Thomas Dreyfus

Charlotte Hardouin

AND Julien Roques
email: roques@math.univ-lyon1.fr

Michael F Singer
email: singer@ncsu.edu

WALKS IN THE QUARTER PLANE: GENUS ZERO CASE

Keywords: Mathematics Subject Classification. 05A15, 30D05, 39A06 Random walks, Difference Galois theory, q-difference equations, Transcendence

We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any nontrivial (possibly nonlinear) algebraic differential equation with rational coefficients.

Introduction

The generating series of lattice walks in the quarter plane have garnered much interest in recent years. In [START_REF] Dreyfus | On the nature of the generating series of walks in the quarter plane[END_REF], we introduced a new method that allowed us to study the nature of the generating series of many lattice walks with small steps (i.e., whose step set is a subset of {-1, 0, 1} 2 \{(0, 0)}) in the quarter plane. In particular, the paper [START_REF] Dreyfus | On the nature of the generating series of walks in the quarter plane[END_REF] is concerned with the differential nature of these generating series, the basic question being: which of them satisfy differential equations? The present paper is a continuation of this research. We will study weighted models of walks with small steps in the quarter plane Z 2 ≥0 . More precisely, let (d i,j ) (i,j)∈{0,±1} 2 be a family of elements of Q ∩ [0, 1] such that i,j d i,j = 1. We encode the eight cardinal directions of the plane by pairs of integers (i, j) with i, j ∈ {0, ±1}. We consider a weighted walk in the quarter plane Z 2 ≥0 satisfying the following properties: • it starts at (0, 0); • it takes steps in a certain subset of the set of cardinal directions, which is called the model of the walk. For (i, j) ∈ {0, ±1} 2 \{(0, 0)} (resp. (0, 0)), the element d i,j is a weight on the step (i, j) and can be viewed as the probability for the walk to go in the direction (i, j) (resp. to stay at the same position). The step set or the model of the walk corresponds the set of directions with nonzero weights, that is, {(i, j) ∈ {0, ±1} 2 \{(0, 0)}|d i,j = 0}. If d 0,0 = 0 and if the nonzero d i,j all have the same value, we say that the model is unweighted.

The weight of the walk is defined to be the product of the weights of its component steps. For any (i, j) ∈ Z 2 ≥0 and any k ∈ Z ≥0 , we let q i,j,k be the sum of the weights of all walks reaching the position (i, j) from the initial position (0, 0) after k steps. We introduce the corresponding trivariate generating series * Q(x, y, t) := i,j,k≥0 q i,j,k x i y j t k .

The typical questions considered in the literature are:

• is Q(x, y, t) algebraic over Q(x, y, t)?

• is Q(x, y, t) x-holonomic (resp. y-holonomic), i.e., is Q(x, y, t), seen as a function of x, a solution of some nonzero linear differential equation with coefficients in Q(x, y, t)? • is Q(x, y, t) x-differentially algebraic (resp. y-differentially algebraic), i.e. is Q(x, y, t), seen as a function of x, a solution of some nonzero (possibly nonlinear) polynomial differential equation with coefficients in Q(x, y, t)? In case of a negative answer, we say that Q(x, y, t) is x-differentially transcendental (resp. y-differentially transcendental) † . Before describing our main result, we will briefly describe the state of the art. In the seminal paper [START_REF] Bousquet | Walks with small steps in the quarter plane, Algorithmic probability and combinatorics[END_REF], Bousquet-Mélou and Mishna studied such questions in the unweighted case (see also [START_REF] Mishna | Classifying lattice walks restricted to the quarter plane[END_REF]). Taking symmetries into consideration and eliminating unweighted models equivalent to models on the half plane (whose generating series is algebraic), Bousquet-Mélou and Mishna first showed that, amongst the 256 possible unweighted models, it is sufficient to study the above questions for an explicit list of 79 unweighted models. Following ideas of Fayolle, Iasnogorodski and Malyshev (see for instance [START_REF] Fayolle | Random walks in the quarter-plane[END_REF][START_REF] Fayolle | Random walks in the quarter plane. Algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics[END_REF]), they associated to each unweighted model a group of birational automorphisms of C 2 and classified the unweighted models accordingly. They found that 23 of the 79 above-mentioned unweighted models were associated with a finite group and showed that for all but one of these 23 models, the generating series was x-, y-and t-holonomic; the remaining one was shown to have the same property by Bostan, van Hoeij and Kauers in [START_REF] Bostan | The complete generating function for Gessel walks is algebraic[END_REF]. In [START_REF] Bousquet | Walks with small steps in the quarter plane, Algorithmic probability and combinatorics[END_REF], Bousquet-Mélou and Mishna conjectured that the 56 unweighted models whose associated group is infinite are not holonomic. Furthermore, following Fayolle, Iasnogorodski and Malyshev, the 56 unweighted models may be gathered into two families according to the genus of an algebraic curve, called the kernel curve, attached to each model:

• 5 of these unweighted models lead to a curve of genus zero; they will be called the genus zero unweighted models • 51 of them lead to a curve of genus one; they will be called the genus one unweighted models In [START_REF] Kurkova | On the functions counting walks with small steps in the quarter plane[END_REF], Kurkova and Raschel showed that the 51 genus one unweighted models with infinite group have nonholonomic generating series (see also [START_REF] Bostan | Non-D-finite excursions in the quarter plane[END_REF][START_REF] Raschel | Counting walks in a quadrant: a unified approach via boundary value problems[END_REF]). Recently, Bernardi, Bousquet-Mélou and Raschel [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method, An extended abstract to appear[END_REF][START_REF]Counting quadrant walks via Tutte's invariant method[END_REF] have shown that 9 of these 51 unweighted models have x-and y-differentially algebraic generating series, despite the fact that they are not x-or y-holonomic.

In [START_REF] Dreyfus | On the nature of the generating series of walks in the quarter plane[END_REF], we introduced a new approach to these problems that allowed us to show that, except for the 9 exceptional unweighted models of [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method, An extended abstract to appear[END_REF][START_REF]Counting quadrant walks via Tutte's invariant method[END_REF], the generating series of genus one unweighted models with infinite groups are x-and y-differentially transcendental. This reproves and generalizes the results of [START_REF] Kurkova | On the functions counting walks with small steps in the quarter plane[END_REF]. Furthermore our results allowed us to show that the 9 exceptional series are not holonomic but are x-and y-differentially algebraic, recovering some of the results of [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method, An extended abstract to appear[END_REF][START_REF]Counting quadrant walks via Tutte's invariant method[END_REF]. It is worth mentioning that there are several results in the literature about the behavior of Q(x, y, t) with respect to the variable t. For instance, in [START_REF] Mishna | Two non-holonomic lattice walks in the quarter plane[END_REF], Mishna and Rechnitzer showed that Q(1, 1, t) is not t-holonomic for two of the 5 genus zero unweighted models and in [START_REF] Melczer | Singularity analysis via the iterated kernel method[END_REF], Melczer and Mishna showed that this remained true for all 5 of the genus zero unweighted models. On the other hand, Bostan, Raschel and Salvy proved in [START_REF] Bostan | Non-D-finite excursions in the quarter plane[END_REF] that Q(0, 0, t) is not t-holonomic for every genus one unweighted model with an infinite group. We also note that, in [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method, An extended abstract to appear[END_REF][START_REF]Counting quadrant walks via Tutte's invariant method[END_REF], it is shown that the generating series of the 9 exceptional genus zero unweighted models mentioned above are differentially algebraic in the variable t as well. Finally, the first two authors proved in [START_REF] Dreyfus | Length derivative of the generating series of walks confined in the quarter plane[END_REF] that, if the generating series is x-or y-differentially transcendental, then it is t-differentially transcendental. Thus, although the present paper focuses on the x-and y-differential properties of Q(x, y, t), it also gives information concerning its t-differential properties.

In the present paper, we start from the 5 unweighted models corresponding to a genus zero kernel curve. These models arise from the following 5 sets of steps.

(S)

We say that a weighted model arises from (S) when this model is obtained by choosing a set of steps in (S) and by assigning nonzero weights to this set of steps. One can show that the kernel curve of a weighted model arising from (S) is still a genus zero curve. Our main result may be stated as follows:

Main Theorem. If 0 < t < 1 is transcendental ‡ and if the weighted model arises from (S), then Q(x, y, t) is x-and y-differentially transcendental.

Our study generalizes the result of Mishna and Rechnitzer on the non holonomy of the complete generating series of the unweighted models of walks {N W, N, SE} and {N W, N E, SE} (see [START_REF] Mishna | Two non-holonomic lattice walks in the quarter plane[END_REF]Theorem 1.1]) and also the one by Melczer and Mishna (for the five cases). Our ‡ This assumption is used repeatedly in our proofs and is crucial in our proof of Proposition 3.4. strategy of proof is inspired by [START_REF] Fayolle | Random walks in the quarter plane. Algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics[END_REF]Chapter 6]. We associate to each of the generating series of these weighted models a function meromorphic on C. These associated functions satisfy first order difference equations of the form y(qs) -y(s) = b(s) for a suitable q ∈ C and b(s) ∈ C(s). The associated functions are differentially transcendental if and only if the generating series are differentially transcendental. We then use criteria stating that if these associated functions were differentially algebraic then the b(s) must themselves satisfy b(s) = h(qs)-h(s) for some rational functions h(s) on C. This latter condition puts severe limitations on the poles of the b(s) and, by analyzing the b(s) that arise, we show that these restrictions are not met. Therefore the generating series are not differentially algebraic, see Theorem 3.1. Note that some unweighted models of walks in dimension three happen to be, after projection, equivalent to two dimensional weighted models of walks [START_REF] Bostan | On 3-dimensional lattice walks confined to the positive octant[END_REF][START_REF] Du | Infinite orders and non-D-finite property of 3-dimensional lattice walks[END_REF]. We apply our theorem in this setting as well. We note that finding the difference equation y(qs) -y(s) = b(s) and the remaining calculations involve only algebraic computations as is true in [START_REF] Dreyfus | On the nature of the generating series of walks in the quarter plane[END_REF]. The general approach followed in the present work is inspired by [START_REF] Dreyfus | On the nature of the generating series of walks in the quarter plane[END_REF] but the details are quite different and justify an independent exposition.

The rest of the paper is organized as follows. In Section 1, we first present the generating series attached to a weighted model of walks and we give some of their basic properties. We then introduce the kernel curves (they are algebraic curves associated to any model of walk in the quarter plane) and we state some of their properties. One of their main properties is that, for the weighted models arising from (S), the kernel curves have genus zero and, hence, can be parameterized by birational maps from P 1 (C). Such parameterizations, suitable for our needs, are given at the end of Section 1. In Section 2, using these parameterizations, we attach to any model some meromorphic functions on C that satisfy simple q-difference equations of the form y(qs) -y(s) = b(s) for some b(s) ∈ C(s). Moreover, we prove that these meromorphic functions are differentially algebraic if and only if the generating series of the associated models are differentially algebraic. In addition, we present necessary conditions on the poles of b when these equations have differentially algebraic solutions. In Section 3, we show that these necessary conditions do not hold for the weighted models arising from (S).

Acknowledgments The authors would like to thank Kilian Raschel for pointing out many references related to this work. In addition, we would like to thank the anonymous referees for many useful comments and suggestions concerning this article.

1. Weighted walks in the quarter plane: generating series, functional equation and kernel curve

In this section, we consider a weighted walk with small steps in the quarter plane Z 2 ≥0 and the corresponding trivariate generating series Q(x, y, t) as in the introduction. We first recall a functional equation satisfied by Q(x, y, t). We then recall the definition of the so-called kernel curve associated to the walk under consideration and give its main properties when the steps set is one of the five steps sets listed in (S).

1.1. Kernel and functional equation. The kernel of a weighted model is defined by

K(x, y, t) := xy(1 -tS(x, y)) where S(x, y) = (i,j)∈{0,±1} 2 d i,j x i y j = A -1 (x) 1 y + A 0 (x) + A 1 (x)y = B -1 (y) 1
x + B 0 (y) + B 1 (y)x, and

A i (x) ∈ x -1 Q[x], B i (y) ∈ y -1 Q[y].
The following result generalizes [BMM10, Lemma 4].

Lemma 1.1. The generating series Q(x, y, t) satisfies the following functional equation:

(1.1) K(x, y, t)Q(x, y, t) = xy -F 1 (x, t) -F 2 (y, t) + td -1,-1 Q(0, 0, t)
where F 1 (x, t) := -K(x, 0, t)Q(x, 0, t), F 2 (y, t) := -K(0, y, t)Q(0, y, t).

Proof. As in [BMM10, Lemma 4], we proceed as follows. First, let us prove that if we do not consider the quadrant constraint, the functional equation would be (1 -tS(x, y))Q(x, y, t) = 1.

Indeed, in this situation, if we write Q(x, y, t)

= ∞ =0 Q (x, y)t , then Q 0 (x, y) = 1 and Q +1 (x, y) = S(x, y)Q (x, y)
. This is exactly (1 -tS(x, y))Q(x, y, t) = 1. However, this formula does not take into account the quadrant constraint. We need to withdraw the walks that leave the x-axis (resp. y-axis), i.e.

ty -1 A -1 (x)Q(x, 0, t) (resp. tx -1 B -1 (y)Q(0, y, t)).
Since we withdraw two times the walks going from (0, 0) in south west, we have to add the term

tx -1 y -1 d -1,-1 Q(0, 0, t). So (1-tS(x, y))Q(x, y, t) = 1-ty -1 A -1 (x)Q(x, 0, t)-tx -1 B -1 (y)Q(0, y, t)+tx -1 y -1 d -1,-1 Q(0, 0, t).
It now suffices to multiply by xy the above equality.

1.2. The algebraic curve defined by the kernel. We recall that the affine curve E t defined by the kernel K(x, y, t) is given by

E t = {(x, y) ∈ C × C | K(x, y, t) = 0}.
In Section 2 we show that the problem of showing that Q(x, y, t) is x-and y-differentially transcendental can be reduced to understanding the relations among the poles of a rational function on E t . When dealing with a rational function b(s) on C, one often needs to consider its behavior "as s goes to infinity". Although this can frequently be finessed, it is convenient to add a point at infinity, constructing the complex projective line as defined below, and consider the behavior at this point. When dealing with rational functions on curves in the affine plane, their behavior, such as the appearance of poles, often depends on missing points "at infinity" and we will see that this is the case in Section 2. To do this we must include the missing points at infinity and so it is useful to compactify such a curve by adding these points. This can be done in several ways (see Remark 1.2 below) but, as in [START_REF] Dreyfus | On the nature of the generating series of walks in the quarter plane[END_REF], it will be useful to consider a compactification E t of E t in P 1 (C) × P 1 (C), which is called the kernel curve.

We first recall that P 1 (C) denotes the complex projective line, which is the quotient of C × C \ {(0, 0)} by the equivalence relation ∼ defined by

(x 0 , x 1 ) ∼ (x 0 , x 1 ) ⇔ ∃λ ∈ C * , (x 0 , x 1 ) = λ(x 0 , x 1 ). The equivalence class of (x 0 , x 1 ) ∈ C × C \ {(0, 0)} is usually denoted by [x 0 : x 1 ] ∈ P 1 (C). The map x → [x : 1] embeds C inside P 1 (C). The latter map is not surjective: its image is P 1 (C) \ {[1 : 0]}; the missing point [1 : 0] is usually denoted by ∞. Now, we embed E t inside P 1 (C) × P 1 (C) via (x, y) → ([x : 1], [y : 1]
). The kernel curve E t is the closure of this embedding of E t . In other words, the kernel curve E t is the algebraic curve defined by

E t = {([x 0 : x 1 ], [y 0 : y 1 ]) ∈ P 1 (C) × P 1 (C) | K(x 0 , x 1 , y 0 , y 1 , t) = 0}
where K(x 0 , x 1 , y 0 , y 1 , t) is the following bihomogeneous polynomial

(1.2) K(x 0 , x 1 , y 0 , y 1 , t) = x 2 1 y 2 1 K( x 0 x 1 , y 0 y 1 , t) = x 0 x 1 y 0 y 1 -t 2 i,j=0 d i-1,j-1 x i 0 x 2-i 1 y j 0 y 2-j 1 . • • • • • P ι 2 (P ) ι 1 (P ) σ(P ) σ -1 (P ) E t Figure 1. The maps ι 1 , ι 2 restricted to the kernel curve E t Since K(x 0 , x 1 , y 0 , y 1 , t)
is quadratic in each of the variables, the curve E t is naturally endowed with two involutions ι 1 , ι 2 , namely the vertical and horizontal switches of E t defined, for any 1). Let us also define

P = (x, y) ∈ E t , by {P, ι 1 (P )} = E t ∩ ({x} × P 1 (C)) and {P, ι 2 (P )} = E t ∩ (P 1 (C) × {y}) (see Figure
σ := ι 2 • ι 1 .
Remark 1.2. There are several choices for the compactification of E t . For instance, we could have compactified the curve E t in the complex projective plane P 2 (C) instead of P 1 (C) × P 1 (C) but, in this case, the compactification is not defined by a biquadratic polynomial so that the construction of the above-mentioned involutions in that situation is not so natural.

Assumption 1.3. From now on, we consider a weighted model arising from (S) and we fix a transcendental real number 0 < t < 1.

φ : P 1 (C) → E t s → (x(s), y(s)).
Proposition 1.5 below gives such an explicit parametrization, which induces a bijection between

P 1 (C) \ φ -1 (Ω) and E t \ {Ω}, where Ω = ([0 : 1], [0 : 1]) ∈ E t .
It is the analogue of [FIM17, Section 6.4.3], where the case t = 1 is considered. The proof is similar for t transcendental and the details are left to the reader. We first introduce some notations. For any [x 0 : x 1 ] and [y 0 : y 1 ] in P 1 (C), we denote by ∆ x [x0:x1] and ∆ y [y0:y1] the discriminants of the degree two homogeneous polynomials given by y → K(x 0 , x 1 , y, t) and x → K(x, y 0 , y 1 , t) respectively. We have

∆ x [x0:x1] = t 2 (- 1 t x 0 x 1 + d 0,0 x 0 x 1 + d 1,0 x 2 0 ) 2 -4d 1,-1 x 2 0 (d -1,1 x 2 1 + d 0,1 x 0 x 1 + d 1,1 x 2 0 ) §
In this paper, we have assumed that the d i,j belong to Q, but everything stays true if we assume that d i,j are positive real numbers and that t is transcendental over the field Q(d i,j ).

and ∆ y

[y0:y1] = t 2 - 1 t y 0 y 1 + d 0,0 y 0 y 1 + d 0,1 y 2 0 ) 2 -4d -1,1 y 2 0 (d 1,-1 y 2 1 + d 1,0 y 0 y 1 + d 1,1 y 2 0 ) . Let us write ∆ x [x:1] = 4 =2 α x
and let a 1 = a 2 = 0, a 3 , a 4 be the four roots of this polynomial. Similarly, let us write

∆ y [y:1] = 4 =2 β x
and let b 1 = b 2 = 0, b 3 , b 4 be the four roots of this polynomial. We have

α 2 (t) = 1 -2td 0,0 + t 2 d 2 0,0 -4t 2 d -1,1 d 1,-1 β 2 (t) = 1 -2td 0,0 + t 2 d 2 0,0 -4t 2 d 1,-1 d -1,1 α 3 (t) = 2t 2 d 1,0 d 0,0 -2td 1,0 -4t 2 d 0,1 d 1,-1 β 3 (t) = 2t 2 d 0,1 d 0,0 -2td 0,1 -4t 2 d 1,0 d -1,1 α 4 (t) = t 2 (d 2 1,0 -4d 1,1 d 1,-1 ) β 4 (t) = t 2 (d 2 0,1 -4d 1,1 d -1,1
). Moreover, a 3 , a 4 , b 3 and b 4 are given by the following formulas

a 3 a 4 α 4 (t) = 0 -α3(t)- √ α3(t) 2 -4α2(t)α4(t) 2α4(t) : 1 -α3(t)+ √ α3(t) 2 -4α2(t)α4(t) 2α4(t) : 1 α 4 (t) = 0 [1 : 0] [-α 2 (t) : α 3 (t)] b 3 b 4 β 4 (t) = 0 -β3(t)- √ β3(t) 2 -4β2(t)β4(t) 2β4(t) : 1 -β3(t)+ √ β3(t) 2 -4β2(t)β4(t) 2β4 (t)
: 1

β 4 (t) = 0 [1 : 0] [-β 2 (t) : β 3 (t)]
Proposition 1.5. An explicit parameterization φ = (x, y) : P 1 (C) → E t is given by

φ(s) = 4α 2 (t) α 3 (t) 2 -4α 2 (t)α 4 (t)(s + 1 s ) -2α 3 (t) , 4β 2 (t) β 3 (t) 2 -4β 2 (t)β 4 (t)( s λ + λ s ) -2β 3 (t)
for a certain λ ∈ C * . Moreover we have (see Figure 2)

x(0) = x(∞) = a 1 , x(1) = a 3 , x(-1) = a 4 , y(0) = y(∞) = b 1 , y(λ) = b 3 , y(-λ) = b 4 , where a 1 = a 2 = [0 : 1] (resp. b 1 = b 2 = [0 : 1]).
Remark 1.6. When t = 1, we recover the uniformization of [FIM17, Section 6.4.3]. Note that if we consider x 3 , x 4 (resp. y 3 , y 4 ) defined in [FIM17, Chapter 6], we have the equality of sets {a 3 , a 4 } = {x 3 , x 4 } and {b 3 , b 4 } = {y 3 , y 4 }, but do not have necessarily

a i = x i , b j = y j , with 3 ≤ i, j ≤ 4.
The number q := λ 2 will be crucial in the rest of the paper. The following lemma determines q up to its inverse.

Proposition 1.7. One of the two complex numbers {q, q -1 } is equal to

(1.3) -1 + d 0,0 t -(1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 -1 + d 0,0 t + (1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 . 0 ∞ 1 -1 λ -λ Ω (a 3 , * ) (a 4 , * ) ( * , b 3 ) ( * , b 4 ) φ : P 1 (C) -→ E t Figure 2

. The uniformization map

Proof. Using the explicit formulas for x(s) and y(s), we get

lim s→0 x(s) y(s) = λα 2 (t) β 3 (t) 2 -4β 2 (t)β 4 (t) β 2 (t) α 3 (t) 2 -4α 2 (t)α 4 (t) and lim s→0 x(1/s) y(1/s) = α 2 (t) β 3 (t) 2 -4β 2 (t)β 4 (t) λβ 2 (t) α 3 (t) 2 -4α 2 (t)α 4 (t) . But, x(1/s) y(1/s) = x(s) y(ι1(s))
. So, the above two limits imply the following:

lim s→0 y(ι 1 (s)) y(s) = q.
Now, let us note that y(s), y(ι 1 (s)) equals to

-x + d 0,0 xt + d 1,0 x 2 t ± (x -d 0,0 xt -d 1,0 x 2 t) 2 -4d 1,-1 x 2 t 2 (d -1,1 + d 0,1 x + d 1,1 x 2 ) -2d -1,1 t -2d 0,1 xt -2d 1,1 x 2 t ,
with the shorthand notation x = x(s). Since x(s) tends to 0 when s goes to 0, we obtain the result.

Remark 1.8. One of the referees remarked that for the special case d 0,0 = 0, and

d 1,-1 = d -1,1 = d, the inverse of (1.3) becomes -1 + √ 1 -4d 2 t 2 -1 - √ 1 -4d 2 t 2 = 1 - √ 1 -4d 2 t 2 1 + √ 1 -4d 2 t 2 = (1 - √ 1 -4d 2 t 2 ) 2 4d 2 t 2 = 1 - √ 1 -4d 2 t 2 2d 2 t 2 -1.
This expression is very similar to the generating series 1-

√ 1-4x 2x
of the Catalan numbers. Regrettably, we do not have, in general, a combinatorial interpretation of q.

Remark 1.9. The uniformization is not unique. More precisely, the possible uniformizations are of the form φ • h, where h is an homography. However, if one requires that h fixes setwise 0, ∞ then q is uniquely defined up to its inverse.

The real q or q -1 specializes for t = 1 to the real ρ 2 in [FIM17, Page 178]. In [FIM17, (7.2.18) and Proposition 7.2.3] it is proved that the ratio of the argument of ρ by π is related to the angle between the tangent lines to the curve E 1 , the kernel curve at t = 1, and the horizontal axis. This relation is obtained by a degeneracy argument from the genus 1 case to the genus 0 case. More precisely, let ω 3 be the period attached to the automorphism of the model of the walk in an elliptic lattice Zω 1 + Zω 2 corresponding the elliptic kernel curve and where ω 2 is a real period. Then, arg(ρ) π is obtained by degeneracy of the fraction ω3 ω2 from the genus 1 to the genus 0 case. It is not completely obvious if these arguments pass to the situation where t varies. In the zero drift situation, this has been done in [START_REF] Fayolle | Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group, Markov Process[END_REF]. In the general situation, it might be interesting to compute the rotation number ω3(t) ω2(t) of the real elliptic fibration ([Dui10, Page 82]) and to study its degeneracy. One could then expect that the ratio of the argument of q by 2π is counting the number of rotations of the curve around the origin induced by the action of the automorphism of the model of the walk.

Corollary 1.10. We have q ∈ R \ {±1}.

Proof. We first claim that (1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 > 0. We know that the d i,j are ≥ 0, that the sum of the d i,j is equal to 1 and that the model is not included in {(0, 0), (1, -1), (-1, 1)}. Therefore, we have 1

> d 0,0 + d 1,-1 + d -1,1 , i.e., 1 -d 0,0 > d 1,-1 + d -1,1 . Since t ∈]0, 1[, we have 1 -d 0,0 t > 1 -d 0,0 . Thus, (1 -d 0,0 t) 2 > (1 -d 0,0 ) 2 > (d 1,-1 + d -1,1 ) 2 and, hence, (1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 > (d 1,-1 + d -1,1 ) 2 -4d 1,-1 d -1,1 t 2 ≥ (d 1,-1 + d -1,1 ) 2 -4d 1,-1 d -1,1 = (d 1,-1 -d -1,1 ) 2 ≥ 0.
This proves our claim. Now Proposition 1.7 implies that q is a real number = 1. Moreover, it also shows that q = -1 if and only if -1 + d 0,0 t = 0. But this is excluded because 1 > d 0,0 t.

In particular, this implies that the birational maps σ and σ have infinite order (see also [START_REF] Bousquet | Walks with small steps in the quarter plane, Algorithmic probability and combinatorics[END_REF][START_REF] Fayolle | Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group, Markov Process[END_REF]). It follows that the group associated with these models of walks, namely the group i 1 , i 2 generated by i 1 and i 2 , has infinite order (because σ is induced on E t by i 1 • i 2 , so if σ has infinite order then i 1 , i 2 has infinite order as well). Note that in [START_REF] Bousquet | Walks with small steps in the quarter plane, Algorithmic probability and combinatorics[END_REF], this was proved using a valuation argument. Using the valuation of the successive elements (i 1 •i 2 ) (f ) for ∈ Z and f ∈ Q(x, y), it was proved that i 1 • i 2 could not be of finite order. Initially, the group of the weighted model was defined as a group of birational transformations of C 2 , generated by two involutions. This is the group studied in [START_REF] Bousquet | Walks with small steps in the quarter plane, Algorithmic probability and combinatorics[END_REF]. It is a finite group if and only if the automorphism of the weighted model σ is of finite order.

Analytic continuation and differential transcendence criteria

The aim of this section is to give differential transcendence criteria adapted to the study of the generating series of the weighted models arising from (S). Let us describe our strategy. In Lemma 1.1, we defined the auxiliary series F 1 (x, t) := -K(x, 0, t)Q(x, 0, t), F 2 (y, t) := -K(0, y, t)Q(0, y, t). Since it is obvious that Q(x, y, t) converges for |x| < 1, |y| < 1, |t| < 1, we have the same conclusion for these former series as well. Using the parameterization φ = (x, y) : P 1 (C) → E t given in the previous section, we can pull back these functions to functions

F 1 (s) = F 1 (x(s), t) and F 2 (s) = F 2 (y(s), t)
analytic in a neighborhood of 0 in P 1 (C). Using the functional equation (1.1), we will prove that F 1 (s) and F 2 (s) each satisfy very simple q-difference equations

F 1 (qs) -F 1 (s) = b1 (s), F 2 (qs) -F 2 (s) = b2 (s),
for suitable b1 , b2 ∈ C(s). This implies in particular that F 1 (s) and F 2 (s) can be continued into meromorphic functions on all of C. A result of Ishizaki, see [START_REF] Ishizaki | Hypertranscendency of meromorphic solutions of a linear functional equation[END_REF], implies that if either F 1 (s) or F 2 (s) are s-differentially algebraic then they must be in C(s) and results from the theory of linear q-difference equations allow us to detect this via the partial fraction decomposition of b1 and b2 . In addition, we will show that F 1 (s) (resp. F 2 (s)) is s-differentially algebraic if and only if Q(x, 0, t) (resp. Q(0, y, t)) is x-differentially algebraic (resp. y-differentially algebraic). We will therefore be able to reduce the question of whether Q(x, 0, t) (resp. Q(0, y, t)) is x-differentially algebraic (resp. y-differentially algebraic) to seeing if the above mentioned conditions on the partial fraction decomposition of b1 and b2 hold. This will be done in Section 3 where we will see that the latter conditions never hold. We now turn to supplying the details of this brief sketch.

In this section, we continue to assume that Assumption 1.3 holds true.

2.1. Functional equation. We let φ = (x, y) : P 1 (C) → E t be the parameterization of E t given in Proposition 1.5. Straightforward calculations show that

• φ(0) = φ(∞) = ([0 : 1], [0 : 1]); • x(ι 1 (s)) = x(s) where ι1 (s) = 1 s ; • y(ι 2 (s)) = y(s) where ι2 (s) = q s = λ 2 s ; • σ(s) = qs where σ = ι2 • ι1 .
In particular, we have that ιk • φ = φ • ι k and σ • φ = φ • σ which will allow the following computations.

Recall the functional equation (1.1):

K(x, y, t)Q(x, y, t) = xy -F 1 (x, t) -F 2 (y, t) + td -1,-1 Q(0, 0, t).
This equation is a formal identity but for |x| < 1 and |y| < 1, the series Q(x, y, t), F 1 (x, t) and F 2 (y, t) are convergent. Using our parameterization of E t , we will show how we can pull back these convergent series and analytically continue them to meromorphic functions on C satisfying simple q-difference equations.

The set

V = {([x : 1], [y : 1]) ∈ E t | |x|, |y| < 1} is an open neighborhood of ([0 : 1], [0 : 1]
) in E t for the analytic topology, and, for all (x, y) ∈ V , we have (2.1) 0 = xy -F 1 (x, t) -F 2 (y, t) + td -1,-1 Q(0, 0, t).

Since φ(0) = φ(∞) = ([0 : 1], [0 : 1]), there exists U ⊂ P 1 (C) which is the union of two small open discs centered at 0 and ∞ such that φ(U ) ⊂ V .

For any s ∈ U , we set F 1 (s) = F 1 (x(s), t) and F 2 (s) = F 2 (y(s), t). Then, F 1 and F 2 are meromorphic functions over U and (2.1) yields, for all s ∈ U ,

(2.2) 0 = x(s)y(s) -F 1 (s) -F 2 (s) + td -1,-1 Q(0, 0, t).
Replacing s by ι2 (s) in (2.2), we obtain, for all s close to 0 or ∞, (in what follows, we use

x(ι 1 (s)) = x(s), y(ι 2 (s)) = y(s), F 1 (ι 1 (s)) = F 1 (s) and F 2 (ι 2 (s)) = F 2 (s)) 0 = x(ι 2 (s))y(ι 2 (s)) -F 1 (ι 2 (s)) -F 2 (ι 2 (s)) + td -1,-1 Q(0, 0, t) = x(ι 1 (ι 2 (s)))y(s) -F 1 (ι 1 (ι 2 (s))) -F 2 (s) + td -1,-1 Q(0, 0, t) = x(q -1 s)y(s) -F 1 (q -1 s) -F 2 (s) + td -1,-1 Q(0, 0, t).
(2.3) Subtracting (2.2) from (2.3), and then replacing s by qs, we obtain, for all s close to 0 or ∞, F 1 (qs) -F 1 (s) = (x(qs) -x(s))y(qs).

(2.4) Remark 2.1. If we set t = 1 and replace F 1 by -F 1 K(0,y,t) , then a similar argument leads to another functional equation which is the one given in [FIM17, Theorem 6.4.1]. Similarly, replacing s by ι1 (s) in (2.2), we obtain, for all s close to 0 or ∞,

0 = x(ι 1 (s))y(ι 1 (s)) -F 1 (ι 1 (s)) -F 2 (ι 1 (s)) + td -1,-1 Q(0, 0, t) = x(s)y(ι 2 (ι 1 (s))) -F 1 (s) -F 2 (ι 2 (ι 1 (s))) + td -1,-1 Q(0, 0, t) = x(s)y(qs) -F 1 (s) -F 2 (qs) + td -1,-1 Q(0, 0, t).
(2.5) Subtracting (2.5) from (2.2), we obtain, for all s close to 0 or ∞, F 2 (qs) -F 2 (s) = x(s)(y(qs) -y(s)). (2.6)

We let F 1 and F 2 be the restrictions of F 1 and F 2 to a small disc around 0. They satisfy the functional equations (2.4) and (2.6) for s close to 0. Since |q| / ∈ {0, 1}, this implies that each of the functions F 1 and F 2 can be continued to a meromorphic function on C with (2.4) satisfied for all s ∈ C. Note that there is a priori no reason why, in the neighborhood of ∞, these functions should coincide with the original functions F 1 and F 2 .

2.2. Application to differential transcendence. In this subsection, we derive differential transcendency criteria for x → Q(x, 0, t) and y → Q(0, y, t). They are based on the fact that the related functions F 1 and F 2 satisfy difference equations. Definition 2.2. Let (E, δ) ⊂ (F, δ) be differential fields, that is, fields equipped with a map δ that satisfies δ(a + b) = δ(a) + δ(b) and δ(ab) = aδ(b) + δ(a)b. We say that f ∈ F is differentially algebraic over E if it satisfies a non trivial algebraic differential equation with coefficients in E, i.e., if for some m there exists a nonzero polynomial P (y 0 , . . . , y m ) ∈ E[y 0 , . . . , y m ] such that

P (f, δ(f ), . . . , δ m (f )) = 0.
We say that f is holonomic over E if in addition, the polynomial is linear. We say that f is differentially transcendental over E if it is not differentially algebraic.

Proposition 2.3. The series x → Q(x, 0, t) is differentially algebraic over (C(x), d dx ) if and only if F 1 is differentially algebraic over (C(s), d ds ). The series y → Q(0, y, t) is differentially algebraic over (C(y), d dy ) if and only if F 2 is differentially algebraic over (C(s), d ds ). Proof. This follows from Lemmas 6.3 and 6.4 of [START_REF] Dreyfus | On the nature of the generating series of walks in the quarter plane[END_REF], since we go from x → Q(x, 0, t) to F 1 by a variable change which is algebraic (and therefore differentially algebraic). The proof for F 2 is similar.

Consequently, we only need to study F 1 and F 2 . Recall that they belong to the field Mer(C) of meromorphic functions on C. Using a result due to Ishizaki [Ish98, Theorem 1.2] (see also [START_REF] Hardouin | Differential Galois theory of linear difference equations[END_REF]Proposition 3.5], where a Galoisian proof of Ishizaki's result is given), we get, for any i ∈ {1, 2}, the following dichotomy ¶ :

• either F i ∈ C(s), or • F i is differentially transcendental over C(s).
Remark 2.4. 1. Note that the fact that F i is meromorphic on C is essential. For instance, if q > 1, the Theta function θ q (s) = n∈Z q -n(n-1)/2 s n is meromorphic on C * , is not rational and is differentially algebraic as it is shown for instance in [HS08, Corollary 3.4]. [START_REF] Mishna | Two non-holonomic lattice walks in the quarter plane[END_REF], and the ¶ Ishizaki's proof of his result proceeds by comparing behavior at various poles and uses growth results from Wiman-Valiron Theory. The approach of [START_REF] Hardouin | Differential Galois theory of linear difference equations[END_REF] avoids the growth considerations and is more algebraic. A slightly weaker result, in the spirit of the considerations of [START_REF] Dreyfus | On the nature of the generating series of walks in the quarter plane[END_REF], would suffice to establish this dichotomy, see [HS08, Corollary 3.2, Proposition 6.4] or [START_REF] Hardouin | Hypertranscendance des systèmes aux différences diagonaux[END_REF].

Combining Ishizaki's dichotomy with the result of Mishna and Rechnitzer

result of Melczer and Mishna [START_REF] Melczer | Singularity analysis via the iterated kernel method[END_REF], on the non holonomy of the complete generating series of the unweighted genus zero walks, one finds that these complete generating series are differentially transcendental, thus proving directly Theorem 3.1 in the five unweighted cases.

So, we need to understand when F i ∈ C(s). We set b 1 (s) = y(qs)(x(qs) -x(s)) and b 2 (s) = x(s)(y(qs) -y(s)), so that the functional equations (2.4) and (2.6) can be restated as

(2.7) F 1 (qs) -F 1 (s) = b 1 (s) and F 2 (qs) -F 2 (s) = b 2 (s)
for s ∈ C.

Lemma 2.5. For any i ∈ {1, 2}, the following facts are equivalent:

• F i ∈ C(s); • there exists f i ∈ C(s) such that b i (s) = f i (qs) -f i (s).
Proof.

If F i ∈ C(s) then (2.7) shows that b i (s) = f i (qs) -f i (s) with f i = F i ∈ C(s).
Conversely, assume that there exists

f i ∈ C(s) such that b i (s) = f i (qs)-f i (s). Using (2.7) again, we find that ( F i -f i )(s) = ( F i -f i )(qs).
Since the function F i -f i is meromorphic over C, we may expand it as a Laurent series at s = 0: F i -f i = ≥ 0 a s . We then have ≥ 0 a s = ≥ 0 a q s and since q is not a root of unity, F i -f i ∈ C. This ensures that F i ∈ C(s).

Remark 2.6. In [START_REF]Counting quadrant walks via Tutte's invariant method[END_REF], the authors introduce the notion of decoupling functions, that is of functions F (x) ∈ Q(x, t) and G(y) ∈ Q(y, t) such that xy = F (x) + G(y) for x, y satisfying K(x, y, t) = 0. It is easily seen that if F and G are decoupling functions, one has

ι 2 (xy) -xy = ι 2 (F (x)) -F (x) and ι 1 (xy) -xy = ι 1 (G(y)) -G(y),
when K(x, y, t) = 0. In our genus zero situation, composing the former identities with the uniformization yields b i (s) = f i (qs) -f i (s) where f 1 (s) = F (x(s)) ∈ C(s) and f 2 (s) = G(y(s)) ∈ C(s). Then, Lemma 2.5 is essentially the same kind of results as [BBMR17, Lemma 2] but in the easier framework of a genus zero kernel curve.

The following lemma is a consequence of the functional equation satisfied by F 1 , F 2 . See [FIM17, Corollary 3.2.5], or [DHRS18, Proposition 3.10], for similar results in the genus one case.

Lemma 2.7. The following properties are equivalent:

• F 1 ∈ C(s); • F 2 ∈ C(s).
Proof. Assume that F 1 ∈ C(s). Lemma 2.5 states that there exists

f 1 ∈ C(s) such that b 1 (s) = f 1 (qs) -f 1 (s). Note that b 1 (s) + b 2 (s) = (xy)(qs) -(xy)(s), so that we have b 2 (s) = f 2 (qs) -f 2 (s), with xy(s) -f 1 (s) = f 2 (s) ∈ C(s). Lemma 2.5 implies that F 2 ∈ C(s).
The converse is proved in a similar way.

Theorem 2.8. The following properties are equivalent:

(1) The series Q(x, 0, t) is differentially algebraic over C(x);

(2) The series Q(x, 0, t) is algebraic over C(x);

(3) The series Q(0, y, t) is differentially algebraic over C(y);

(4) The series Q(0, y, t) is algebraic over C(y);

(5) There exists

f 1 ∈ C(s) such that b 1 (s) = f 1 (qs) -f 1 (s); (6) There exists f 2 ∈ C(s) such that b 2 (s) = f 2 (qs) -f 2 (s).
Proof. Assume that (1) holds true. Proposition 2.3 implies that F 1 is differentially algebraic over C(s). Ishizaki's Theorem ensures that F 1 ∈ C(s). But x : P 1 (C) → P 1 (C) is locally (for the analytic topology) invertible at all but finitely many points of P 1 (C) and the corresponding local inverses are algebraic over C(x). It follows that F 1 (•, t) can be expressed as a rational expression, with coefficients in C, of an algebraic function, and, hence, is algebraic over C(x). Hence (2) is satisfied. The fact that (2) implies (1) is obvious. The fact that (3) is equivalent to (4) can be shown in a similar manner to the equivalence of (1) and (2). The fact that (1) to (4) are equivalent now follows from Lemma 2.7 combined with [Ish98, Theorem 1.2]. The remaining equivalences follow from Lemma 2.5.

So, to decide whether Q(x, 0, t), Q(0, y, t) are differentially transcendental, we are led to the following problem:

Given b ∈ C(s), decide whether there exists f ∈ C(s) such that b(s) = f (qs) -f (s).
When such an f exists, we say that b is q-summable in C(s). This problem is known as a q-summation problem and has been solved by Abramov [Abr95]. This procedure was recast in [START_REF] Chen | Residues and telescopers for bivariate rational functions[END_REF] in terms of the so-called q-residues of b, which we now define.

We begin by defining the q-orbit of β ∈ C * to be βq Z = {β • q i | i ∈ Z}. Given a rational function b(s) ∈ C(s) we may rewrite its partial fraction decomposition uniquely as

(2.8) b(s) = c + sp 1 + p 2 s r + m i=1 ni j=1 ri,j =0 α i,j, (s -q • β i ) j , where c ∈ C, p 1 , p 2 ∈ C[s], m, n i ∈ Z ≥0 are nonzero, r, r i,j ∈ Z ≥0 , α i,j, , β i ∈ C
and the β i 's are nonzero and in distinct q-orbits. Definition 2.9. (cf. [CS12, Definition 2.7]) Let b ∈ C(s) be of the form (2.8). The sum ri,j =0 q -•j α i,j, is called the q-residue of b at the q-orbit of β i of multiplicity j (this is called the q-discrete residue in [START_REF] Chen | Residues and telescopers for bivariate rational functions[END_REF]) and is denoted by qres(b, β i , j). In addition, we call the constant c the q-residue of b at infinity and denote it by qres(b, ∞).

Example 2.10. Let q = 2 and b(s

) = 1 + s + s + 2 s 2 + 3 (s -1) 2 - 12 (s -2) 2 + 1 s -5 .
We have qres(b, ∞) = 1, qres(b, 1, 2) = 2 0 • 3 + 2 -1•2 (-12) = 0, and qres(b, 5, 1) = 1. All other q-residues are 0.

One has the following criterion for q-summability.

Proposition 2.11.

(c.f. [CS12, Proposition 2.10]) Let b = f /g ∈ C(x) be such that f, g ∈ C[x]
with gcd(f, g) = 1. Then b is q-summable in C(s) if and only if the q-residues qres(b, ∞) = 0 and qres(b, β, j) = 0 for any multiplicity j and any β = 0 with g(β) = 0, g(q β) = 0 for every < 0.

Applying this criteria to the above example we see that b is not q-summable because qres(b, ∞) = 0 as well as qres(b, 5, 1) = 0. In fact, whenever an element b ∈ C(x) has a pole of order m ≥ 1 at a point β and no other pole of order ≥ m in the q-orbit of β, then a q-residue of multiplicity m will be nonzero. We therefore have the following corollary (also a consequence of results in [START_REF] Abramov | Indefinite sums of rational functions[END_REF]) which plays a crucial role in the next section.

Corollary 2.12. If β ∈ C * is a pole of b ∈ C(x) of order m ≥ 1 and if b has no other pole of order ≥ m in the q-orbit of β, then b is not q-summable, i.e., there is no f (s) ∈ C(s) such that b(s) = f (qs) -f (s).

Using the parameterization φ : P 1 (C) → E t , we can translate this to give a criterion for the differential transcendence of x → Q(x, 0, t) and y → Q(0, y, t) over C(x) and C(y) respectively. We set (see Section 1 for notations)

b 1 = ι 1 (y)(ι 2 (x) -x) and b 2 = x(ι 1 (y) -y), so that we have b 1 = b 1 • φ and b 2 = b 2 • φ.
Proposition 2.13. We suppose that Assumption 1.3 holds true and recall that |q| = 1. Let b ∈ C(x, y) be a rational function on E t . Assume that P ∈ E t \ {Ω} is a pole of b of order m ≥ 1 such that none of the σ i (P ) with i ∈ Z\{0} is a pole of b of order ≥ m, then b = σ(g) -g

has no solution g ∈ C(x, y) which restricts to a rational function on E t .

In particular, if b 2 = x(ι 1 (y) -y) satisfies this condition, then x → Q(x, 0, t), (resp. y → Q(0, y, t)) is differentially transcendental over C(x) (resp. differentially transcendental over C(y)).

Proof. We know that the parameterization φ = (x, y) : P 1 (C) → E t that we have constructed, induces an isomorphism between P 1 (C) \ {0, ∞} and E t \ {Ω}. If s 0 ∈ P 1 (C) \ {0, ∞} is such that φ(s 0 ) = P , then s 0 is a pole of order m ≥ 1 of b • φ such that none of the σi (s 0 ) = q i s with i ∈ Z\{0} is a pole of b • φ of order ≥ m. If g ∈ C(x, y) restricts to a rational function on E t and satisfies b = σ(g) -g, then f = g • φ would satisfy b(s) = f (qs) -f (s) contradicting Lemma 2.12.

If b 2 = x(ι 1 (y) -y) satisfies the condition of the Proposition, then b 2 = σ(g) -g has no solution g that is a rational function on E t . Pulling this back to P 1 (C), we see that for b 2 (s) = b 2 • φ(s) = x(s)(y(1/s) -y(s)), the equation b 2 (s) = f (qs) -f (s) has no solution in C(s). Theorem 2.8 yields our conclusion.

Finally we note that given a fixed family of probabilities (d i,j ), the algorithms [START_REF] Abramov | Indefinite sums of rational functions[END_REF][START_REF] Chen | Residues and telescopers for bivariate rational functions[END_REF] permit us to decide if the generating series is differentially algebraic or not. In Section 3, we will prove an unconditional statement, that is, for every set of probabilities d i,j , the generating series is differentially transcendental. Note that this kind of result may a priori not be obtained via the above mentioned algorithms, since the generating series depends on parameters (the probabilities d i,j ) and it is not clear how to make the algorithms give information about arbitrary specializations of the parameters.

Differential transcendence: main result

In this section, we will prove the main result of this paper: Theorem 3.1. We suppose that Assumption 1.3 is satisfied. Then, the functions x → Q(x, 0, t) and y → Q(0, y, t) are differentially transcendental over C(x) and C(y) respectively. Remark 3.2. (i) Models of walks in three dimensions in the octant have been recently studied. In [START_REF] Bostan | On 3-dimensional lattice walks confined to the positive octant[END_REF][START_REF] Du | Infinite orders and non-D-finite property of 3-dimensional lattice walks[END_REF], the authors study such unweighted models having at most six steps. Among the non trivial 35548 models, 527 are equivalent to weighted models of walks in the quarter plane, in the sense of [BBMKM16, Definition 2] and Assumption 1.3 is satisfied for 69 such models, see [DHW16, Section 3]. For these models of two dimensional walks our results apply. For example, in [START_REF] Du | Infinite orders and non-D-finite property of 3-dimensional lattice walks[END_REF], the authors prove that one of the three dimensional unweighted models of a walk in the octant is equivalent to the following weighted model of a two dimensional walk of genus zero:

1/2 1/4 1/4
(ii) Combining Theorem 3.1 with Remark 2.6, we have proved that in the genus zero situation there are no decoupling functions.

The proof of Theorem 3.1 will be given at the very end of this section. Our strategy will be to use Proposition 2.13. So, we begin by collecting information concerning the poles of b 2 = x(ι 1 (y) -y).

3.1. Preliminary results concerning the poles of b 2 . We write

b 2 = x(ι 1 (y) -y) in the projective coordinates ([x 0 , x 1 ], [y 0 , y 1 ]) with x = x0
x1 and y = y0 y1 . We note that Ω = ([0 : 1], [0 : 1]) is not a pole of b 2 . Since we want to compute the poles of b 2 , it is natural to start with the poles of xy. Therefore let us focus our attention on the points ([x 0 : x 1 ], [y 0 : y 1 ]) of E t corresponding to the equation x 1 y 1 = 0, namely:

P 1 = ([1 : 0], [β 0 : β 1 ]),P 2 = ι 1 (P 1 ) = ([1 : 0], [β 0 : β 1 ]), Q 1 = ([α 0 : α 1 ], [1 : 0]),Q 2 = ι 2 (Q 1 ) = ([α 0 : α 1 ], [1 : 0]).
Since P 1 , P 2 ∈ E t , to compute [β 0 : β 1 ] and [β 0 : β 1 ], we have to solve K(1, 0, y 0 , y 1 , t) = 0. We then find that [β 0 : β 1 ] and [β 0 : β 1 ] are the roots in P 1 (C) of the homogeneous polynomial in y 0 and y 1 given by d 1,-1 y 2 1 + d 1,0 y 0 y 1 + d 1,1 y 2 0 = 0. Similarly, the x-coordinates [α 0 : α 1 ] and [α 0 : α 1 ] of Q 1 and Q 2 are the roots in P 1 (C) of the homogeneous polynomial in x 0 and x 1 given by

d -1,1 x 2 1 + d 0,1 x 0 x 1 + d 1,1 x 2 0 = 0.
Although the following Lemma already appears in [DHRS18, Lemma 4.11], we give its proof to be self-contained.

Lemma 3.3. The set of poles of b

1 = ι 1 (y) (σ(x) -x) in E t is contained in S 1 = {ι 1 (Q 1 ), ι 1 (Q 2 ), P 1 , P 2 , σ -1 (P 1 ), σ -1 (P 2 )}.
Similarly, the set of poles of b

2 = x(ι 1 (y) -y) in E t is contained in S 2 = {P 1 , P 2 , Q 1 , Q 2 , ι 1 (Q 1 ), ι 1 (Q 2 )} = {P 1 , P 2 , Q 1 , Q 2 , σ -1 (Q 1 ), σ -1 (Q 2 )}.
Moreover, we have

(3.1) (b 2 ) 2 = x 2 0 ∆ x [x0:x1] x 2 1 ( 2 i=0 x i 0 x 2-i 1 td i-1,1 ) 2 .
Proof. The proofs of the assertions about the location of the poles of b 1 and b 2 are straightforward. Let us prove (3.1). By definition, the y coordinates of ι 1 ( y0 y1 ) and y0 y1 are the two roots of the polynomial y → K(x 0 , x 1 , y, t). The square of their difference equals to the discriminant divided by the square of the leading term, that is,

ι 1 ( y 0 y 1 ) - y 0 y 1 2 = ∆ x [x0:x1] ( i x i 0 x 2-i 1 td i-1,1 ) 2 . Therefore, we find b 2 x 0 x 1 , y 0 y 1 2 = x 2 0 ∆ x [x0:x1] x 2 1 ( i x i 0 x 2-i 1 td i-1,1 ) 2 .
To apply Proposition 2.13 we now need to separate the orbits. Let us begin with P 1 and P 2 (resp. Q 1 and Q 2 ). In what follows, we will use the equivalent relation ∼ on E t defined, for P, Q ∈ E t , by

P ∼ Q ⇔ ∃ ∈ Z, σ (P ) = Q.
Proposition 3.4. If P 1 = P 2 , then one of the following properties holds:

• P 1 ∼ P 2 ; • d 0,1 = d 1,1 = 0. If Q 1 = Q 2 ,
then one of the following properties holds:

•

Q 1 ∼ Q 2 ; • d 1,0 = d 1,1 = 0.
Proof. We only prove the statement for the P i , the proof for the Q j being similar. Let p 1 , p 2 ∈ C * be such that φ(p 1 ) = P 1 and φ(p 2 ) = P 2 . Recall that Proposition 1.5 ensures that α 2 = 1 -2td 0,0 + t 2 d 2 0,0 -4t 2 d -1,1 d 1,-1 α 3 = 2t 2 d 1,0 d 0,0 -2td 1,0 -4t 2 d 0,1 d 1,-1 α 4 = t 2 (d 2 1,0 -4d 1,1 d 1,-1 ) and that, according to Proposition 1.7, one of the two complex numbers q or q -1 is equal to -1 + d 0,0 t -(1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 -1 + d 0,0 t + (1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 .

The explicit formula for φ given in Proposition 1.5 shows that p 1 and p 2 are the roots of -α 2 3 -4α 2 α 4 X 2 + 2α 3 X -α 2 3 -4α 2 α 4 = 0.

So, we have (for suitable choices of the complex square roots )

p 1 = -α 3 -2 √ α 2 α 4 -α 2 3 -4α 2 α 4 and p 2 = -α 3 + 2 √ α 2 α 4 -α 2 3 -4α 2 α 4 .
Assume that P 1 ∼ P 2 . Then, there exists ∈ Z * such that p1 p2 = q ( = 0 because P 1 = P 2 ). Using the above formulas for p 1 , p 2 and q and replacing by -if necessary, this can be rewritten as:

(3.2) -α 3 -2 √ α 2 α 4 -α 3 + 2 √ α 2 α 4 = -1 + d 0,0 t -(1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 -1 + d 0,0 t + (1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 .
Since p i is chosen so that φ(p i ) = P i , we need to take the square roots consistent with this selection.

Recall that t is transcendental. We shall treat t as a variable and both sides of (3.2) as functions of the variable t, algebraic over Q(t). Formula (3.2) shows that these algebraic functions coincide at some transcendental number, therefore they are equal.

We now consider these algebraic functions near 0 (we choose an arbitrary branch) and will derive a contradiction by proving that they have different behaviors at 0.

If d 1,1 = 0, then, considering the Taylor expansions at 0 in (3.2), we obtain, up to replacing by -if necessary:

d 1,0 -∆ 1 d 1,0 + ∆ 1 + O(t) = 1 t 2 1 d 1,-1 d -1,1 + O(1/t)
where ∆ 1 is some square root of d 2 1,0 -4d 1,1 d 1,-1 , and d 1,0 -∆ 1 and d 1,0 + ∆ 1 are not 0 because d 1,1 = 0 (note that, by Assumption 1.3, we have d 1,-1 d -1,1 = 0). This equality is impossible.

If d 1,1 = 0, then (3.2) gives

t d 0,1 d 1,-1 d 1,0 + O(t 2 ) = 1 t 2 1 d 1,-1 d -1,1 + O(1/t)
(note that we have d 1,0 = 0 because P 1 = P 2 ). This implies d 0,1 = 0 and concludes the proof.

Proposition 3.5. Assume that d 1,1 = 0. Then, for any i, j ∈ {1, 2}, we have P i ∼ Q j .

Proof. Let p i , q j ∈ C * be such that φ(p i ) = P i and φ(q j ) = Q j . As seen at the beginning of the proof of Proposition 3.4, we have (for suitable choices of the square roots)

p i = -α 3 -2 √ α 2 α 4 -α 2 3 -4α 2 α 4
.

Similarly, we have (for suitable choices of the square roots)

q j = λ -β 3 -2 √ β 2 β 4 -β 2 3 -4β 2 β 4 .
Suppose to the contrary that P i ∼ Q j . The condition d 1,1 = 0 yields that P i = Q j . Then, there exists ∈ Z * such that pi qj = q . Using the above formulas for p i and q j , using Proposition 1.7 and replacing by -if necessary, this can be rewritten as:

(3.3) α 3 + 2 √ α 2 α 4 α 2 3 -4α 2 α 4 β 2 3 -4β 2 β 4 β 3 + 2 √ β 2 β 4 = -1 + d 0,0 t -(1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 -1 + d 0,0 t + (1 -d 0,0 t) 2 -4d 1,-1 d -1,1 t 2 + 1 2 .
As in the proof of Proposition 3.4, we can treat t as a variable and both sides of (3.3) as functions of the variable t algebraic over Q(t), the above equality shows that they coincide, and we shall now consider these algebraic functions near 0 (we choose an arbitrary branch). Considering the Taylor expansions at 0 in (3.3), we obtain:

-d 1,0 -∆ 1 d 2 1,0 -∆ 1 2 d 2 0,1 -∆ 2 2 -d 0,1 -∆ 2 + O(t) = 1 t 2 1 d 1,-1 d -1,1
+ O(t)

+ 1 2
where ∆ 1 and ∆ 2 are suitable square roots of d 2 1,0 -4d 1,1 d 1,-1 and d 2 0,1 -4d 1,1 d -1,1 respectively, and none of the numbers -d 1,0 -∆ 1 , d 2 1,0 -∆ 1 2 , d 2 0,1 -∆ 2 2 , -d 0,1 -∆ 2 2 is zero because d 1,1 = 0. This equality is impossible.

T. DREYFUS, C. HARDOUIN, J. ROQUES, M.F. SINGER

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under the Grant Agreement No 648132. The second author would like to thank the ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-0 for its partial support. The second author's work is also supported by ANR Iso-Galois. The work of the third author has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d'avenir. The work of the fourth author was partially supported by a grant from the Simons Foundation (#349357, Michael Singer). not assumed that i,j d i,

3.2. Proof of Theorem 3.1. We shall use the criteria of Proposition 2.13 applied to b 2 . From the expression of a 3 and a 4 given in Section 1, we may deduce that a 3 = a 4 and therefore ∆ x

seen as a function on P 1 (C) has at most a simple zero at P 1 and P 2 . With (3.1) we find that P 1 and P 2 are poles of b 2 .

If d 1,1 = d 1,0 = 0 (and d 0,1 = 0 by Assumption 1.3), then a direct calculation shows that the polar divisor of b 2 * * on

where

The result is now a direct consequence of Proposition 2.13 because P 1 is a pole of order three of b 2 , and all the other poles of b 2 have order 1.

The case d 1,1 = d 0,1 = 0 is similar. Assume that d 1,1 = 0 and d 1,0 d 0,1 = 0. In this case, we have

). Note that these four points are two by two distinct (since d 0,1 = 0 and t is transcendental, the quantity d 0,1 (1 -td 0,0 ) + td 1,0 d -1,1 does not vanish). A direct computation shows that the polar divisor of b 2 on E t is 2P 1 + 2P 2 + Q 2 + ι 1 (Q 2 ). Proposition 3.4 ensures that P 1 ∼ P 2 . So, P = P 1 or P 2 is such that none of the σ i (P ) with i ∈ Z\{0} is a pole of order ≥ 2 of b 2 . The result is now a consequence of Proposition 2.13.

Last, assume that d 1,1 = 0. Then, combining Proposition 3.4 and Proposition 3.5, and using the fact that the set of poles of b 2 is included in {P 1 , P 2 , Q 1 , Q 2 , σ -1 (Q 1 ), σ -1 (Q 2 )}, we get that P 1 is such that none of the σ i (P 1 ) with i ∈ Z\{0} is a pole of b 2 . The result is now a consequence of Proposition 2.13.


domain: spi chim phys

text:
Grégoire Martouzet

Choongyeop Lee

Christophe Pirat

Christophe Ybert

Anne-Laure Biance

Drag reduction on drop during impact on multiscale superhydrophobic surfaces

Liquid drop impact dynamics depends on the liquid-substrate interaction. In particular, when liquid-solid friction is decreased, the spreading of the impacting drop lasts longer. We characterise this effect by using two types of superhydrophobic surfaces, with similar wetting properties but different friction coefficients. It is found that, for large enough impact velocities, a reduced friction delays the build-up of a viscous boundary layer, and leads to an increase of the time required to reach the maximal radius of the impacting drop. An asymptotic analysis is carried out to quantify this effect, and agrees well with the experimental findings. Interestingly, this novel description complements the general picture of drop impact on solid surfaces, and more generally addresses the issue of drag reduction in the presence of slippage for non-stationary flows.

Introduction

Despite being microscopic in nature, the liquid-solid interface can have a strong influence on various macro scale dynamics such as for liquid drop impact on solid surfaces [START_REF] Yarin | Drop impact dynamics: Splashing, spreading, receding, bouncing[END_REF], solid body impact into a liquid pool [START_REF] Duez | Making a splash with water repellency[END_REF], and inertial separation between liquid flow and solid [START_REF] Duez | Wetting controls separation of inertial flows from solid surfaces[END_REF]. Drop impact on a solid surface has been of particular interest for understanding the complex physical phenomena involved, as well as for its consequences in many industrial processes [START_REF] Josserand | Drop impact on a solid surface[END_REF]. Despite its huge interest, the effect of surface wettability on droplet impact dynamics remains debated. On the one hand, experiments show its pronounced role on spreading [START_REF] Bayer | Contact angle dynamics in droplets impacting on flat surfaces with different wetting characteristics[END_REF][START_REF] Vadillo | Dynamic contact angle effects onto the maximum drop impact spreading on solid surfaces[END_REF][START_REF] Liu | Pancake bouncing on superhydrophobic surfaces[END_REF] or bouncing [START_REF] Richard | Bouncing water drops[END_REF] at low impact velocity, whereas others show no influence at larger impact velocities [START_REF] Clanet | Maximal deformation of an impacting drop[END_REF][START_REF] Tsai | Microscopic structure influencing macroscopic splash at high weber number[END_REF]. Local characterisation even points the crucial effect of trapped air film during impact [START_REF] De Ruiter | Wettability-independent bouncing on flat surfaces mediated by thin air films[END_REF]. On the other hand, a large number of theoretical, numerical and experimental studies have evidenced that surface friction can play an important role during drop impact at large impact velocity, mainly by considering the two limiting cases of frictionless impact [START_REF] Clanet | Maximal deformation of an impacting drop[END_REF][START_REF] Tsai | Drop Impact upon Microand Nanostructured Superhydrophobic Surfaces[END_REF][START_REF] Lastakowski | Bridging local to global dynamics of drop impact onto solid substrates[END_REF], and fully frictional impact with a zero velocity condition at the solid substrate [START_REF] Roisman | Inertia dominated drop collisions. I. On the universal flow in the lamella[END_REF][START_REF] Roisman | Inertia dominated drop collisions. ii. an analytical solution of the navierstokes equations for a spreading viscous film[END_REF][START_REF] Eggers | Drop dynamics after impact on a solid wall: Theory and simulations[END_REF][START_REF] Schroll | Impact of a viscous liquid drop[END_REF][START_REF] Lagubeau | Spreading dynamics of drop impacts[END_REF]). In the latter case, drop spreading after impact is governed by the build-up of a viscous boundary layer in the vicinity of the substrate. The case of partial friction has however hardly been explored.

With the development of micro and nanofabrication techniques, it is possible to engineer surfaces with well controlled topography. Particularly, hydrophobic surfaces with micro or nanoscale features (so-called superhydrophobic (SHPo) surfaces) are found to promote unique liquid dynamic characteristics due to the reduced liquid-solid contact [START_REF] Callies | On water repellency[END_REF]. For example, previously unseen phenomena such as crystallographic splashing [START_REF] Reyssat | Dynamical superhydrophobicity[END_REF][START_REF] Tsai | Microscopic structure influencing macroscopic splash at high weber number[END_REF], pancake bouncing [START_REF] Liu | Pancake bouncing on superhydrophobic surfaces[END_REF] and reduced contact time [START_REF] Bird | Reducing the contact time of a bouncing drop[END_REF] were observed upon impact on such surfaces. On SHPo surfaces, the presence of a stress-free liquid-gas interface affects not only the wetting properties of the droplet but also results in a frictional drag reduction, which is commonly quantified using the effective slip length denoted b, defined as the ratio of slip velocity to the shear rate at the wall [START_REF] Joseph | Slippage of water past superhydrophobic carbon nanotube forests in microchannels[END_REF][START_REF] Choi | Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface[END_REF][START_REF] Rothstein | Slip on superhydrophobic surfaces[END_REF][START_REF] Ybert | Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries[END_REF]. This effect has been ignored for impacts of simple fluid drops, as it has been observed to have no consequences on droplet spreading [START_REF] Clanet | Maximal deformation of an impacting drop[END_REF][START_REF] Tsai | Microscopic structure influencing macroscopic splash at high weber number[END_REF], although a giant drag reduction has been observed with complex fluids [START_REF] Luu | Giant drag reduction in complex fluid drops on rough hydrophobic surfaces[END_REF]. Determining how the boundary layer -and the overall impact dynamics -is affected by an intermediate friction is the goal of this study. We focus on a regime at high impact velocities, where the effect of friction on the substrate, quantified by the so-called slip length, is relevant for drop spreading dynamics.

One of the main concerns in conducting quantitative experimental studies on SHPo surfaces is that it can sustain a low frictional Cassie-Baxter state as long as the drop interacts with the top region of surface structures without intruding into them [START_REF] Lee | Structured surfaces for a giant liquid slip[END_REF]). However, during drop impact, the dynamic pressure in the liquid drop might overcome the resistance upon imbibition provided by surface tension, leading to the so-called Cassie-to-Wenzel transition [START_REF] Bartolo | Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces[END_REF][START_REF] Reyssat | Bouncing transitions on microtextured materials[END_REF]. It was proposed that the impalement during impact might be avoided by using SHPo surface with nanometric features [START_REF] Deng | Nonwetting of impinging droplets on textured surfaces[END_REF][START_REF] Tsai | Drop Impact upon Microand Nanostructured Superhydrophobic Surfaces[END_REF]), but it would compromise the slip length, as the slip length is proportional to the structural pitch of surface features. This problem can be addressed by employing multi-scale superhydrophobic surfaces [START_REF] Lee | Underwater restoration and retention of gases on superhydrophobic surfaces for drag reduction[END_REF], where microscale features ensure a large slip length, while nanoscale features prevent the detrimental Cassie-to-Wenzel transition.

Experiments

We designed two types of multiscale superhydrophobic surfaces (Fig. 1a-b, Appendix A), noted as B and C, resulting in both cases in similar large static contact angles, but in different friction characteristics and so different slip lengths for water (Table 1). A bare smooth silicon wafer was used as a reference (surface A).

Drop impact is captured from the side with a high speed camera (Photron SA-4) at up to 30,000 fps (Fig. 1c). The liquids used are mixtures of ultra pure water and glycerol, with a kinematic viscosity ν ranging from 1 cS (pure water) to 630 cS (pure glycerol). Drops of radius R ≈1 mm are used, with an impact velocity 0.1 < U < 3.5 m/s. We introduce as relevant non-dimensional numbers the Weber number We = ρRU 2 /γ, that gives the relative importance of inertia compared to surface tension, and the Reynolds number defined here as Re = U R/ν. Accordingly, We varies between 40 and 1000, and Re between 8 and 11000. τ , the time to reach the maximum spreading radius, (t = 0 when the drop reaches the substrate), is a relevant parameter to probe the dynamics. U , R and τ were detected automatically via image analysis using python (Appendix B). For each set of parameters, three experiments are recorded and τ is averaged.

To confirm that multiscale surfaces ensure robustness towards the Cassie-to-Wenzel transition, a top view image of impact on SPHo surface B is reported in Fig. 1d). One can observe that the region where the transition might have occurred, characterised by small dark dots on the picture, is very limited. More interestingly, dynamical wetting properties seem to be very similar in the three cases considered. In Fig. 1c) and 1e) left an right, corresponding to spreading on surfaces C, A, and B respectively, a dynamic contact angle can be measured and is around 140 o ± 10 o in both cases, so fully in an hydrophobic regime. We can notice that in our range of impact parameters, lamella lift-off is never observed.

Analysis: scaling laws

The different regimes of drop impact are summarised in Fig. 2 in the Re-We phase diagram. In the frictionless case, two regimes can be identified, depending on We values. At low We, the inertial timescale to reach the pressure-gradient-free regime (τ i ∼ R/U ) is larger than the timescale of capillary recoil (τ c ∼ ρR 3 /γ) and a pancake regime is observed as described in [START_REF] Clanet | Maximal deformation of an impacting drop[END_REF], defined by the orange zone of Fig. 2.. At larger We, the rim dynamic limits the drop spreading and τ ∼ τ c = τ i We 1/2 [START_REF] Eggers | Drop dynamics after impact on a solid wall: Theory and simulations[END_REF][START_REF] Lastakowski | Bridging local to global dynamics of drop impact onto solid substrates[END_REF], whatever the impact velocity [START_REF] Biance | Drop fragmentation due to hole formation during Leidenfrost impact[END_REF]. This regime is defined by the yellow region in Fig. 2. We limit our study here to We > 40, avoiding pancake regimes and coupling between intrusion timescale and spreading [START_REF] Liu | Pancake bouncing on superhydrophobic surfaces[END_REF].

When friction comes into play, a viscous boundary layer builds up from the substrate. It has been demonstrated numerically [START_REF] Josserand | Droplet impact on a dry surface: triggering the splash with a small obstacle[END_REF][START_REF] Eggers | Drop dynamics after impact on a solid wall: Theory and simulations[END_REF]) and experimentally [START_REF] Roisman | Inertia dominated drop collisions. I. On the universal flow in the lamella[END_REF][START_REF] Lagubeau | Spreading dynamics of drop impacts[END_REF][START_REF] Lastakowski | Bridging local to global dynamics of drop impact onto solid substrates[END_REF]) that the spreading is stopped when the viscous boundary layer meets the upper interface of the thinning lamella. Since pressure-free drop thinning follows h t ∼ R 3 /U 2 t 2 (Lastakowski et al. 2014), whereas viscous boundary layer development reads h BL ∼ √ νt, the spreading dynamics is stopped for a critical lamella thickness satisfying

R 3 U 2 τ 2 v √ ντ v , (3.1)
thus at a viscous time τ v ∼ τ i Re 1/5 . From the comparison of τ v with τ c , the viscous regime is dominant over the capillary regime if We Re 2/5 [START_REF] Lagubeau | Spreading dynamics of drop impacts[END_REF]. This corresponds to the grey region of Fig. 2. We now consider the case of a slippery surface. Introducing the slip length b, the no-slip boundary condition is changed for bu z = u| z=0 , where u is the fluid velocity tangential to the surface and the subscript (z) denotes partial differentiation with respect to z. From the unsteady 1D momentum diffusion equation ∂ t u = ν∂ zz u, it can be shown that a self similar solution for u can be approximated by u(z, t, b) U f z+b √ πνt in the vicinity of the surface (z ∼ b), where U f is the fluid velocity far from it (Appendix C). Asymptotically, the effect of slippage is to shift by ∼ b the viscous boundary layer toward the surface, and τ s , the value of τ in the slippery case, is thus given by

R 3 U 2 τ 2 s √ ντ s -b. (3.2)
Assuming that the slippage has a weak influence on the dynamics, a perturbation of the viscous case readily leads to

τ s τ v (1 + 2α), with α = b 5 √ ντ v = b 5R Re 2/5 . (3.3)
Comparing τ s with τ c , a new critical transition line can be defined in the Re-We phase space when slippage effects appear. It results that the critical domain for which the capillary-driven regime is observed is extended in the viscous regime, and is given by We ∼ Re 2/5 (1 + 4α). This is shown in Fig. 2 where the classical boundary (no-slip) between capillary and viscous regimes is presented by a dotted line, while the new boundary in presence of a slip length b appears as a solid line. We focus then on a new regime that should be caused by a slip-delayed viscous boundary layer development.

Discussion

We now discuss our experimental results concerning the spreading time reported in Fig. 3. Deviations between the two cases (impact on smooth and SPHo substrates) are observed whatever the Reynolds number. When the spreading time is larger than the capillary time τ c , indicated in Fig. 3 by an orange band, spreading on SPHo surface is shorter than on smooth substrate, a surprising behaviour but which is not in our region of interest. On the contrary, when the spreading time is shorter than the capillary time τ c , the spreading time observed for impact on smooth surfaces A is always shorter than the one on slippery SHPo surfaces B. Even though significant, this increase could be attributed to rim shape or contact angle differences in both cases. To disentangle the effects of wetting and slippage, we performed similar experiments on surface C, which remains superhydrophobic but with a larger friction coefficient. In case of total friction, the limit between the capillary regime and viscous regime is determined by the dashed line (We = 8Re 2/5 ). In case of partial friction, the capillary regime is extended toward the grey region until the plain line following We = 8Re 2/5 (1+0.024Re 2/5 ), corresponding to b=59 µm. Data points correspond to experiments performed on smooth (A) and rough superhydrophobic surface (B). Color scales correspond to the value of τ (SPHo)/τ (Smooth). increases with Reynolds number. On the contrary surface C shows a spreading essentially identical to a classical no-slip wetting surface, with even a slightly shortened spreading, a behavior we can attribute to either contact line effects [START_REF] Laan | Maximum diameter of impacting liquid droplets[END_REF] or fragmentation [START_REF] Wachters | Heat Transfer From A Hot Wall To Impinging Water Drops In Spheroidal State[END_REF]. To show a quantitative agreement, a fit of our data with eq.3.3 is performed allowing us to define an average slip length b of the order of 59±8 µm. This value is fully consistent with the slip length obtained by Torque measurements (30-40 µm) for water on the same surfaces [START_REF] Lee | Underwater restoration and retention of gases on superhydrophobic surfaces for drag reduction[END_REF]. However, considering that our analysis is performed for a 2D stationary flow, that the slip length varies with the liquid viscosity and nature (here water-glycerol mixtures instead of water), the applied pressure [START_REF] Lee | Underwater restoration and retention of gases on superhydrophobic surfaces for drag reduction[END_REF]) (here dynamical pressure during impact), as well as a partial transient impalement on the microstructure, the quantitative comparison should be taken cautiously.

To define more generally the range of parameters for which these effects are relevant, we report experimental observations on the Re-We diagram defined above in Fig. 2. The colour bar corresponds to the experimental values of the ratio τ (SPHo)/τ (Smooth). One can indeed observe an increase of this ratio in the viscous regime. A new frontier between the capillary and slippery regimes can be defined by balancing τ s and τ c (Fig. 2).

Conclusion

We propose here a semi-analytical analysis that accounts for drag reduction on viscous impact. A phase diagram is built to better identify the physical parameters for which these effects on the overall dynamics are relevant. It shows the consequences of a weak friction on the spreading dynamics, which can be crucial in many situations. To test it, impacts are performed on two types of multiscale SHPo surfaces, that both resist the Cassie-Baxter-to-Wenzel transition and that have similar wetting properties but different friction characteristics. Longer spreading is indeed observed for the lower friction.

Taking into account this effect would be particularly relevant for the design of efficient cooling or anti-icing systems built upon such substrates, where increasing heat transfer through extended contact time and on an extended surface is of crucial interest. On a more fundamental prospect, we propose here an experimental demonstration and a simple analysis of the effect of slippage on the build-up of a viscous boundary layer in a non-stationary flow, a complex mathematical problem [START_REF] Matthews | A note on the boundary layer equations with linear slip boundary condition[END_REF][START_REF] Gie | Boundary layer analysis of the navier-stokes equations with generalized navier boundary conditions[END_REF], still investigated recently [START_REF] Fujioka | Letter: New boundary layer structures due to strong wall slippage[END_REF]. each experiment at the spreading radius and spreading time respectively. Finally, the spreading time uncertainty is 0.17 ms.

The effect of slippage can be captured by estimating z(b, t, k → 1) -z(0, t, k → 1). When t → ∞, this value is converging to b. The boundary layer is shifted up by a length b.

C.2. Consequences on droplet dynamics upon impact: timescales

We consider a drop impacting on a solid substrate.We distinguish here the three cases of drop impact without friction (i.e. Leidenfrost case), with large friction (smooth solid substrate), and with slippage (superhydrophobic substrate). For each regime, we identify the characteristic time to reach the maximal radius and these times will be compared to identify which regime is predominant as a function of the Weber We = ρR 3 γ and Reynolds Re = U R ν numbers.

C.2.1. Frictionless case

In this regime, two cases can be determined depending only on the Weber number. A deceleration phase over a time τ i ∼ R/U [START_REF] Clanet | Maximal deformation of an impacting drop[END_REF]) is followed by a capillary oscillation over τ c ∼ ρR 3 γ . If τ c is smaller than τ i , only the deceleration is observed and we reach a pancake regime whereas on the contrary a capillary recoil [START_REF] Roisman | Inertia dominated drop collisions. I. On the universal flow in the lamella[END_REF][START_REF] Eggers | Drop dynamics after impact on a solid wall: Theory and simulations[END_REF][START_REF] Lastakowski | Bridging local to global dynamics of drop impact onto solid substrates[END_REF]) is observed. The transition is given by τ c = τ i , which implies a critical Weber number We 1 . In the following analysis, we assume that We is always above We 1 .

C.2.2. Large friction

In this case, it has been shown that the development of a boundary layer sets the maximal attained radius, i.e. when the droplet thickness is comparable to this developed boundary layer, the droplet stops spreading and thinning. For We larger than We 1 , this criterion reads [START_REF] Roisman | Inertia dominated drop collisions. I. On the universal flow in the lamella[END_REF][START_REF] Eggers | Drop dynamics after impact on a solid wall: Theory and simulations[END_REF][START_REF] Lastakowski | Bridging local to global dynamics of drop impact onto solid substrates[END_REF])

R 3 U 2 (τ v ) 2 k ν(τ v ) (C 6)
and then, the spreading viscous time scales as τ v ∼ τ i Re 1/5 . This regime is relevant if this time is shorter than τ c , i.e. We Re 2/5 .

C.2.3. Slippage

Following previous asymptotic analysis, the characteristic time τ s to develop the boundary layer over the distance h satisfies:

R 3 U 2 τ 2 s k √ ντ s -b. (C 7)
Let's now assume that the effect of slippage is small compared to viscous effects. The characteristic viscous slippery time τ s can then be written

τ s = τ v + t s (C 8)
with t s = o(τ v ) is the slippage contribution. Eq. (C 7) can then be developed as

- R 3 U 2 τ 2 v 2t s τ v k √ ντ v t s 2τ v -b, (C 9) τ s τ v 1 + 2b 5R Re 2/5 . (C 10)
The capillary region is then extended to We < ∼ Re 2/5 1 + 2b 5R Re 2/5 (Fig. 2).

Figure 1 .

 1 Figure 1. Scanning electron microscopy images of : (a) the textured SHPo surfaces B, (b) the disordered SHPo surface C. (c) Snapshot of a 80% mixture of water and glycerol drop impact on surface B (U = 1.5 m/s, time interval 2 ms). (d) Top view of a water droplet impact on surface B (U =1.5 m/s). e) Snapshots of the impact at maximal spreading for impact on surface A (left) and B (right). Scale bar corresponds to 1 mm. (U =2.6 m/s, η=21 cS).

Figure 2 .

 2 Figure2. Schematic of the drop impact regimes in the We-Re phase diagram. Grey, yellow and orange areas correspond respectively to the viscous, capillary and inertial regimes (see text). In case of total friction, the limit between the capillary regime and viscous regime is determined by the dashed line (We = 8Re 2/5 ). In case of partial friction, the capillary regime is extended toward the grey region until the plain line following We = 8Re 2/5 (1+0.024Re 2/5 ), corresponding to b=59 µm. Data points correspond to experiments performed on smooth (A) and rough superhydrophobic surface (B). Color scales correspond to the value of τ (SPHo)/τ (Smooth).

Figure 3 .

 3 Figure 3. Time τ to reach the maximal radius versus the Reynolds number on smooth (A, open symbols) and on SHPo (B, plain symbols) substrates with ν=505 cS (triangle), ν=103 cS (diamond), ν=40.5 cS (circle), ν=21 cS (square), ν=9.4 cS (hexagon), ν=4.3 cS (star), ν=2.8 cS (cross) and ν=1.9 cS (pentagon). Plain lines correspond to τ = (R/U )Re 1/5 and horizontal band corresponds to the interval containing τ = 0.41 ρR 3 /γ.

Figure 4 .

 4 Figure 4. Time τ (SHPo) to reach the maximal radius on slippery surfaces normalised by the time τ (Smooth) to reach the maximal radius on smooth surface. Blue plain symbols: ordered rough surface B; Open red symbols: disordered rough surface C. For ordered rough surfaces, viscosities are ν=505 cS (triangle), ν=103 cS (diamond), ν=40.5 cS (circle) and ν=21 cS (square). For disorder rough surface, viscosity used are ν 250 cS (triangle), ν=67 cS (cross) and ν=27 cS (star). Plain line corresponds to adjustment by eq.3.3 with b = 59 µm.

Figure 5 .

 5 Figure 5. Analysed images from a video of a spreading droplet of a 70% glycerol/water mixture on C. (a) and (c), images with substracted background; (b) and (d), binary images.

Table 1 .

 1 The three types of surface used and their characteristics.

	Name	Type	Contact angle	slip length b
	A B C	Smooth Multiscale ordered Multiscale disordered	30 • 160 • 160 •	0 µm 35 ± 5 µm (Lee & Kim 2011) 8 ± 1 µm

† Email address for correspondence: anne-laure.biance@univ-lyon1.fr

We thank H. Lastakowski for preliminary experiments. G. Martouzet acknowledges ENS Paris-Saclay for funding, and this work is supported by the Basic Research Program (2017R1A2B4008028) through the National Research Foundation of Korea funded by the Ministry of Science and ICT. This work was sponsored by the French National Agency for Research (Freeflow project ref. ANR-11-B504-001-01). The authors report no conflict of interest.

Appendix A. Surface fabrication and characterization

Surface B is a well structured surface with multiscale pillars. First, micro-sized cylindrical pillars were obtained using photo-lithography, followed by deep reactive ion etching on silicon. Pillars are arranged in a square lattice with a fixed structural pitch L (centre-to-centre distance of two adjacent pillars) of 50 µm. The top pillar area fractions φ s = πD 2 /(4L 2 ), with D the pillar diameter, is set to 0.1. Then, over micro-scale pillars, nano-scale pillars with a structural pitch of 0.5-1 µm and very sharp tips were generated using a self-masking nanofabrication method known as a black silicon method [START_REF] Sun | Fabrication of micro/nano dual-scale structures by improved deep reactive ion etching[END_REF][START_REF] Lee | Underwater restoration and retention of gases on superhydrophobic surfaces for drag reduction[END_REF]. The surface is made hydrophobic with a few nanometer thick Teflon layer coating. Using torque measurements, the slip length of water on this surface is found to be around 35 ± 5 µm.

Surface C is a randomly textured surface made of CuO obtained by oxidation of a copper sample in a mixture of NaCl0 2 , NaOH and Na 3 PO 4 . It is made hydrophobic by silane vapour deposition. Images of the surface are reported in Fig. 1. Slip length from water Poiseuille flow rate measurements in a microfluidic channel is found to be 8 ± 1 µm. Note that even if slip length measurement methods are different for the two surfaces, they both involve a relatively low liquid pressure and a laminar flow condition, they can be reasonably compared [START_REF] Lee | Universal rescaling of drop impact on smooth and rough surfaces[END_REF].

Static angles on both surfaces were found to be similar and above 160 • . Dynamical contact angle during the spreading are also similar in the three cases of smooth substrate (A) and superhydrophobic surfaces (B,C), larger than 160 • .

Appendix B. Spreading time detection and accuracy

Typical snapshots of impact recording are reported in Fig. 1. The droplet contour and position is determined by substracting the background and thresholding (Fig. 5). Results are hardly dependent on the choice of the threshold (1 pixel). The contact time origin is automatically determined by looking at the presence of a bright pixel below the drop. The droplet maximum radius is determined by fitting the evolution of the droplet radius versus time and fitting the curve in the vicinity of the maximum by a second order polynomial. The maximum of the function and the time to reach this maximum correspond then for Navier condition

C.1. 2D analysis-Infinite surface

We consider the motion of a liquid near a solid infinite plane located at z = 0. The liquid and the plate are initially at rest and then the plate is moved at a velocity U f , in the x direction. We investigate the development of a viscous boundary layer in the z direction taking into account this effect. The momentum diffusion equation locally reads:

where ν is the kinematic viscosity of the liquid. Introducing ξ = z/ √ νt and the two constants A and B, the solution of eq.C 1 is:

Boundary conditions are that (i) the fluid is at rest far from the surface (lim u x (x → +∞) = 0), (ii) in the vicinity of the surface (z = 0), a slip condition defined by a slip length b, (b ∂ux ∂z | z=0 = u x | z=0 -U f ). Then, the velocity profile reads:

where we recover its usual form for b = 0. The deviation due to slippage is then

This term is negligible when t b 2 /ν. As the boundary layer develops from the substrate, it reaches the altitude z after a characteristic time z 2 /ν, the effect of slippage will be important if b z. We define the effect of slip on the altitude at which viscous friction is felt versus time. Considering that the viscous friction is felt when u x (z, t)/U f = k, with k comprised between 0 and 1, the time-dependent z at which viscous effects are important reads

).

(C 5)