 0 = 0. By Proposition 3.1 the rational functions R k (z/z 0 ) can be used to reconstruct any polar part in C * , so we can find a finite linear combination of functions R k (z/z 0 ) such that

R 1 -α 0 P 0 (1/z) -α ∞ P ∞ (z) - z0 k(z0) k=1 α k,z0 R k (z/z 0 )
has no poles in the Riemann sphere, hence it is a constant. Therefore, we have that f.

k,z0

e -α k,z 0 R k (z/z0) e -α0P0(1/z) e -α∞P∞(z) ∈ C(z) * is a non-zero rational function and the result follows.

Extension of the eñe product to T .

In view of the previous factorization given by Theorem 3.5, and the definition of the eñe product from [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF], we already have the eñe product in the subgroup of T of elements without a pole or singularity at 0. Using the projective invariance from Theorem 11.4 from [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF] for rational functions,

f (1/z) g(1/z) = f g(1/z)
we can extend the exponential form of the eñe product to Laurent developments in the exponential at 0, i.e. if

f (z) = e F (z) g(z) = e G(z) with F (z) = k∈Z F k z k G(z) = k∈Z G k z k then we have f g(z) = e H(z) with H(z) = - k∈Z kF k G k z k = F e G(z)
where we denote e the extension of the linearized exponential form of the eñe product

F e G(z) = - k∈Z kF k G k z k
With this definition we extend the eñe product to the full group T /C * = T , i.e. T modulo non-zero constants. Note that the eñe product with the constant 1 is the constant 1 that plays the role of additive zero in the following ring structure, Theorem 3.6. ( T , ., ) is a commutative ring.

The eñe product is clearly continuous on the generators of the transalgebraic class for the topology defined in Section 1. We get Remark on further algebraization.

We can observe that if we consider a subfield Q ⊂ K ⊂ C (for example a number field K), we can define T (P 1 (K)) as the sub-group of T with functions with exponential singularities at places in P 1 (K) ⊂ P 1 (C) such that d log f ∈ Ω 1 /K (P 1 (K)). Then, since the Euler rational functions have rational coeffidints, R k (z) ∈ Q(z), we can check that the proof of Theorem 3.5 goes through (the coefficients (α k,z0 ) are elements of K), and the eñe product extends to T (P 1 (K)) = T (P 1 (K))/K * and ( T (P 1 (K)), ., ) is a graded topological commutative ring.

For an algebraic curve X defined over a field K we can also define T (X, K), the transalgebraic class over K, and so on.

One of the magic in Euler's computations is its symbiotic analytic-algebraic content. Thus, it is not surprising that the eñe structures allow such algebraizations. We leave these rich algebraic extensions for future articles.

Eñe poles.

Once we have defined zeros of infinite order at finite places, it is natural to ask for the definition of poles of infinite order at finite places. Polylogarithm functions appear then naturally. Recall that polylogarithms are defined for k ≥ 1 by Li k (z) = +∞ n=1 n -k z n .

We have Li k (0) = 0, the series has radius of convergence 1, and we have a singularity at 1 which is a branching point. On the other sheets of its log-Riemann surface we have branchings at 0 and 1 for k ≥ 2, and only at 1 for k = 1 since Li 1 (z) = -log(1 -z) (see for example [START_REF] Oesterl É | [END_REF] for these geometric information and more properties of polylogarithms).

We can complete the sequence (R k ) k≥1 of Euler rational function to indexes k < 0 using the differential recurrence 2 and the condition R k (0) = 0, and we get R 0 (z) = Li 1 (z) = -log(1 -z) and for k ≤ 0, R k (z) = Li k (z) Proposition 3.9. We have e R k (z) e -Li k+1 (z) = 1 -z .

Proof. This follows directly from the exponential form, for k ≥ 0, R k e (-Li k+1 ) = log(1 -z) .

  proof, assuming known the existence of the elementary limit for e = 2.71828182846 . . .

Theorem 3 . 7 .

 37 ( T , ., ) is a topological commutative ring.Also it is clear that we have for n, m ≥ 0, Tn Tm ⊂ Tnm hence we have Theorem 3.8. ( T , ., ) is a graded topological commutative ring.

Proposition 3.1. For k ≥ 0, we have

where P k ∈ Z[z] and for k ≥ 1, (4) P k+1 (z) = (1 + (k -1)z)P k (z) + z(1 -z)P k (z)

and P 1 = 1. We have for k ≥ 2, deg P k = k -2, P k (1) = (k -1)!, P k (0) = 1, and the functional equations,

Hence, for k ≥ 2, R k vanishes at 0 and ∞, and has only one pole of order exactly k at 1.

Proof. By induction we get the recurrence (3), and the polynomial recurrence (4) follows from it. Making z = 1 in (4) we have for k ≥ 1, P k+1 (1) = kP k (1) and

Now, it is clear that for k ≥ 2, R k vanishes at 0 and ∞, and has only one pole of order exactly k at 1. Following Euler's intuition described in section 1, and the results from the previous section, it is natural to define a zero of infinite order at a finite place z 0 ∈ C "à la Euler": Definition 3.2. We define symbolically for

With this notation, the convolution formula can be rewritten as,

Now we define higher order zeros, Definition 3.3. For z 0 ∈ C * , we define symbolically,

is the function of exponential type with a single point transcen-

∞ Note the new fact that taking the eñe power of a simple infinite order zero at z 0 , changes the support of the new infinite zero of order k to z k 0 . From the exponential form of the eñe product we get Proposition 3.4. We have for k ≥ 0 and

and this can be written à la Euler

More generally, for n ≥ 1 we have, e R k 1 (z/z1) . . . e R kn (z/zn) = e R k 1 +...+kn (z/(z1...zn)) .

or, à la Euler,

The proof is clear from the definitions. It is satisfactory to check that this is the expected result from Euler heuristics. For example, we have the following formal computation à la Euler, for

Now, it easy to see that these transalgebraic functions generate the multiplicative group of transalgebraic functions on the Riemann sphere: Theorem 3.5. The multiplicative group (T (C), .) is generated by the non-zero meromorphic functions M(C) * , e P (z) , e P (1/z) , with P ∈ C[z], and for

and it is natural to define the eñe-pole of infinite order k ≥ 1 at 1 as

More generally, we define the pole of infinite order k "at" a finite place z 0 = 0, ∞ as Definition 3.10. For z 0 ∈ C * , we symbolically define

as the function with divisor a single pole of infite order k at z 0 ∈ C * (but the singularity is at z -k 0 ).

Note that this time these functions don't have exponential singularities, but branching singularities with non-trivial monodromy. The function with a single k-infinite pole at z 0 has a branching point located at z -k 0 . These functions take the value 1 at 0 (in their principal determination) and the eñe product is well defined through the exponential form. The important observation is that these type of singularities do appear naturally. Thus, it is natural to extend the eñe product to functions with singularities with non-trivial monodromy. This will be treated in subsequent articles.

With these definition we conclude, always à la Euler, Theorem 3.11. For k, l ∈ Z, z 1 , z 2 ∈ C, z 1 , z 2 = 0, ∞, we have

.


Evangelia Adamou

Annotation of information structure in an Ixcatec text

Information on Ixcatec

Name and ISO code: ʃ h wa 2 ni 3 , better known in the literature under the name Ixcatec (IXC), ixcateco (in Spanish), based on Nahuatl ichcatl 'cotton' + -teca/tecatl 'inhabitant of a place (whose name ends in -tlan or -lan)'.

Speakers: Ten identified speakers, of whom only four are fluent. Most of themwith one exception-are in their late 80s. All are bilingual in Spanish. They have had little formal education in Spanish and no formal education in Ixcatec.

Region: Ixcatec is spoken in the municipality of Santa María Ixcatlán in the state of Oaxaca, in Mexico. Today, Santa María Ixcatlán has some 400 inhabitants but at the time of the arrival of the Spaniards in 1522 it was an important centre for the Mixteca zone with an estimated population of 10,000 to 30,000 people.

Classification: Ixcatec belongs to the Popolocan branch of the Otomanguean stock together with Ngiba/Ngigua (also known as Chocho), Popoloc, and Mazatec.

Dialectology: There are no known dialects.

Status: Ixcatec is a critically endangered language, with less than ten speakers. An orthography was developed in the 1950s by a native Ixcatec speaker, Doroteo Jiménez, in collaboration with linguists of the Instituto Lingüístico de Verano, the Mexican branch of the Summer Institute of Linguistics. Doroteo Jiménez's orthography uses the Latin script and relies on the graphic correspondences with Spanish with some additions when necessary.

Main typological features: Ixcatec is a tone language, with three lexically contrastive tones: a high tone, transcribed with a superscripted¹, a mid tone, transcribed with², and a low tone, transcribed with³. Its phonology is complex and not yet well understood. The existence of stress is under discussion. Consonant inventory ranges from 24 to 52 depending on whether glottalized and aspirated consonants are analyzed as clusters of two segments, complex single segments or simple onsets followed by simple nuclei. It has five vowels which may be oral, /a e i o u/, or nasal /ã ẽ ĩ õ ũ/. Ixcatec makes a clear distinction between verbs and nouns; some adjectives may also function as predicates. It is a head-marking language, i.e., grammatical relations are marked on the verb. It has accusative alignment in indexing (A = S ≠ P), i.e., only the single argument of intransitive verbs (S) and the agent-like argument of transitive verbs (A) are indexed on the verb through suffixes. A dozen experience predicates take a different coding, namely through possessive suffixes. Ixcatec is a pro-drop language, i.e., free pronouns are optionally used for all functions, and NPs are generally omitted. It has a VS/SVO unmarked order. When an S argument is moved to the preverbal position, a cross-reference morpheme is suffixed on the verb. The Ixcatec crossreference morphemes (-da² 'male', -k w a² 'female', and -βa³ 'animal') corefer to nouns formed with the noun classifiers, di²-'man', k w a²-'woman', ʔu²-'animal', to some animate nouns even though they have no classifier, and to the masculine and feminine third singular pronouns which bear the same suffixes as those used for the cross-reference morphemes, i.e., su¹wa¹-da² 'he' and su¹wa¹-kwa² 'she'.

Noun classifiers are distinct from so-called class terms which partake in word formation for inanimates but are not associated with any cross-reference morphemes.

Annotation of information structure in an Ixcatec spoken text

For the annotation of Information Structure in Ixcatec (Otomanguean) I apply the "Annotation guidelines for questions under discussion and information structure" by [START_REF] Riester | Annotation guidelines for Questions under Discussion and information structure[END_REF]. The Question under Discussion (QUD) approach aims to specify how the listener recovers the information structure from a text. QUD relies on three principles:

Q-A-Congruence:
QUDs must be answerable by the assertion(s) that they immediately dominate.

Maximize-Q-Anaphoricity:

Implicit QUDs should contain as much given material as possible.

Q-Givenness:

Implicit QUDs can only consist of given (or, at least, highly salient) material.

Questions under discussion are annotated in the Ixcatec ELAN file as follows: {What are you doing?}

In the Ixcatec ELAN file, the following IS categories were annotated:  Focus: Definition: "The part of a clause that answers the current QUD" (Riester et al. 2018: 441).

In Ixcatec: Contrastive and corrective focus can be marked in Ixcatec with the focus marker -na² and prosodic marking [START_REF] Adamou | Prosodic and morphological focus marking in Ixcatec (Otomanguean)[END_REF].

 Restrictive focus:

Definition: It restricts the scope of X (e.g. referent, location, temporal span) in relation to a set of alternatives.

In Ixcatec: Marked with an adverb ʰngwa² 'just'.

 Topic:

Definition: A discourse referent identifying what the sentence is about (Riester et al. 2018: 417).

 Frame setting topic: Definition: A topic that sets the location and time.

 Abouteness topic: Definition: "A referential entity ("term") in the background which constitutes what the utterance is about" (Riester et al. 2018: 441).

 Contrastive topic:

Definition: "The instantiation of a variable within the background, which signals the existence of a superquestion-subquestion discourse structure. CTs are backgrounded w.r.t the subquestion and focal with respect to the superquestion" (Riester et al. 2018: 441).

In Ixcatec: CTs are constituents related to alternative questions and are optionally marked in Ixcatec by the focus marker -na² [START_REF] Adamou | Prosodic and morphological focus marking in Ixcatec (Otomanguean)[END_REF].

 Antitopic:

Definition: Serves to reactivate a previously active discourse-topic [START_REF] Lambrecht | Dislocations[END_REF].

In Ixcatec: Always in last position, separated by prosodic phrasing.

 Afterthought:

Definition: Right-detached constituents which express focus. This is a tentative annotation that needs to be improved with further work on IS categories in Ixcatec. Bibliography Adamou, Evangelia. (2014). L'antipassif en ixcatèque. Bulletin de la Société de Linguistique de Paris 109(1): 373-396. Adamou, Evangelia. (2017a). Subject preference in Ixcatec relative clauses.

Studies in Language 41(4): 872-913.

  and Corpus Annotation of Information Structure and Grammatical Relations" (Axe 3/GD1) of the excellence cluster Empirical Foundations of Linguistics (Labex EFL) funded by the French National Research Agency (Investments for the Future, ANR-10-LABX-0083)


Julien Mozziconacci
email: mozziconacci@lptmc.jussieu.fr

Mélody Merle

Annick Lesne
email: lesne@lptmc.jussieu.fr

The 3D genome shapes the regulatory code of developmental genes

Keywords: gene regulatory code, adaptor, transcription factor, enhancer, insulator, gene domain, topologically associating domain topologically associating domain, TF, transcription factor, tRNA, transfer RNA

We revisit the notion of gene regulatory code in embryonic development in the light of recent findings about genome spatial organisation. By analogy with the genetic code, we posit that the concept of code can only be used if the corresponding adaptor can clearly be identified. An adaptor is here defined as an intermediary physical entity mediating the correspondence between codewords and objects in a gratuitous and evolvable way. In the context of the gene regulatory code, the encoded objects are the gene expression levels, while the concentrations of specific transcription factors in the cell nucleus provide the codewords. The notion of code is meaningful in the absence of direct physicochemical relationships between the objects and the codewords, when the mediation by an adaptor is required. We propose that a plausible adaptor for this code is the gene domain, that is, the genome segment delimited by topological insulators and comprising the gene and its enhancer regulatory sequences. We review recent evidences, based on genome-wide chromosome conformation capture experiments, showing that preferential contact domains found in metazoan genomes are the physical traces of gene domains. Accordingly, genome 3D folding plays a direct role in shaping the developmental gene regulatory code.

Introduction

The development of a multicellular organism from a unique cell, the fertilised egg, centrally relies on cell differentiation. In this process, local differences in gene expression lead to the formation of distinct cell types in different places in the embryo. Each gene responds to the local concentrations of transcription factors (TFs) in a way that determines its own transcriptional output. The initial symmetry breaking that leads to different TF concentrations in different cells of the embryo varies from one organism to another [START_REF] Davidson | Spatial mechanisms of gene regulation in metazoan embryos[END_REF]. In Drosophila, for instance, the diffusion of maternal factors from one pole of the embryo to the other generates gradients that account for a first spatial heterogeneity in TF concentrations, along the anterior-posterior axis. Once this first patterning is formed, a cascade of genes is activated resulting in robust spatial distribution of gene expression. This distribution, which is the basis of the future body plan, thus critically depends on regulatory dependencies between genes. A complex combinatorics relates TF concentrations and the target gene expression [START_REF] Reiter | Combinatorial function of transcription factors and cofactors[END_REF][START_REF] Scholes | Combinatorial gene regulation through kinetic control of the transcription cycle[END_REF][START_REF] Inukai | Transcription factor DNA binding: beyond binding site motifs[END_REF]. Several studies have addressed the potential existence of a gene regulatory code, which would regulate gene expression at the transcriptional level. The very term of gene regulatory code has been introduced in studies looking for an association between regulatory DNA sequences and spatial gene expression patterns [START_REF] Michelson | Deciphering genetic regulatory codes: a challenge for functional genomics[END_REF][START_REF] Markstein | A regulatory code for neurogenic gene expression in the Drosophila embryo[END_REF]. The term cis-regulatory code has more recently been used to sum up the influence on gene expression of regulatory sequences and their linear organisation along the genome [START_REF] Yáñez-Cuna | Deciphering the transcriptional cis-regulatory code[END_REF]. We will here critically re-evaluate these notions and propose that a carefully defined notion of code is instrumental for interpreting most recent experimental data.

Figure 1---Definition of a semantic code by its adaptors. A) As an emblematic example of a semantic code, the genetic code is mediated by tRNAs, each bridging a codon (triplet of nucleotides, in blue) with an amino acid (red oval). Named 'adaptor' by

Crick [START_REF] Crick | On protein synthesis[END_REF], a tRNA is a chimeric entity, joining a part (the anticodon, in blue-green) recognising the codon and a part (in yellow) specifically binding the amino acid. B) The general scheme of an adaptor emphasises the evolutionary-devised assembly, within a single molecular entity, of a site (in bluegreen) specifically binding the codeword (in blue) and a distant site (

in yellow) specifically binding the encoded object (in red). This adaptor defines the code. C) The adaptor of the gene regulatory code must comprise the distal regulatory sequences (enhancers, blue-green) specifically binding transcription factors (in blue), and the gene promoter (yellow) binding the active polymerase (red).

Our starting point is the definition of a code that will be used in the present text. Different meanings of this word are encountered in science, from the secret codes in cryptography and the source codes in computer science to the neural codes and the genetic code. The latter is the emblematic example of a semantic code, in a biological context (Fig. 1A). The definition of a semantic code relies on three ingredients, namely codewords, objects, and adaptors (Fig. 1B):

•

codewords are inputs to be interpreted;

• a single object is associated to each codeword;

• adaptors are physical entities that implement the association of each codeword with a unique object, which can be seen as 'the object encoded by the codeword'.

Semantic codes go beyond a catchword, as their ability to evolve and their gratuity, in the sense introduced by Monod [START_REF] Monod | Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology[END_REF], provide evolutionary and mechanistic insights. To clarify the aforementioned relationships between TF concentrations, enhancers and gene expression, we will revisit the notion of gene regulatory code, taken with a semantic meaning. In this context, the codewords would be TF concentrations while the objects would be the expression status of a gene, with the idea that a particular set of TF concentrations elicits a defined transcriptional output. The missing link required to determine a gene regulatory code is the identification of the adaptor. We will base our analysis on recent experimental quantifications of gene expression during Drosophila embryogenesis and insights coming from the detailed study of 3D genome organisation in the surroundings of genes.

A semantic code is characterised by its adaptors

During the early years of the discovery of the genetic code, Crick [START_REF] Crick | On protein synthesis[END_REF] proposed that some molecular entities that relate codons to amino-acids should exist. They are known today to be tRNAs. Crick originally called them adaptors (and actually argued that they could be made of RNA). This term of adaptor is particularly meaningful, as it emphasises the mediating role of this set of molecules: each tRNA binds a codon and the corresponding amino acid at two different sites (Fig. 1A). The emblematic example of the genetic code actually meets the definition of a semantic code, as a conventional correspondence between a set of objects and a set of codewords, without any particular correspondence being imposed by the laws of nature [START_REF] Barbieri | Biosemiotics: a new understanding of life[END_REF]. In a biological context, a correspondence between two molecular entities thus deserves to be termed a code when mediated by molecular adaptors that are capable of recognising separately each of these entities (Fig. 1B). By its very existence, the adaptor circumvents the need of direct physical, chemical or stereo-chemical interactions between the two entities. Which one of them is the codeword (the other being the encoded object) follows from the functional status of the correspondence. For instance, in the case of the genetic code, the codewords encoding the amino acids at the genomic level are the codons.

Identifying the adaptors demonstrates that speaking of a code is not a mere metaphor: it offers mechanistic insights on the evolutionary origin of the code. The code is conventional in the sense that the relationship between codewords and objects does not result from some direct and unavoidable physicochemical interaction, but from the mediation of an adaptor. In principle, any pair composed of a codeword and an object could be related once the proper adaptor has been produced in the course of molecular coevolution. The arbitrariness of a code implies its ability to evolve. Indeed, the mapping relating codewords and encoded objects can be changed by a change in the adaptor. This offers a way to devise synthetic variants, as done for the genetic code [START_REF] Wang | Expanding the genetic code[END_REF].

Given this operational definition, it appears that the word code is often misleadingly used in the scientific literature. In particular, several so-called 'codes' have been introduced in a biological context at the molecular level, for instance sequence codes [START_REF] Trifonov | The multiple codes of nucleotide sequences[END_REF] or nucleosome codes [START_REF] Segal | A genomic code for nucleosome positioning[END_REF]. Sequence codes are based on the affinity between base pairs A/T and G/C, i.e. the physically prescribed Watson-Crick pairing rule, and not on a gratuitous association, as required for a semantic code. Similarly, nucleosome codes are based on the physicochemical affinity between DNA sequence and the histone core, and do not involve any adaptor. In these cases, speaking of a code is only metaphoric, based on the plain meaning of the word in common speech. In short, calling some association a code is pointless when a modification in the laws of physics and chemistry would be required to change this association.

A domain including the gene promoter and its enhancers is a plausible adaptor of the gene regulatory code

Prompted by the above caveats about the relevance (or irrelevance) of naming a relationship a code, we will examine published facts on transcriptional regulation, in the context of development. The discussion in the previous section provides a critical guideline: the existence of a gene regulatory code would be assessed by identifying its adaptor.

Figure 2 ---Transcriptional control at an enhancer.

The transcription of a gene can be jointly activated by the binding at the enhancer of some TF (blue star) and inhibited by the binding of some other TF (blue disk). Considering an all-or-none control for simplicity, the pair of TF concentrations acts on the enhancer in a way that triggers transcription when the value given by the truth table is 1, while the gene remains silent when the value is 0. Transcription regulation at the enhancer is thus encoded in a combinatorial codeword, here [START_REF] Davidson | Spatial mechanisms of gene regulation in metazoan embryos[END_REF]0). In real situations, the regulation involves TF concentration values (not only TF presence or absence as sketched here) and the transcriptional response is gradual.

Pioneering studies investigating the early development of sea urchin have led to the discovery of genomic sequences, lying beside a developmental gene, which are necessary and sufficient to determine the transcription of this gene in response to the proper external cues. These sequences spanned a 2kb region, which was then referred to as the gene regulatory domain in the seminal publication [START_REF] Thézé | Sequences of the Cyiiia actin gene regulatory domain bound specifically by sea urchin embryo nuclear proteins[END_REF]. Many such gene regulatory domains were later identified in different organisms. Their genetic dissection led to the identification of specific short sequences of ~100 bp that are rich in TF binding sites and essential for gene expression. These sequences were shown to coincide with sequences today known as enhancers, originally discovered in a different context [START_REF] Müller | Enhancer sequences and the regulation of gene transcription[END_REF]. Enhancers are now recognised as the basic ingredient of the developmental gene regulation in metazoans (see [START_REF] Ong | Enhancer function: new insights into the regulation of tissue-specific gene expression[END_REF] for a review). The current understanding of their action involves the formation of an enhancerpromoter loop [START_REF] Pombo | Three-dimensional genome architecture: players and mechanisms[END_REF][START_REF] Rickels | Enhancer logic and mechanics in development and disease[END_REF]. While the simplest scheme involves only one TF, clusters of TF binding sites are usually found at an enhancer [START_REF] Yáñez-Cuna | Deciphering the transcriptional cis-regulatory code[END_REF]. Some TFs can be activators, i.e. their binding to the enhancer is a prerequisite to transcription initiation, whereas other TFs are inhibitors, i.e. their binding to the en-hancer prevents transcription initiation [START_REF] Yáñez-Cuna | Deciphering the transcriptional cis-regulatory code[END_REF]. TF can even have dual functions [START_REF] Bauer | Dual-functioning transcription factors in the developmental gene network of Drosophila melanogaster[END_REF]. Different combinations of TF concentrations are then associated with different transcription outputs. However, singleenhancer action is not sufficient to explain gene patterning during development. Most often genes are under the control of several enhancers [START_REF] Perry | Multiple enhancers ensure precision of gap geneexpression patterns in the Drosophila embryo[END_REF]. In parallel, it has been observed in Drosophila embryo that different combinations of TF concentrations can be associated with the same level of expression for the target developmental gene [START_REF] Fowlkes | A quantitative spatio-temporal atlas of gene expression in the Drosophila blastoderm[END_REF][START_REF] Petkova | Optimal decoding of cellular identities in a genetic network[END_REF].

Figure 3 --Gene domain as a plausible adaptor of a gene regulatory code during development A) In general, the transcription of genes that are differentially expressed in space and time, typically developmental genes, is controlled by multiple enhancers (in blue-green) within a gene-centred genomic region delineated by insulators (magenta crosses) which we call the 'gene domain'. Contact frequencies between the different regions of the domain, determined by genome-wide chromosome conformation capture experiments (Hi-C), are depicted with a pink colour-scale on top of the domain (to be read as the diagonal of a Hi-C contact map). The analysis of these contact frequencies reveals the 3D folding of the gene domain in the nuclear space and the absence of physical contacts between genomic loci separated by insulators, preventing an action of the enhancers outside the domain. B) The gene domain acts as an adaptor. Each enhancer responds to TF concentrations according to a specific combinatorics, summarised (in the simplest case of an all-or-none control) in a truth table, as detailed in Fig. 2. The truth tables of the different enhancers are then integrated at the domain level, as sketched by the truth table in the middle. In this example, each TF can be either activating or inhibiting C) Sketch of the gene expression phase diagram in the TF concentration space (here 2-dimensional). The axes represent the TF levels, whose combination defines a codeword. Gene product concentration (generally normalised mRNA level) is represented with a colourscale for each combination of TF concentrations. A clear delineation of high-expression regions ('active phases') and no-expression regions ('inactive phases') would support the existence of a gene regulatory code and unravel its codewords.

We thus propose that a plausible adaptor of a gene regulatory code is provided by an extension of the primary gene regulatory domain notion, which we will simply name the gene domain (Figs. 123). In contrast to the primary notion, the gene domain contains the gene enhancers, and also its promoter, as well as the linker sequences between them, overall forming a single composite unit able to both read TF concentrations and trigger gene transcription. The parallel with the general notion of adaptor is underlined by using the same colours in Figs. 123.

Additional players, the insulators, set the limits of the gene domain along the genome sequence (Fig. 3). An insulator has been operationally defined as a sequence located between a gene and an enhancer, which prevents the spurious action of the enhancer onto this gene [START_REF] Gall | Roles of chromatin insulators in the formation of longrange contacts[END_REF]. Initially discovered in Drosophila, insulators have then been characterised in many, if not all, metazoans, including human [START_REF] Gerasimova | Chromatin insulators and boundaries: effects on transcription and nuclear organization[END_REF].

Gene domains can be physically identified as preferential contact domains

Additional insights into the regulation of developmental genes came in the last years from chromosome conformational capture experiments (3C). This ensemble of techniques has been devised to investigate the three-dimensional (3D) folding of the genome in living cells [START_REF] Dekker | Capturing chromosome conformation[END_REF]. They consist in a chemical crosslinking of protein-mediated contacts between pairs of genomic loci, followed by a molecular dissection of the crosslinks and a sequencing-based identification of the genomic loci involved. They accordingly provide contact frequencies between genome regions, reflecting the in-vivo 3D organisation of these regions in the nucleus [START_REF] Pombo | Three-dimensional genome architecture: players and mechanisms[END_REF][START_REF] Bonev | Organization and function of the 3D genome[END_REF]. In particular, they have shown that, most often, insulators prevent physical contacts between the genomic loci they insulate [START_REF] Gall | Roles of chromatin insulators in the formation of longrange contacts[END_REF][START_REF] Rao | A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping[END_REF]. 3D genome folding reflects the proper physical communication between the enhancers and the gene promoter, while it prevents enhancer-promoter contacts across the insulators [START_REF] Sun | Promoter-enhancer communication occurs primarily within insulated neighborhoods[END_REF]. An important experiment in this context has been to use hundreds of random insertions of an enhancer at different positions along the genome and to measure gene expression levels in the vicinity of the insertion. The results showed that the activity of the enhancer is homogeneous within wide domains separated by insulators [START_REF] Symmons | Functional and topological characteristics of mammalian regulatory domains[END_REF]. Accordingly, as outlined in Fig. 3A, the gene domain can consistently be defined as the genome domain, located between two insulators, which contains the gene and its enhancer(s).

Preferential and prevented contacts can be mapped along the genome using the genome-wide derivative of 3C (Hi-C). The first preferential contact domains identified with Hi-C were called Topologically Associating Domains (TADs) [START_REF] Dixon | Topological domains in mammalian genomes identified by analysis of chromatin interactions[END_REF][START_REF] Nora | Spatial partitioning of the regulatory landscape of the X-inactivation centre[END_REF][START_REF] Sexton | Three-dimensional folding and functional organization principles of the Drosophila genome[END_REF]. These domains are on average 1Mb long (in mammals) and usually contain several gene domains. With the increase of the resolution of Hi-C maps, sub-TADs and even smaller micro-TADs have been identified within TADs [START_REF] Rao | A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping[END_REF][START_REF] Hsieh | Resolving the 3D landscape of transcription-linked mammalian chromatin folding[END_REF][START_REF] Krietenstein | Ultrastructural details of mammalian chromosome architecture[END_REF]. These smaller contact domains would correspond to the gene domains as defined here. An ever-increasing number of functional evidences indeed suggests an important role of TAD boundaries in regulating genes in space and time during development.

Assessing the effect of preferential contact domains on gene transcription

To investigate the general role of TADs and sub/micro-TADs on gene expression, two different strategies have been employed. In global strategies the domains are disrupted genome-wide through the perturbation of one of the players that drive domain formation. In local strategies, a specific genomic element, usually containing a domain boundary, is modified.

In mammals, two important players have been identified for domain formation. The first is CTCF, which binds at domain borders. The second is cohesin, which induces the formation of a contact domain between two CTCF sites. Several studies investigated the effect of cohesin and/or CTCF depletion on gene expression at the genome-wide level [START_REF] Rao | Cohesin loss eliminates all loop domains[END_REF][START_REF] Schwarzer | Two independent modes of chromatin organization revealed by cohesin removal[END_REF][START_REF] Haarhuis | The cohesin release factor WAPL restricts chromatin loop extension[END_REF]. Their results quantitatively depend on the cell type used and the strategy for protein depletion but all these studies pointed out an effect of the contact domains on gene regulation. While this effect is small or non-existent for housekeeping genes, it is stronger for cell-specific genes, suggesting that gene domains as defined by contact domains are more relevant to describe developmental gene regulation [START_REF] Seitan | Cohesin-based chromatin interactions enable regulated gene expression within preexisting architectural compartments[END_REF].

The prevalent role of contact domains in regulating genes during development has been evidenced on specific examples identified using genomic variants known to induce pathogenic phenotypes [START_REF] Lupiáñez | Breaking TADs: how alterations of chromatin domains result in disease[END_REF][START_REF] De Laat | Topology of mammalian developmental enhancers and their regulatory landscapes[END_REF][START_REF] Franke | Formation of new chromatin domains determines pathogenicity of genomic duplications[END_REF]. Disruption of a contact domain boundary may induce ectopic enhancer-promoter interactions causing developmental anomalies in mouse [START_REF] De Laat | Topology of mammalian developmental enhancers and their regulatory landscapes[END_REF][START_REF] Kraft | Serial genomic inversions induce tissue-specific architectural stripes, gene misexpression and congenital malformations[END_REF]. As a paradigmatic example, deletions, inversions, or duplications of genomic segments containing a contact-domain boundary induce the mis-regulation of genes at the Epha4 locus [START_REF] Lupiáñez | Disruptions of topological chromatin domains cause pathogenic rewiring of gene-enhancer interactions[END_REF]. A similar effect has been demonstrated at the level of a single developmental gene, where changes in 3D chromatin conformation may restrict the enhancer activity and result in gene mis-expression [START_REF] Kragesteen | Dynamic 3D chromatin architecture contributes to enhancer specificity and limb morphogenesis[END_REF]. In a very different context, the hypermethylation of a specific insulator has been shown to prevent CTCF function, leading again to gene misexpression and cancer [START_REF] Flavahan | Insulator dysfunction and oncogene activation in IDH mutant gliomas[END_REF]. This link between domain alteration and cancer has been further exploited to provide pan-cancer candidate genes [START_REF] Weischenfeldt | Pan-cancer analysis of somatic copy-number alterations implicates IRS4 and IGF2 in enhancer hijacking[END_REF]. More and more developmental loci are now under scrutiny and it is becoming increasingly clear that while the underlying mechanisms still need to be investigated [START_REF] Despang | Functional dissection of TADs reveals non-essential and instructive roles in regulating gene expression[END_REF]. contact domains have an important role in regulating genes at the right time and in the right place during development. On the other hand, the functional impact of contact domains appears less clear when considering gene regulation at a genome-wide level in cell cultures or tissues (see other reviews on that matter in this special issue [START_REF] De Wit | TADs as the caller calls them[END_REF][START_REF] Li-Hsin | TADs and their borders: free movement or building a wall?[END_REF].

In the gene domain model, genes are supposed to be completely insulated from others. TADs (on average 1Mb long in mammals) usually contain several gene domains, appearing in Hi-C maps as sub-TADs. Exploring the detailed relationship between TADs and gene domains thus offers a way towards a better understanding of gene co-expression. In this line, it has been proposed that TADs form functional units, each composed of co-regulated genes [START_REF] Lonfat | Structure, function and evolution of topologically associating domains (TADs) at HOX loci[END_REF][START_REF] Hubner | Chromatin organization and transcriptional regulation[END_REF].

Gene domains allow for an epigenetic control of the developmental code

The association between codewords and transcriptional responses is achieved by the gene domain with some epigenetic plasticity. Indeed, this association can be modified, following either a change in the borders of the gene domain that would add enhancers in (or remove from) the domain [START_REF] Cubeñas-Potts | Topologically associating domains: an invariant framework or a dynamic scaffold?[END_REF], or a change in the sequence of an enhancer affecting its affinity for specific TFs. A dramatic example of this latter possibility is given by the case of the ZRS enhancer of the Sonic hedgehog (Shh) gene, where a synthetic restauration of a single ETS1 binding site can re-establish a normal phenotype of otherwise serpentinised mouse (i.e. a mouse with truncated limbs) [START_REF] Kvon | Progressive loss of function in a limb enhancer during snake evolution[END_REF]. An example of the first possibility is observed in the case of HoxD gene cluster in mice, which lies between two TADs [START_REF] Lonfat | Structure, function and evolution of topologically associating domains (TADs) at HOX loci[END_REF][START_REF] Andrey | A switch between topological domains underlies HoxD genes collinearity in mouse limbs[END_REF]. A shift in the contacts established by Hox genes is observed at different positions in the embryo and different times during development. The gene domains are accordingly modified possibly impacting the expression level of the target genes. Similar changes can be observed by engineering appropriate deletions in the gene domain boundary [START_REF] Rodriguez-Carballo | The HoxD cluster is a dynamic and resilient TAD boundary controlling the segregation of antagonistic regulatory landscapes[END_REF]. The increasing resolution of Hi-C maps should help in the future to precisely delineate gene domains and quantitatively confirm these mechanisms.

Gene domains allow for gene regulation by multiple enhancers

The notion of gene domain as an adaptor encoding the gene transcriptional regulation goes beyond the current picture of enhancer-based regulation, presented in Fig. 2. Indeed, in eukaryotes, it appears that most developmental genes are under the control of several enhancers [START_REF] Yáñez-Cuna | Deciphering the transcriptional cis-regulatory code[END_REF], which confers pleiotropy and multi-functionality to the genes [START_REF] De Laat | Topology of mammalian developmental enhancers and their regulatory landscapes[END_REF]. In contrast, in prokaryote unicellular organisms, transcriptional regulation relies on regulatory sequences located close to the gene transcription start site [START_REF] Van Hijum | Mechanisms and evolution of control logic in prokaryotic transcriptional regulation[END_REF]; genomes are compact, with no need of a code or gene domains to integrate multiple cues.

In multicellular organisms, which have longer genomes, gene domains embed several enhancers and integrate their regulatory activities. Fig. 3B illustrates the combinatorics of molecular cues and associated responses in the simple case of two TFs, two enhancers and an all-or-none control. In the case presented there, the activity of the two enhancers is coupled with an exclusive OR logical function, so that the gene is active if one and only one of the two TFs is present. In real situations, the response is gradual. The corresponding case of a continuous control is sketched in Fig. 3C. We introduce a 2D heat map where each axis represents a TF concentration and the colour scale represents the gene expression level (typically, the normalised mRNA level), and propose to call such a plot the expression phase diagram of a gene. This diagram is expected to display well-defined regions in which the target gene is significantly expressed (active phases) and regions where the gene remains silent (inactive phases), each of these regions are defined by a set of codewords. The expression phase diagram is thus the most relevant piece of information to explicitly describe the gene regulatory code.

The way enhancers interact with each other is sometimes unclear. Any kind of logical or analytical function could in principle be envisioned, and efforts have been made both experimentally [START_REF] Bothma | Enhancer additivity and non-additivity are determined by enhancer strength in the Drosophila embryo[END_REF] and theoretically [START_REF] Barr | A sequence level model of an intact locus predicts the location and function of nonadditive enhancers[END_REF] to determine whether enhancers act additively, competitively or in another way. As a special case, shadow enhancers have been proposed to act as redundant enhancers to add robustness to the decoding process [START_REF] Cannavò | Shadow enhancers are pervasive features of developmental regulatory networks[END_REF].

Figure 4 ---Encoding of the early patterning of Drosophila embryo (scheme). A) Spatially extending the abstract logical scheme of Fig. 3, TF concentrations are now varying along the anterior-posterior axis of an embryo (oval shape). A star-shaped TF (dotted concentration curve) and a disk-shaped TF (bold concentration curve) jointly encode the local regulation of a downstream developmental gene according to the integrated truth table in Fig. 3B. B) Accordingly, the gene expression level (in general measured as a mRNA level) follows the variation of the two TF concentrations and takes different value in successive regions of the embryo (red stripes, and red curve along the anterior-posterior axis). As in Fig. 3, an all-or-none control has been considered for simplicity, however what occurs in reality is a gradual regulation based on continuous concentration values (see panel C) C)

The expression phase diagram of the gene, Fig. 3C, can be drawn experimentally by scanning the various combinations of TF levels. See Fig. 6 for a similar diagram based on real data.

Codewords can be used to read positional information within the embryo.

In the discussion above, we posit that codewords for the gene regulation code are concentrations of few specific TFs, as this fits in the general scheme of a semantic code. This idea that local TF concentrations regulate gene expression during development goes back to the concept of positional information introduced in 1968 by Lewis Wolpert. He proposed that some local feature (in his words, a positional value) could underlie the spatio-temporal pattern of cell differentiation in the Drosophila embryo [START_REF] Wolpert | Positional information and the spatial pattern of cellular differentiation[END_REF][START_REF] Wolpert | Positional information revisited[END_REF][START_REF] Wolpert | Positional information and patterning revisited[END_REF]. Since then, the concept of positional information has been extensively studied and its application to complex gene expression patterns was put forward early on [START_REF] Howard | Decoding positional information: regulation of the pair-rule gene hairy[END_REF]. Nevertheless, the quantitative analysis of positional information only began fifteen years later. Using fine in-vivo biophysical measures, it was shown that local concentrations can be defined sharply enough in the embryo to play the role of codewords (in this case simple codewords as only one TF was considered) [START_REF] Gregor | Probing the limits to positional information[END_REF]. A remarkable finding was that the accuracy with which the concentration profiles are established is remarkably good, with less noise than previously expected.

The combination of the two notions of positional information and expression phase diagram has practical implications when considering spatial gene regulation within the developing embryo. Fig. 4 sketches a gene regulation scenario for a simplified embryo, in which the target gene is controlled by two TFs according to the truth table in Fig. 3B. If one assumes the spatial distributions of TF concentrations depicted in Fig. 4A, the red gene regulated by these two TFs is expressed in three stripes (Fig. 4B). In stripes 1 and 2, the activation results from the use of the same codeword whereas in stripe 3 it results from the use of a different codeword as sketched on the expression phase diagram reproduced in Fig. 4C.

A real-world example: the gene regulatory code of Eve in the early patterning of Drosophila embryos.

In Drosophila early development, the nucleus of the fertilised egg undergoes several rounds of division forming a syncytium, i.e. a large multinucleate cell in which about 6000 nuclei occupy distinct positions in space. The first layer of differentiation is based on the gradient of maternal factor concentrations, which are already asymmetrically distributed in the egg. They impact the transcription of four elementary genes, Krüppel, Hunchback, Giant and Knirps, globally known as gap genes. This in turn produces a heterogeneous distribution of the associated proteins. These proteins act as TFs for other developmental genes, known as pair-rule genes, which exhibit a stripe-patterned expression [START_REF] Reinitz | Stripe forming architecture of the gap gene system[END_REF]. The even-skipped (eve) gene belongs to the family of pair-rule genes and displays a seven-striped pattern (Fig. 5). While this emblematic gene has been so far the most studied, other pair-rule genes behave in a similar way [START_REF] Schroeder | How to make stripes: deciphering the transition from non-periodic to periodic patterns in Drosophila segmentation[END_REF]. The eve gene domain, which contains five enhancers [START_REF] Sackerson | The even-skipped locus is contained in a 16-kb chromatin domain[END_REF], is able as a whole to properly respond to the different combinations of gap-protein concentrations encountered in the embryo. Two recent studies used the Drosophila melanogaster embryo to show that the spatial expression (protein concentration) of the eve gene can be quantitatively predicted from the concentration patterns of the four gap TFs Krüppel, Hunchback, Giant and Knirps [START_REF] Petkova | Optimal decoding of cellular identities in a genetic network[END_REF][START_REF] Dubuis | Positional information, in bits[END_REF]. Being robustly defined and associated with a unique eve expression response, the set of these four TFs concentrations could thus provide in principle the input of the gene regulatory code involved in eve early patterning.

To illustrate further this assertion, we performed a complementary analysis of other recent experimental data (Figs. 5 and6). We used the quantitative measurements of mRNA levels in each nucleus of the Drosophila embryo at stage 14 provided by the Berkeley 3D gene expression atlas [START_REF] Fowlkes | A quantitative spatio-temporal atlas of gene expression in the Drosophila blastoderm[END_REF][START_REF]Berkeley Drosophila Transcription Network Project[END_REF]. For our purpose, the spatial distribution of mRNAs in the embryo offers a way to sample at different time points the various combinations of TF concentrations as well as the associated expression of eve. Indeed, when both the protein and mRNA levels of a developmental gene were available, we have checked that they display a high correlation of 0.89, which could even reach 0.95 when optimally tuning the time delay between the protein and mRNA measurements. We thus extracted the (normalized) mRNA concentrations of Krüppel, Hunchback, Giant and Knirps (Fig. 5, top) in the nuclei located in each of the seven stripes of eve expression (Fig. 5, bottom). The result is presented in the form of a box-plot for each stripe (Fig. 5, middle). We checked its robustness with respect to the expression threshold used to delineate eve stripes. The plot shows quantitatively that each stripe is associated with a well-defined codeword. A more precise test of the gene expression code scenario could be done in a near future using space-and time-resolved eve transcription data [START_REF] Zoller | Diverse spatial expression patterns emerge from unified kinetics of transcription[END_REF] instead of averaged concentrations of the gene products (mRNA). Joint spatial imaging of genome folding and gene expression at the single-cell level will be instrumental to quantitatively delineate the cell-to-cell variability of the relationship between 3D domains and transcriptional activity [START_REF] Cattoni | Single-cell absolute contact probability detection reveals chromosomes are organized by multiple low-frequency yet specific interactions[END_REF][START_REF] Bintu | Super-resolution chromatin tracing reveals domains and cooperative interactions in single cells[END_REF][START_REF] Mateo | Visualizing DNA folding and RNA in embryos at single-cell resolution[END_REF]. This should ultimately explain how intrinsic stochasticity of molecular processes can be canalized into a robust average gene expression at the organism level.

Figure 5 ---Gene regulatory code for eve gene expression during Drosophila embryogenesis

Experimental measures of the concentrations of the four gap-gene products (Krüpple-Kr, Hunchback-Hb, Giant-Gt and Knirps-Kni, top) and the striped-pattern of eve gene product in Drosophila embryo (bottom). The four gap proteins act as TFs for eve. From these data, we quantified the distribution of the four gap-gene products in each of the seven eve stripe (middle). The result for each stripe is presented in the form of a boxplot (average, standard deviation and outliers). This analysis shows that each stripe is encoded by a specific combination of the four gap-gene levels, acting as a codeword in the eve gene regulatory code. Data from [START_REF] Fowlkes | A quantitative spatio-temporal atlas of gene expression in the Drosophila blastoderm[END_REF], normalised mRNA levels.

Dedicated experiments have shown that one of the enhancers is predominantly involved in the formation of each stripe. However, these experiments also revealed that the additive modularity and one-to-one correspondence between enhancers and stripes is not exact. For instance, Lim et al. removed stripe 1 using a CRISPR/Cas9-mediated deletion on the corresponding enhancer, but this modification also induced a displacement of stripe 2 [START_REF] Lim | Temporal dynamics of pair-rule stripes in living Drosophila embryos[END_REF]. Kim et al. constructed several eve loci variants containing only the enhancers for stripe 2 and the pair of stripes 3 and 7. They showed that the order of the enhancers in the fusion construct significantly affects the expression levels [START_REF] Kim | Rearrangements of 2.5 kilobases of noncoding DNA from the Drosophila even-skipped locus define predictive rules of genomic cis-regulatory logic[END_REF]. The gene domain is thus more than the sum of its enhancers and we need to consider its coordinating action to account for the target gene regulation.

An integrated view of the gene regulatory code associated with the gene domain of eve is presented on Fig. 6. We again used the normalised mRNA concentrations of Krüppel, Hunchback, Giant, Knirps and Eve, Fig. 5, to draw a regulatory phase diagram similar in principle to the one shown in Fig. 4C, now using real data. In this case, the TF concentration space is now four-dimensional, which requires the use of a projection method, known as multidimensional scaling [START_REF] Torgerson | Multidimensional scaling: I. Theory and method[END_REF], for the representation in the two dimensions of a paper sheet. This diagram captures the full landscape of eve regulation, clearly delineating regions where the gene is expressed and regions where the gene is silent. It thus supports the existence of a regulatory code and accurately determines its codewords.

Figure 6 ---Encoding of the early patterning of Drosophila embryo (gene eve, real data).

A refined view of the regulatory code of the gene eve is provided by its expression phase diagram, describing as a heat map its expression level as a function of the four gap-gene mRNA levels (Kr, Hb, Gt, Kni, see Fig. 5), measured in each nucleus of the embryo. Compared to the notion sketched in Fig. 4C, the TF concentration space is now four-dimensional. To be represented on this page, the phase diagram has been projected into a 2-dimensional representation using a classical multidimensional scaling method. Accordingly, the four bold lines are not coordinate axes; they only correspond to codewords with only one of the four TFs being present. The spatial resolution of the measures enables an extensive scan of the TF concentration space (overall about 6000 nuclei, i.e. 6000 dots). This expression phase diagram delineates regions ('phases') in the TF concentration space where eve is expressed (stripes numbered from 1 to 7) and regions where this gene is inactive. Data from [START_REF] Fowlkes | A quantitative spatio-temporal atlas of gene expression in the Drosophila blastoderm[END_REF], normalised mRNA levels.

Conclusion

By identifying a plausible adaptor, the gene domain, our analysis supports the existence of a gene regulatory code translating specific combinations of TF concentrations into gene expression level during embryogenesis, and possibly other contexts. This viewpoint markedly differs from previously introduced gene regulatory codes that would lie in the enhancer sequence, with no adaptor involved [START_REF] Michelson | Deciphering genetic regulatory codes: a challenge for functional genomics[END_REF][START_REF] Markstein | A regulatory code for neurogenic gene expression in the Drosophila embryo[END_REF][START_REF] Yáñez-Cuna | Deciphering the transcriptional cis-regulatory code[END_REF]. The code adaptor, i.e. the gene domain forms a segregated spatial entity, in which all enhancers can at some point interact with the gene promoter, depending on surrounding TF concentrations. The encoding of gene expression achieved by the gene domain thus centrally involves genome 3D organisation. As codewords, i.e. TFs concentrations, in turn depend on the position of the nucleus in the embryo, the notion of adaptor-mediated gene regulatory code offers a novel viewpoint to investigate spatio-temporal gene patterning during development. A full understanding of this spatio-temporal gene patterning is still an open challenge attracting huge research efforts. Many mechanistic questions remain.. How does a single enhancer operate within a gene domain and how multiple enhancers cooperate ? How do TF binding sites at an enhancer allow the proper reading of TFs concentrations ? What is the expected transcriptional outcome if two enhancers were jointly active ? To this end, designing both modified cis-regulatory sequences [START_REF] Crocker | TALE-mediated modulation of transcriptional enhancers in vivo[END_REF] and totally synthetic enhancers [START_REF] Crocker | A fully synthetic transcriptional platform for a multicellular eukaryote[END_REF] offer promising paths that can be supplemented with evolutionary perspectives [START_REF] Combs | Spatially varying cis-regulatory divergence in Drosophila embryos elucidates cis-regulatory logic[END_REF].

Another challenge for the next years will be to characterise the expression phase diagram for the many genes involved in development. Drosophila is a paradigmatic example as embryos are easy to image and to genetically manipulate [START_REF] Gregor | The embryo as a laboratory: quantifying transcription in Drosophila[END_REF]. While microscopy allows for the determination of some gene transcription in space and time, new techniques such as single cell transcripition profiling can now be used to get a genome-wide readout [START_REF] Karaiskos | The Drosophila embryo at single-cell transcriptome resolution[END_REF]. These high throughput approaches have now been used in a large variety of context, inclusing human and mouse [START_REF] Rizvi | Single-cell topological RNA-seq analysis reveals insights into cellular differentiation and development[END_REF]. We anticipate that a combination of data analysis and modelling on these results can be used to derive quantitative information on gene expression diagrams and decipher the developmental gene regulatory code.

Acknowledgments

The authors thank Mathieu Coppey and Thomas Gregor for their critical reading of an initial version of this paper and their insightful comments. AL was funded by the Agence Nationale de la Recherche, grant ANR-16-CE15-0018-04 and the Cancéropôle Grand-Sud-Ouest, grant 2018-E08