Sylvie Le Hégarat-Mascle

Emanuel Aldea
email: emanuel.aldea@u-psud.fr

Spectral analysis of masked signals in the context of image inpainting

Keywords: Spectral analysis, Fourier domain, image inpainting

This paper proposes a computationally efficient algorithm for evaluating a sum of squared differences in the image domain in the presence of arbitrary mask configurations. Among the many potential applications of this algorithm, we consider for illustration an image inpainting task. The results show that on a diverse sample of hundreds of simulated holes in the tested images, the proposed technique is more effective than the baseline normalized cross-correlation, even when the masks are properly dealt with by the baseline.

Introduction

In image processing, a lot of tasks have been performed in the Fourier domain, given the capability of the Discrete Fourier Transform (DFT) to provide a discriminative spectral representation of a uniformly spaced discrete signal. Computationally wise, the DFT has been appealing as well due to the Fast Fourier Transform algorithm which allowed for addressing a wide range of applications in a practical manner. Specifically, image registration is a particularly suited task, since Fourier domain representations allow for recovering the translation offset, as well as for inferring the relative rotation and scale up to some extent [START_REF] Sheng | Experiments on pattern recognition using invariant fourier-mellin descriptors[END_REF][START_REF] Reddy | An fft-based technique for translation, rotation, and scale-invariant image registration[END_REF][START_REF] Pan | An adaptable-multilayer fractional fourier transform approach for image registration[END_REF][START_REF] Konstantinidis | Phase amplified correlation for improved sub-pixel motion estimation[END_REF]. The classification task may also be addressed with great success, as the spectra of images with rich content featuring real-world environments often contain diverse frequencies that may be exploited for robust discrimination [START_REF] Chan | Fabric defect detection by fourier analysis[END_REF][START_REF] Tsai | Defect detection of solar cells in electroluminescence images using fourier image reconstruction[END_REF][START_REF] Kliangsuwan | Fft features and hierarchical classification algorithms for cloud images[END_REF].

One important but often irrelevant drawback of Fourier domain analysis is the requirement for the uniform sampling of the signal domain (in our case, the image domain). Although this does not raise particular concerns for most applications due to the widespread use of standard image sensors which sample uniformly the surrounding environment, in some cases either the support for the signal of interest might not satisfy the above constraint, or it is otherwise detrimental to process the entire domain indiscriminately. The most common situations are when 1) only some categories of objects in a scene should be considered for Fourier analysis and/or some categories of objects should be entirely ignored, 2) some context-specific masks underline the sub-part of interest in the field of view of the sensor, and 3) a part of multiple disjoint parts of the object of interest are visible. In all these circumstances, the absent area or the area containing irrelevant data would in fact contribute as well to the spectrum of the signal, and thus bias its representation and directly impact the subsequent analysis.

The community proposed various strategies for accounting for masked areas in a manner consistent with the Fourier transform, however these algorithms usually fail to preserve faithfully the spectrum of the regions of interest [START_REF] Thevenaz | A pyramid approach to subpixel registration based on intensity[END_REF][START_REF] Kaneko | Using selective correlation coefficient for robust image registration[END_REF][START_REF] Orchard | Efficient least squares multimodal registration with a globally exhaustive alignment search[END_REF]. In contrast, Padfield [START_REF] Padfield | Masked object registration in the fourier domain[END_REF] introduced an algorithm which directly and explicitly integrates the masking into the FFT algorithm steps. A normalized crossedcorrelation (NCC) is computed in the Fourier domain, which is thereafter used for solving image registration. We propose to generalize this idea by extending the initial approach and demonstrate that 1) it may be used to compute other similarity criteria related to norms and 2) it may be applied to other tasks beyond registration.

Use of the masked Fourier transform

To measure the consistency between two areas of two different images, several criteria have been proposed, among them the NCC and the sum of squared differences (SSD). Both are computed on image subparts defined as spatial domains. To be able to compute a map giving the value of the considered consistency measure in one pass (in contrast with using a sliding window), we aim at formulating the consistency measure based on the Fast Fourier Transforms (FFT). Besides, we assume that the considered measure has to be evaluated on a domain which is not necessarily rectangular.

Let us consider f 1 and f 2 as the two considered images, with f 2 being the moving image. Let us denote by D 1 the f 1 2D domain and by D 2 (u, v) the f 2 translated/shifted by the 2D vector of coordinates (u, v). Then, D u,v = D 1 ∩ D 2 (u, v) denote the intersection (or overlap) between the two domains, so that the sums involved in the consistency measures are computed on D u,v .

In [START_REF] Padfield | Masked object registration in the fourier domain[END_REF], Padfield expressed the sums (x,y)∈Du,v 1, (x,y)∈Du,v f 1 (x, y) and (x,y)∈Du,v f 1 (x, y) 2 (and the corresponding expressions for f 2 ) after introducing the masks m 1 and m 2 corresponding to the indicator functions of the D 1 and D 2 domains, and their Fourier transforms

M i = F(m i ), i ∈ {1, 2}: |D u,v | = (x,y)∈Du,v 1 = F -1 (M 1 • M * 2 )(u, v), (1)
f 1 = 1 |D u,v | (x,y)∈Du,v f 1 (x, y) = F -1 (F 1 • M * 2 )(u, v) F -1 (M 1 • M * 2 )(u, v) , (2)
(x,y)∈Du,v f 1 (x, y) 2 = F -1 (F(f 1 • f 1 ) • M * 2 )(u, v), (3)
with

F i = F(f i ), i ∈ {1, 2}
, and all the multiplications in these equations being elementwise (Hadamard product), and M * 2 denotes the complex conjuguate of the Fourier transform of M 2 (that is also the Fourier transform of the transposed of m 2 ).

Average of Squared Differences with masks

In this work, we derive the formulation of the Average of Squared Differences either centered or not centered, using the Fourier transform, i.e. without resorting to a sliding window. Indeed, conversely to the NCC measure, the SSD has been very popular due to its convenient summation properties, which are valuable for considering multichannel images such as color ones. Note that in the standard case, this measure is not normalized since the support has a set size and the normalization would be useless. In our case however, we will explicitly focus on the normalized (i.e., averaged with respect to the number of valid pixels) version since the masks will modify the support size.

Using the previous notations, we express the sum of the squared differences as follows:

SSD(u, v) = (x,y)∈Du,v f 1 (x, y) -f 1 -f 2 (x -u, y -v) -f 2,u,v 2 (4) = (x,y)∈Du,v f 1 (x, y) 2 + (x,y)∈Du,v f 2 (x -u, y -v) 2 (5)
-2

(x,y)∈Du,v f 1 (x, y)f 2 (x -u, y -v) -|D u,v | f 1 -f 2,u,v 2 = F -1 (F(f 1 • f 1 ) • M * 2 )) + F -1 (M 1 • F(f ′ 2 • f ′ 2 ))- (6)
2F -1 (F 1 • F * 2 ) -F -1 (F 1 • M * 2 ) -F -1 (F 2 • M * 2 ) 2 (u, v),
with f ′ 2 being the f 2 image flipped (central symmetry). The previous equation stands for the SSD with centered differences. In some cases, it might be advisable to consider the uncentered differences, and this version of the SSD is even simpler to express:

SSD(f 1 , f 2 ) = F -1 (F(f 1 • f 1 ) • M * 2 )) + F -1 (M 1 • F(f ′ 2 • f ′ 2 )) -2F -1 (F 1 • F * 2 ). (7)
Note that the computation of the uncentered SSD involves two less inverse Fourier transforms than that of the centered SSD. Then, the map of the average of squared differences (ASD) is provided by dividing pixel per pixel SSD map values (Eq. ( 6) or ( 7)) by

F -1 (M 1 • M * 2 ) map values: ASD(f 1 , f 2 ) = SSD(f 1 , f 2 ) F -1 (M 1 • M * 2 ) . ( 8
)

Normalized Cross Correlation with masks

In [START_REF] Padfield | Masked object registration in the fourier domain[END_REF], following this reformulation of the intermediate terms which accounts for the impact of the masks m 1 and m 2 on the Fourier transform, the final evaluation of the NCC becomes:

N CC(f 1 , f 2 ) = F -1 (F1•F * 2 )- F -1 (F 1 •M * 2 )•F -1 (M 1 •F * 2 ) F -1 (M 1 •M * 2 ) F -1 (F (f1•f1)•M * 2 )- (F -1 (F 1 •M * 2 )) 2 F -1 (M 1 •M * 2 ) F -1 (M1•F (f ′ 2 •f ′ 2 ))- (F -1 (M 1 •F * 2 )) 2 F -1 (M 1 •M * 2 ) (9)
3 Inpainting with the masked Fourier transform

Inpainting is a radical form of image restoration in which pixels inside a missing region of the image are filled with information provided by the surrounding areas, and potentially by the entire domain. The concept of self-similarity is central to this task, since the main underlying assumption is that repeating patterns from other parts of the image will be relevant for the missing area, and will thus be imported there as well. In the spatial domain, the inpainting task has been traditionally addressed, as an ill-posed problem, by total variation regularization, by dictionary based approaches [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (mca)[END_REF][START_REF] Fadili | Inpainting and zooming using sparse representations[END_REF], by diffusion with partial differential equations [START_REF] Masnou | Disocclusion: a variational approach using level lines[END_REF][START_REF] Tschumperlé | Vector-valued image regularization with pdes: A common framework for different applications[END_REF] or by some hybrid strategy [START_REF] Criminisi | Region filling and object removal by exemplarbased image inpainting[END_REF][START_REF] Bugeau | A comprehensive framework for image inpainting[END_REF]. Compared to these commonly encountered approaches, methods which use explicitly the Fourier transform of the deteriorated input image are less common. Some works focus on some specific cases of inpainting which are particularly suited for spectral analysis (e.g. removal of text overlay [START_REF] Sridevi | Image inpainting based on fractional-order nonlinear diffusion for image reconstruction[END_REF]). In [START_REF] Wang | Exemplar-based image inpainting using structure consistent patch matching[END_REF], the FFT is used as an accelerating step for patch matching in exemplar based image inpainting, while in [START_REF] Mousavi | A new algorithm for image inpainting in fourier transform domain[END_REF] the inpainting task is performed in the Fourier space of the image representation (i.e. some coefficients are assumed to be missing). In [START_REF] Thai | Simultaneous inpainting and denoising by directional global three-part decomposition: connecting variational and fourier domain-based image processing[END_REF], the image is decomposed in decomposed in texture and non-texture components, then texture inpainting and denoising is performed in a subsequent step. In [START_REF] Hasegawa | Image inpainting on the basis of spectral structure from 2-d nonharmonic analysis[END_REF], the authors employ an alternative to the FFT, namely nonharmonic analysis, in order to minimize the impact of the side lobes appearing on truncated data. None of these works try to cope explicitly with the impact of the mask on the extracted frequencies, nor they propose a fully spectral, computationally efficient approach.

In this work, we propose to take advantage of the fact that we are able to compute a consistency criterion only considering the available pixels (i.e., fully disregarding the missing ones) thanks to the use of Fourier transform based expressions introduced in Section 2. Based on these latter, we are then able to propose an efficient strategy for solving the inpainting problem in which the mask represents the area to be filled. The proposed algorithm searches, for each area including missing pixels, another area which presents similar structures and colors than the ones near the missing pixels. This search can be performed globally or locally depending on the assumptions. We then expect such an approach to be all the more performing that the image presents repetitive structures or textures.

The global outline is given by Algorithm 1. In the considered algorithm, as well as in the conducted experiments in Section 4, we consider RGB images. However, it is straightforward to apply the proposed approach in another color space, or even to multispectral (or hyperspectral) images. The size l of the search f1 ← rectangular tile centered on (x, y) and having side length equal to l 7:

Cut a box of side length d at the center of f1 8:

n ← ⌈log2(d)⌉ 9:

f2 ← rectangular tile centered on (x, y) and having side length equal to 2 n 10:

m1 ← binary mask of f1 valid pixels 11:

M1 ← FFT(m1) 12:

m2 ← binary mask of f2 valid pixels 13:

M2 ← FFT(m2) 14:

M12 ← IFFT(M1 • M * 2 ) 15:
Initialize map S to 0 in every pixel 16:

for each channel k of f1 do 17:

for i ∈ 1, 2 do 18:

f i,k ← channel k of fi 19: Fi ← FFT(f i,k ) 20: f 2 i,k ← f i,k • f i,k 21: Gi ← FFT(f 2 i,k ) 22:
end for 23:

F12 ← IFFT(F1 • F * 2 ) 24: Ȟ12 ← IFFT(G1 • M * 2 ) 25: Ȟ21 ← IFFT(M1 • G * 2 ) 26: if bu then 27: J12 ← IFFT(F1 • M * 2 ) 28: J21 ← IFFT(M1 • F * 2 ) 29: end if 30: S ← S + Ȟ12 + Ȟ21 -2 F12 + bu J12 -J21 M12 31: end for 32: S ← S M12 33: (û, v) ← arg min (u,v) S(u, v) 34:
Fill the missing values in Ĩ by pasting the values of I around (x + û, y + v) 35: end for area is an input parameter left to the user. Note that using a global search, the Fourier transforms associated to the search area (the whole image in this case) can be computed once at the beginning of the algorithm. However, in this case, we lose the support of a locality constraint. Note also that, to benefit from fast Fourier transform algorithms, l has to be a power of 2. In the extraction of f 1 and f 2 the original image can be padded with 0 (masked pixels) if necessary. Finally, since the patch that is researched in the image (around the area to fill) is extracted from the image itself, it is necessary to mask the pixels of the patch (to avoid to reselect the original patch location), which is simply done by "cutting" a box of patch size in original image (and filling it with black pixels).

Experimental results

We performed experiments related to the inpainting task on images selected from the publicly available DAFNE challenge dataset [START_REF] Dondi | Dafne: A dataset of fresco fragments for digital anastlylosis[END_REF]. As previously stated, the proposed algorithm is generally applicable to grayscale or color images, but the samples which are present in DAFNE offer a good diversity in terms of appearance and style of the content, with a good balance between repeating patterns and more singular structures. The hole creation process is performed by randomly and uniformly removing disk patches of content from the selected images. Evaluation metrics Two widely used metrics are considered for the numerical evaluation. First, the Root Mean Squared Error (RMSE) is considered, then we also compute a metric which is more specific for benchmarking image reconstruction tasks, namely the peak signal-to-noise ration (PSNR). Although PSNR is partly related to the RMSE, it highlights better the method performance independently of the numerical range of the studied signal values. Algorithm variants Based on the general idea for inpainting with the masked Fourier transform, we consider three variants depending on the similarity measure which is employed in searching for the most visually close data: (1) the normalized cross correlation (NCC, cf. Eq. ( 9)) computed on the intensity image, (2) a weighted sum of the three intensity-based criteria that may be potentially used: NCC, centered ASD (cf. Eq. ( 6) and Eq. ( 8)) and uncentered ASD (cf. Eq. ( 7) and Eq. ( 8)) called uASD, (3) the uASD computed on the three color channels (benefiting from good summation property), called 3D uASD.

Tables 1 and2 show the obtained results in terms of evaluation metric statistics computed on 100 holes per fresco. We clearly see that the NCC is not sufficient to find a good image patch to fill a given hole, mainly since it is "only' based on intensity relative variations. Adding the two criteria based on squared differences significantly improves the results (between 2 and 8 dB depending on the considered fresco) while significantly reducing the standard deviation. Finally, the benefit of color information is also visible, varying however with the content of the fresco: the Lanzani fresco presents rather homogeneous colors (cf. Figure 1, second line) whereas the Dellafrancesca fresco has a variety of colors (cf. Figure 1, third line) allowing for a more significant improvement of performance. Finally, let us have a qualitative look at the obtained results. Figure 1 shows the whole frescoes with simulated holes and the inpainting results from 3D uASD, while Figure 2 shows some selected subareas. Indeed, Figure 1 allows us to check quickly that holes have been filled with consistent values since at first glance it is difficult to see the location of the simulated holes (without the help of the left image) whereas Figure 2 allows us to evaluate the visual quality of the local reconstruction (fresco subareas of size 128 × 128 pixels), as well as to point out some remaining imperfections. Specifically, in Figure 2, for each line considering details from a different fresco, left side shows a subarea that is rather well reconstructed (it is difficult to guess where the holes were even focusing on this subarea), while the right side shows a subarea with at least one partially flawed reconstruction: one hole filled with an incorrect shade of white (first line), a discontinuity in the leg of the soldier (second line) and appearing inconsistencies in the drapery of tunics (third line). However, all these imperfections are explainable and we hope to reduce them by searching in future works a patch candidate for the filling not only in translation but also in rotation.

Conclusion

In this work, we extended the strategy initially proposed by [START_REF] Padfield | Masked object registration in the fourier domain[END_REF] beyond NCC, to more consistency measures between masked image subparts. We have also addressed with success a novel task, namely inpainting, which has not been considered until now with these techniques, despite its suitability with the method assumptions. For future work, we intend to fully exploit the other properties of the Fourier transform (e.g. following Fourier-Mellin approach) which would allow us to perform more complex rotation and scaling invariant queries, and to characterize more in detail the computational advantage of the proposed algorithm.

Algorithm 1

 1 Inpainting by research of similar areas; input: RGB image I, search area side length l, boolean b u indicating if ASD is centered; output: RGB image Ĩ. FFT stands for the Fast Fourier Transform and IFFT for the Inverse Fast Fourier Transform. 1: L ← list of I areas with missing pixels 2: Initialize Ĩ to I 3: for each element Aj of L do 4: (x, y) ← Aj center coordinates 5: d ← Aj min square bounding box side length 6:

Fig. 1 .

 1 Fig. 1. Reconstructed frescoes: original image with simulated holes (left) and inpainting result (right); first line: Tiepolo fresco ("The Institution of the Rosary"), second line: Lanzani fresco ("Sant Antonio protegge Pavia"), and third line: Della Francesca fresco ("Exaltation of the Cross").

Fig. 2 .

 2 Fig. 2. Details of the reconstructed frescoes: simulated holes (left) and inpainting result (right); first line: Tiepolo fresco, second line: Lanzani fresco, and third line: Della Francesca fresco.

Table 1 .

 1 RMSE statistics obtained on three frescoes for three algorithms based on different consistency measures; statistics (mean, median and standard deviation after ± symbol) are derived from 100 simulated holes. Best results are highlighted in green.

	Algorithm	Tiepolo mean median	Lanzani mean median	DellaFrancesca mean median
	NCC	35.64	31.76	36.32	33.92	35.85	30.01
		±27.67		±17.15		±21.73
	NCC+ASD	25.86	20.56	29.09	28.14	29.74	27.28
	+uASD	±19.20		±11.52		±14.41
	3D uASD	24.74	20.13	28.69	26.99	27.40	25.28
		±19.02		±12.03		±13.81

Table 2 .

 2 PSNR statistics obtained on three frescoes for three algorithms based on different consistency measures; statistics (mean, median and standard deviation after ± symbol) are derived from 100 simulated holes. Best results are highlighted in green.

	Algorithm	Tiepolo mean median	Lanzani mean median	DellaFrancesca mean median
	NCC	47.61	41.66	41.58	40.35	42.76	42.79
		±21.87		±11.92		±13.14
	NCC+ASD	52.85	50.36	45.41	44.08	45.79	44.70
	+uASD	±19.79		±10.66		±12.16
	3D ASD	54.0	50.78	45.88	44.91	47.66	46.23
		±19.92		±11.01		±12.56


Noriko Hirata-Kohno

Makoto Kawashima

Anthony Poëls

Yukiko Washio

S-unit equation in two variables and Padé approximations

Keywords: S-unit equation, Padé approximation, Binomial function, Hypergeometric function

.

Introduction and main result

The main purpose of the present article is to use explicit construction of Padé approximations, derived from those constructed in [START_REF] Kawashima | Padé approximations for shifted functions and parametric geometry of numbers[END_REF] for binomial functions, in order to improve the upper bound for the number of the solutions of the S-unit equation in two variables.

Let K be a number field with degree m := [K : Q] < ∞ and fix non-zero elements λ, µ ∈ K. Consider a finite set S of places of K, which contains all the archimedean ones. Denote by s its cardinality and by U S the set of the S-units in K. It is known that U S is a finitely generated abelian group of rank s -1.

We deal with the S-unit equation λx + µy = 1 in unknowns x, y ∈ U S .

(1)

The finiteness of the number of solutions of Eq. (1) dates back to C. L. Siegel [START_REF] Siegel | Approximation algebraischer Zahlen[END_REF], who implicitly reduced the problem to a system consisting of a finite number of Thue equations.

In 1984, J. -H. Evertse proved that the equation λx + µy = 1 has at most 3 × 7 m+2s solutions [12, Theorem 1]. His proof relies on quantitative arguments on Diophantine inequalities of Roth type, with certain Padé approximations. Up to now this result has been the best one.

In a more general setting, F. Beukers and H. P. Schlickewei [START_REF] Beukers | The equation x + y = 1 in finitely generated groups[END_REF] considered a finitely generated subgroup Γ of rank r in (K \ {0}) 2 to show that the equation x + y = 1 in (x, y) ∈ Γ has at most 256 r+1 solutions. They used Padé approximations for hypergeometric functions, whereas E. Bombieri and W. Gubler [6, Chap. 5] followed a different approach in a geometrical framework (see also [START_REF] Bombieri | Diophantine Equations in Low Dimensions[END_REF] [START_REF] Evertse | Effective Results and Methods for Diophantine Equations over Finitely Generated Domains[END_REF]).

It is known that the theory of the linear forms in logarithms gives effective bounds for the height of the solutions in the two variables case (confer [START_REF] Evertse | Effective Results and Methods for Diophantine Equations over Finitely Generated Domains[END_REF]). Nevertheless, in this paper, we restrict ourselves to counting the solutions of the S-unit equation.

We prove the following statement.

Theorem 1.1. Let λ, µ ∈ K \ {0}. Then the equation Eq. (1) has at most

(3.1 + 5 (3.4) m ) • 45 s (2) solutions (x, y) ∈ U 2 S .
Mathematics Subject Classification (2020): 11D45 (primary); 11J68, 11J61, 33C05 (secondary).

3 • 7 m • 49 s .
Note that a more precise bound is shown at the page 583, line -8 of [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]:

2 + 5 • (3.26396) m • 49 s ≤ 2 + 5 • 2e 24/49 m • 49 s . ( 4
)
Our numerical bound Eq. ( 3) is better than Eq. ( 4) for any positive integers m, s.

The structure of our proof is similar to that of Evertse, however, we obtain significant improvement, especially for the estimates coming from the Padé approximation part. See Lemma 6.3 which replaces [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]Lemma 8]. For that purpose, starting with constructions from [16, §9], we first establish new explicit identities for Padé approximants related to binomial functions (see Sections 2 and 3). In Section 4, by using an argument relying on Mahler measure we are able to estimate precisely the coefficients of the polynomials involved, both in the arichimedean and the non-arichimedean cases. This leads us in Section 5 to several new inequalities needed for Lemma 6.3 (the main result of Section 6). In the proof of this lemma, we use the local height as much as possible instead of rough upper bounds, which allows us to refine inequalities in [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]. Section 7 is devoted to some numerical computation for a certain choice of parameters, which leads us to the final estimate Eq. (3).

Padé approximants for a cubic binomial function

Throughout the article, we denote by N the set of non-negative rational integers and by Z ≤0 the set of non-positive rational integers.

Given ℓ ∈ N, we write (1/z ℓ ) the ideal of the ring of Laurent series Q[[1/z]], generated by 1/z ℓ .

Consider rational parameters a, b, c ∈ Q with c / ∈ Z ≤0 . Let us recall Gauss' hypergeometric function 2 F 1 defined by

2 F 1 a, b c z = ∞ k=0 (a) k (b) k (c) k • z k k!
where (a) k = Γ(a + k)/Γ(a) is the k-th Pochhammer symbol. The radius of convergence is 1, unless a ∈ Z ≤0 or b ∈ Z ≤0 , in which case the function is a polynomial in z. It is known that we can extend the domain of definition of 2 F 1 by analytic continuation (confer [START_REF] Beukers | Gauss' hypergeometric function[END_REF] [11] for details).

A cubic binomial function is a special case of Gauss' hypergeometric function. It belongs to the class of functions studied in [START_REF] Kawashima | Padé approximations for shifted functions and parametric geometry of numbers[END_REF], for which we construct explicit Padé approximants.

The next result can be deduced directly from Proposition 3.5 and Corollary 3.6 of [START_REF] Kawashima | Padé approximations for shifted functions and parametric geometry of numbers[END_REF]. Note that it is in general difficult to explicitly construct Padé approximants (P 1 (z), P 2 (z)), whereas the existence is obvious.

Lemma 2.1. Consider the cubic binomial function given by the Laurent series

f (z) = 1 z 1 - 1 z 1/3 = ∞ k=0 (-1/3) k k! 1 z k+1 ∈ 1/z • Q[[1/z]].
For each n, define the polynomials P n,0 (z),

P n,1 (z) ∈ Q[z] by P n,0 (z) = n k=0 (-1) n-k n + k -1 k n -4/3 n -k z k , P n,1 (z) = n-1 k=0 (-1) n-1-k n + k k n + 1/3 n -1 -k z k
(with the convention that P 0,1 (z) = 0), and set

R n (z) = P n,0 (z)f (z) -P n,1 (z) . (5)
Then the pair of polynomials (P n,0 , P n,1 ) forms Padé approximants, and R n (z) is a Padé approximation of f (z), namely,

R n (z) ∈ 1/z n+1 • Q[[1/z]].
Explicitly, we have

R n (z) = ∞ k=n k n (-1/3) k (4/3) n (n + k)! 1 z k+1 = (4/3) n (-1/3) n (2n)!z n+1 • 2 F 1 n + 1, n -1/3 2n + 1 1 z ∈ (1/z n+1 ) . (6)
Note that our approximants (P n,0 , P n,1 ) are the same as those due to M. A. Bennett in [2, Eq. ( 5) and Eq. ( 6)] (with the parameters a = -1 and ℓ = 1), although the formulaes are different.

Next we establish several identities for the above approximants.

Proposition 2.2. Define the polynomials

A n (1 -z) = z n-1 P n,1 (1/z) and B n (1 -z) = z n P n,0 (1/z).
Then, we have

A n (1 -z) = n-1 ℓ=0 n + 1/3 ℓ n -4/3 n -1 -ℓ (1 -z) ℓ , B n (1 -z) = n ℓ=0 n -4/3 ℓ n + 1/3 n -ℓ (1 -z) ℓ .
The proof of Proposition 2.2 is based on the following simple combinatorial lemma.

Lemma 2.3. Let ω ∈ C. (i) Let ℓ be a non-negative integer with n ≥ ℓ. Then we have n-ℓ k=0 (-1) n-ℓ-k n + k -1 k n -ω -ℓ -1 n -k -ℓ = n + ω n -ℓ .
(ii) Let ℓ be a non-negative integer with n -1 ≥ ℓ. Then we have

n-1-ℓ k=0 (-1) n-1-k-ℓ n + k k n + ω -ℓ n -1 -k -ℓ = n -1 -ω n -ℓ -1 . (iii) We have n k=0 n -1 -ω k n + ω n -k = 2n -1 n .
Proof. We first prove (i). Note that we have

(1 -z) -ω-ℓ-1 = ∞ λ=0 -ω -ℓ -1 λ (-z) λ = ∞ λ=0 λ + ω + ℓ λ z λ and (1 -z) -n (1 -z) n-ω-ℓ-1 = ∞ λ=0 n + λ -1 λ z λ ∞ λ=0 n -ω -ℓ -1 λ (-z) λ = ∞ λ=0 λ k=0 (-1) λ-k n + k -1 k n -ω -ℓ -1 λ -k z λ . Since (1 -z) -ω-ℓ-1 = (1 -z) -n (1 -z) n-ω-ℓ-1
, the coefficients of z n-ℓ in the two above formulae must be the same, hence (i). We deduce (ii) from (i) by replacing (n, ℓ, ω) by (n + 1, ℓ + 2, -ω -2) and (iii) is just a special case of Vandermonde identity.

Proof of Proposition 2.2. By the definition, we have

A n (1 -z) = n-1 k=0 n + k k n + 1/3 n -1 -k (1 -z -1) n-1-k = n-1 k=0 n + k k n + 1/3 n -1 -k n-1-k ℓ=0 n -1 -k ℓ (-1) n-1-k-ℓ (1 -z) ℓ = n-1 ℓ=0 n-1-ℓ k=0 (-1) n-1-k-ℓ n + k k n + 1/3 n -1 -k n -1 -k ℓ (1 -z) ℓ .
Combining the above with the identity

n + 1/3 n -1 -k n -1 -k ℓ = n + 1/3 ℓ n + 1/3 -ℓ n -1 -k -ℓ
and Lemma 2.3 (ii) (for ω = 1/3), we obtain the expected formula for A n (1 -z). Similarly,

B n (1 -z) = n k=0 n + k -1 k n -4/3 n -k (1 -z -1) n-k = n k=0 n + k -1 k n -4/3 n -k n-k ℓ=0 n -k ℓ (-1) n-k-ℓ (1 -z) ℓ = n ℓ=0 n-ℓ k=0 (-1) n-k-ℓ n + k -1 k n -4/3 n -k n -k ℓ (1 -z) ℓ ,
and combining the above with the identity

n -4/3 n -k n -k ℓ = n -4/3 n -k n -4/3 -ℓ n -k -ℓ
and Lemma 2.3 (i) (for ω = 3), we get the formula for B n (1-z). This completes the proof of Proposition 2.2.

3 Remainder function

Notation 3.1. Let A n (1 -z), B n (1 -z) be the polynomials defined as in Proposition 2.2. Define the polynomial V n (T ) by V n (T ) := A n (T 3 ) -T B n (T 3 ) ∈ Q[T ].
Below we show that the polynomials A n (T ), B n (T ) and V n (T ) have the following properties, thanks to the fact they are related to Padé approximation.

Lemma 3.2. (i) For each n ≥ 0, there exists W n (T ) ∈ Q[T ] such that V n (T ) = (1 -T ) 2n W n (T ).
(ii) For any β ∈ C \ {1}, we have

A n (β)B n+1 (β) -A n+1 (β)B n (β) = 0 .
Proof. Let R n (z) be the series defined in Eq. ( 5). Since R n (z) ∈ (1/z n+1 ) (see Eq. ( 6)), we have

S n (z) := -z n-1 R n (1/z) = -z n-1 [P n,0 (1/z)f (1/z) -P n,1 (1/z)] = A n (1 -z) -(1 -z) 1/3 B n (1 -z) ∈ (z 2n ) . Set g(z) = (1 -z) 1/3 . Noting that 3T -3T 2 + T 3 = 1 -(1 -T ) 3 , the formal series g(3T -3T 2 + T 3 ) converges T -adically to 1 -T , from which we deduce the formal identity (in Q[[T ]]) S n (3T -3T 2 + T 3 ) = A n (1 -T ) 3 -(1 -T )B n (1 -T ) 3 = V n (1 -T ).
Since 3T -3T 2 +T 3 is divisible by T and S n (z) ∈ (z) 2n , we deduce that T 2n divides the polynomial V n (1-T ). This completes the proof of (i).

By definition of A n and B n , we have

A n (1 -z)B n+1 (1 -z) -A n+1 (1 -z)B n (1 -z) = z 2n (P n,0 (1/z)P n+1,1 (1/z) -P n+1,0 (1/z)P n,1 (1/z)) = (n + 1) n+1 (-1/3) n (4/3) n (n + 1)!(2n)! • z 2n ,
the last identity coming from [16, Lemma 3.7] with (α, γ, δ) = (1, -1, 4/3). Assertion (ii) follows by evaluating at z = 1 -β. The length L(P ) of P is the sum of the moduli of its coefficients. It is known (see for example [20, §1 ex. 1.2]) that M (P ) ≤ L(P ) ≤ 2 deg P M (P ). [START_REF] David | Can polylogarithms at algebraic points be linearly independent ?[END_REF] Note that for any z ∈ C we have

|P (z)| ≤ L(P ) • max(1, |z|) deg P . (8) Lemma 4.1. Let z ∈ C. For each n ≥ 1 we have max |A n (z)|, |B n (z)| ≤ 4 n max(1, |z|) n , (9)
and for each n ≥ 2 we have

L(A n (z)) + L(B n (z)) ≤ 4 n /2 . ( 10
)
Proof. All the coefficients of the polynomial A n (z) are positive and that A n (1), the constant coefficient of A n (1 -z), corresponds to the leading coefficient of P n,1 (z), hence

L(A n (z)) = A n (1) = 2n -1 n -1 ≤ 4 n-1 (n ≥ 1).
Similarly, since all the coefficients but the leading one of the polynomial B n (z) are positive, and observing that the constant coefficient of B n (1 -z) equals to the leading one of P n,0 (z), we obtain

L(B n (z)) = B n (1) -2 n -4/3 n = 2n -1 n -2 n -4/3 n ≤ 4 n (n ≥ 1). (11)
We get Eq. ( 9) by using Eq. ( 8) with the above estimates. Eq. ( 10) follows from the above and by noting that in Eq. ( 11), we can replace the upper bound 4 n by 4 n-1 for each n ≥ 2.

Lemma 4.2. Let z ∈ C. Recall that V n (z) = (1 -z) 2n • W n (z). We have for each n ≥ 1, |W n (z)| ≤ 8 n max(1, |z|) n+1 . ( 12
)
Proof. Since W n (T ) ∈ Q[T ] has degree n + 1, Eq. ( 7) implies that

2 -n-1 L(W n ) ≤ M (W n ) = M (V n ) ≤ L(V n ) = L(A n (T 3 )) + L(B n (T 3 )) ≤ 2 -1 • 4 n , for each n ≥ 2, hence L(W n ) ≤ 4 • 8 n .
The lemma follows by Eq. ( 8). We have W 1 (z) = -(z 2 + 2z + 3)/3, thus Eq. ( 12) still holds in the case of n = 1.

Remark. Eq. ( 6) gives an explicit expression of R n (z) that corresponds to the degenerate case of Gauss' hypergeometric function. It is possible to deduce asymptotic estimates of the remainder function R n (z) when n is large enough (confer [START_REF] Erdélyi | Higher Transcendental Functions, Vols. I, II and III[END_REF] [17] [START_REF] Watson | Asymptotic expansions of hypergeometric functions[END_REF]). However, we will not use is in our proof.

Also note that we could slightly improve our inequalities (and thus Theorem 1.1) by estimating the common denominators of the coefficients of approximants as Bennett does in [2, Lemma 3.2] (the proof is based on Dirichlet's prime theorem in arithmetic progressions). For simplicity, we will just combine the above argument relying on Mahler measure with properties of the local height in Section 6. It will be sufficient to get our refinement.

Heights and estimates in the archimedean case

Recall that K is a number field of degree m. We denote the set of places of K by M K (respectively by M ∞ K for archimedean places, and by M f K for finite places).

Definition 1. Given a place v of K, we define the normalized absolute value | • | v by

|p| v = p - [Kv:Qp] [K:Q] if v ∈ M f K and v|p , |x| v = |σ v (x)| [Kv:R] [K:Q] if v ∈ M ∞ K ,
where p is a rational prime number in the first case and σ v is the embedding K ֒→ C corresponding to v in the second case. Following the notation of [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF], for each v ∈ M K and each β ∈ K, we define the exponential local height of β by

H v (β) = max{1, |β| v } ,
and its exponential absolute height by

H(β) = v∈M K H v (β) .
We also set

s(v) :=      1/m if v archimedean real , 2/m if v archimedean complex , 0 if v non-archimedean .
Given v ∈ M K and β 1 , . . . , β r ∈ K, we have

|β 1 + • • • + β r | v ≤ r s(v) max (|β 1 | v , . . . , |β r | v ) . ( 13
)
When v is archimedean, we deduce from Lemmata 4.1 and 4.2 the following property. Proposition 5.1. Let v be an archimedean place and α ∈ K. For each n ≥ 1 we have

max |A n (α 3 )| v , |B n (α 3 )| v ≤ 4 ns(v) H v (α) 3n and |V n (α)| v ≤ |1 -α| 2n v • 8 ns(v) H v (α) n+1 .

Quantitative argument

In this section, we closely follow the argument in [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]. We collect the main properties we need and which are already proven in [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]. We first introduce some notation used by Evertse. Fix ρ a primitive cubic root of the unity and set L = K(ρ). Given x, y ∈ U s solutions of λx + µy = 1, we put ξ = ξ(x, y) = λx -ρµy, η = η(x, y) = λx -ρ 2 µy, and ζ = ζ(x, y) = ξ(x, y) η(x, y) .

We denote by 1 L the set of the root of the unity in L and define

V 0 = ζ(x, y) ∈ L x, y ∈ U S with λx + µy = 1 and λx µy / ∈ 1 L . ( 14
)
Let T ⊂ M L be the set of places of L consisting of those lying above the places in S. Put

A = V ∈T |3| V 1 2 V ∈T |λµ| V V ∈T max(|λ| V , |µ| V ) 3 .
We may suppose that A ≥ 1 by simple properties of absolute values [12, page 571].

The next two results correspond respectively to [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]Lemma 5] and [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]Lemma 7]. Lemma 6.1. Let B be a real number with

1 2 ≤ B < 1 and set R(B) = (1 -B) -1 B B B-1 . Then, there is a set W 0 of cardinality at most 3 s R(B) s-1 , consisting of tuples (ρ V ) V ∈T , (Γ V ) V ∈T , with V ∈T Γ V = B and such that ρ 3 V = 1 and Γ V ≥ 0 for each V ∈ T , which satisfies the following property. For each ζ ∈ V 0 , there exists (ρ V ) V ∈T , (Γ V ) V ∈T ∈ W 0 with min(1, |1 -ρ V ζ| V ) ≤ 8A H(ζ) 3 Γ V for V ∈ T. (15)
In the following we fix a tuple (ρ V ) V ∈T , (Γ V ) V ∈T ∈ W 0 , where W 0 is as in Lemma 6.1. Lemma 6.2. Let B be a real number with

2 3 < B < 1, and let ζ 1 , • • • , ζ k+1 ∈ V 0 be distinct elements with H(ζ 1 ) ≤ • • • ≤ H(ζ k+1
) and which satisfy inequality Eq. ( 15). Then we have

H(ζ k+1 ) ≥ A 1-B 6 × 8 B (3B-1) k -1 3B-2 H(ζ 1 ) (3B-1) k . (16)
The lemma below is an improved version of [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]Lemma 8]. Lemma 6.3. Let B be a real number with 5 6 < B < 1. Let r 0 and k be integers with

r 0 > 6 + 3B 3B(6B -5)
and (3B -1) k+1 > 3r 0 + 4.

Define five exponents as follows

f 1 (B, r 0 ) = 2r 0 B(3B -1) + B 3r 0 B(6B -5) -6 -3B , f 2 (B, r 0 ) = 3r 0 B + 3 3r 0 B(6B -5) -6 -3B , g 1 (B, k, r 0 ) = B -(1 -B)(3B -1) (3B-1) k -1 3B-2 (3B -1) k+1 -3r 0 -4 , g 2 (B, k, r 0 ) = r 0 + 1 + (3B -1) (3B-1) k -1 3B-2 (3B -1) k+1 -3r 0 -4 , g 3 (B, k, r 0 ) = r 0 (3B -1) k+1 -3r 0 -4 .
Then there exist at most k elements ζ ∈ V 0 satisfying Eq. ( 15) and

H(ζ) > max (8A) f 1 (B, r 0 ) (24) f 2 (B, r 0 ) , (8A) g 1 (B, k, r 0 ) (48) g 2 (B, k, r 0 ) 2 -g 3 (B,k,r 0 ) . ( 17
)
where the last inequality follows from Eq. ( 15). Given V ∈ T , we simply use the triangle inequality to get the estimate

|U n | V ≤ 2 s(V ) max |ζ 2 | V |A n (ζ 1 3 )| V , |ζ 1 | V |B n (ζ 1 3 )| V ≤ 2 s(V ) 24 ns(V ) H V (ζ 1 ) 3n+1 H V (ζ 2 ).
Combining all those estimates with the product formula and the identities T ∈V Γ T = B and v s(v) = 1 (and since v∈M K H v (ζ) = H(ζ) by definition of the height), we obtain

1 = V ∈T |U n | V × V / ∈T |U n | V ≤ 2 • 24 n × max (8A) B h 1-3B 2 h 3n+1 1 , (8A) 2nB h 2 h 3n+1-6nB 1 . (20)
First, note that by Eq. ( 19) with n ≤ ℓ, we have 2

• 24 n (8A) B h 1-3B 2 h 3n+1 1 < 1. Combined with the above, we deduce that 1 ≤ 2 • 24 n × (8A) 2nB h 2 h 3n+1-6nB 1 .
Rising the above inequalities to the power 3B -1 and using the right-hand side of Eq. ( 19) together with ℓ ≤ n + 1, we obtain 1 ≤ 2 3B-1 24 (3B-1)n (8A) 2nB(3B-1) h ,n) . Since f 1 (B, r), f 2 (B, r) are positive and decreasing in r on r ≥ r 0 and since r 0 ≤ n this contradicts Eq. ( 17). Consequently, the number of ζ ∈ V 0 satisfying Eq. ( 17) is at most k.

(3B-1)(3n+1-6nB) 1 h 3B-1 2 ≤ 2 3B 24 (3B-1)n+n+2 (8A) 2nB(3B-1)+B h (3B-1)(3n+1-6nB)+3n+7 1 < (8A) 2nB(3B-1)+B (24) 3nB+3 h 6+3B-3nB(6B-5) 1 , hence h 1 < (8A) f 1 (B,n) (24) f 2 (B

Numerical optimization

We adapt the strategy in [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]Section 8] to estimate the number of solutions (x, y) of the S-unit equation Eq. ( 1). Since such a couple (x, y) is entirely determined by the corresponding ζ(x, y), the first step is to count the elements ζ ∈ V 0 which satisfies a fixed system Eq. ( 15). We call f -part and g-part of Eq. ( 17) the quantities (8A) f 1 (B, r 0 ) (24) f 2 (B, r 0 ) and (8A) g 1 (B, k, r 0 ) (48) g 2 (B, k, r 0 ) 2 -g 3 (B,k,r 0 ) respectively. Fix C > 0 (we will specify this parameter later).

Step 1. We choose parameters B, r, k satisfying the conditions of Lemma 6.3 and we compute the associated quantities f 1 , f 2 , g 1 , g 2 and g 3 . Generally, with our choices we have f -part > g-part, so that the maximum in Eq. ( 17) is given by the f -part. Then, by Lemma 6.3 there are at most k numbers ζ ∈ V 0 satisfying Eq. ( 15) with H(ζ) > f -part.

Step 2. We estimate the number t of ζ ∈ V 0 satisfying a fixed Eq. [START_REF] Evertse | Linear equations in variables which Lie in a multiplicative group[END_REF]

A 1-B 6 × 8 B 1/(3B-2) × f -part ≥ A 1-B 6 × 8 B 1/(3B-2) × C (3B-1) t-1
.

We deduce that

(3B -1) t-1 ≤ a + b log A c + d log A where                  a = log (6 • 8 B ) -1/(3B-2) 8 f 1 24 f 2 b = 1 -B 3B -2 + f 1 c = log C(6 • 8 B ) -1/(3B-2) d = 1 -B 3B -2 .
Our choice of B, r, C will imply that a, b, c, d > 0. Since A ≥ 0 and since the function x → (a + bx)/(c + dx) is increasing (resp. decreasing) on [0, +∞) if a/c < b/d (resp.if a/c > b/d), the quantity (3B -1) t-1 is bounded from above by max{a/c, b/d}. Hence an upper bound N (which does not depend on A) for t.

Step 3. There are therefore at most N +k numbers ζ ∈ V 0 satisfying a fixed system Eq. ( 15) with H(ζ) ≥ C. Moreover, according to Lemma 6.1, we may assume that the given system Eq. ( 15) is one of the 3 s R(B) s-1 ones associated to the elements of W 0 . On the other hand, Evertse proved (see [12, Eq. ( 36)]) that for any solution (x, y) of Eq. ( 1), we have

H λx µy ≤ 2H ζ(x, y) .
Since each solution (x, y) of Eq. ( 1) is completely determined by ζ(x, y), we deduce from the above that the number of solutions (x, y) of the S-unit equation Eq. ( 1) with H(λx/µy) ≥ 2C is at most 3 s (N + k)R(B) s-1 .

Step 4. It remains to estimate the number of solutions (x, y) with H(λx/µy) < 2C. Studying the group structure of the S-units, Evertse shows that given a non-zero n ∈ N, there exists at most 5n s (2(2C) 3/n ) m solutions (x, y) of Eq. ( 1) with H(λx/µy) < 2C. Thus, the total number of solutions of the S-unit equation is at most 5n s (2(2C) 3/n ) m + 3 s (N + k)R(B) s-1 . Combining the two upper bounds together, we get Eq. ( 3) and Theorem 1.1 follows.

Choice

4 1 0

 1 Mahler measure and estimates in the archimedean case In this section, denote by | • | the usual absolute value in C. For a polynomial P (z) ∈ C[z], recall the definition of Mahler measure M (P ) of P given by M (P ) = exp log |P (e 2iπθ | dθ .

I

  Take B = 0.834, r 0 = 1600 and C = e 7.8 /2. Then the condition on k is k > 19.8389 • • • , hence we choose k = 20. A short computation givesf 1 (B, r 0 ) = 533.814 • • • f 2 (B, r 0 ) = 533.391 • • • g 1 (B, k, r 0 ) = -5.20814 • • • g 2 (B, k, r 0 ) = 36.3095 • • • g 3 (B, k, r 0 ) = 4.91666 • • • We have f -part= e 2805.183••• × A 533.814••• , and g-part= e 126.323••• × A -5.20814••• . Since A ≥ 1, the maximum in Eq. (17) is given by the f -part. A short computation show that the upper bound N coming from Step 2 is N = 26, so that k + N = 46. Finally, in the last step we choose n = 45. With this choice, the total number of solutions of the S-unit equation is at most 5 (3.36406) m • 45 s + 3.06759 • (44.9866) s .Choice II Take B = 0.84, r 0 = 100 and C = e 7.5 /2. Then the condition on k is k > 12.6539 • • • , so we choose k = 13. A short computation givesf 1 (B, r 0 ) = 164.230 • • • f 2 (B, r 0 ) = 163.461 • • • g 1 (B, k, r 0 ) = -2.25320 • • • g 2 (B, k, r 0 ) = 16.3237 • • • , g 3 (B, k, r 0 ) = 2.1093 • • • Once again we find f -part > g-part. The upper bound N coming from Step 2 is N = 31, so that k + N = 44. In step 4 we choose n = 47. The total number of solutions of the S-unit equation is at most 5 × 3.22803 m • 47 s + 2.81864 × 46.8312 s .

1

  More precisely, the number of solutions is bounded from above by min 2.81864 • (46.8312) s + 5 (3.22803) m • 47 s , 3.06759 • (44.9866) s + 5 (3.36406) m • 45 s . (3) In [12, Theorem 1], Evertse established the following upper bound:

  with C ≤ H(ζ) ≤ f -part. For that, it suffices to apply Lemma 6.2 for those numbers ζ 1 , . . . , ζ t . Assuming H(ζ 1 ) ≤ • • • ≤ H(ζ t ) and combining Eq. (16) with our hypothesis H(ζ t ) ≤ f -part and H(ζ 1 ) ≥ C, we obtain

Acknowledgments

The third author was supported by JSPS Postdoctoral Fellowship No. PE20746 and Research Support Allowance Grant for JSPS Fellows, during his stay 2021-2022 in Japan. The first author is also supported by JSPS KAKENHI Grant no. 18K03225 and Grant no. 21K03171. Thanks to these Fellowship and Grants, the authors could meet to work together.

Remark. The structure of the proof is the same as that of [START_REF] Evertse | On equations in S-units and the Thue-Mahler equation[END_REF]Lemma 8], but we will use sharper estimates. It allows us to replace the constant 96 √ 3 of Evertse by 24 and 48 (in the first and second term of the maximum respectively) and to get the extra term 2 -g 3 (B,k,r 0 ) . For the numerical computation of Section 7, the maximum in Eq. ( 17) is the term (8A) f 1 (B, r 0 ) (24) f 2 (B, r 0 ) . Proof. By contradiction, suppose that there exist k + 1 elements ζ 1 , . . . , ζ k+1 ∈ V 0 satisfying (15) and [START_REF] Luke | The special functions and their approximants, Vols. I and II[END_REF]. For simplicity, write h i = H(ζ i ) for i = 1, . . . , k+1. We may assume that

Note that the function r → f (B, r) is decreasing on [r 0 , ∞[ and tends to 2(3B -1)/(3(6B -5)) > 4/3. We therefore have

In particular, ζ 1 is not a root of unity since h 1 > 1.

We claim that there exists an integer ℓ ≥ r 0 + 1 such that

To prove this, it is sufficient to show that the left-hand inequality holds for ℓ = r 0 + 1. According to Lemma 6.2, it is even sufficient to show that,

, which, by definition of g 1 , g 2 and g 3 , is equivalent to (8A) g 1 (B,k,r 0 ) (48) g 2 (B,k,r 0 ) 2 -g 3 (B,k,r 0 ) < h 1 . This last inequality holds by Eq. ( 17), hence our claim.

For each integer n, define the polynomials A n (z) := 3 n A n (z), B n (z) := 3 n B n (z) as well as

where A n (z) and B n (z) are as in Proposition 2.2. Note that A n (z) and B n (z) have integer coefficients. In particular, for each finite place v ∈ T and each ζ ∈ K, we have 3n . Combined with Proposition 5.1, we deduce that for each place v ∈ T (finite or archimedean) and each ζ ∈ K, we have

Now, for each integer m, set

Note that by Lemma 3.2 (ii) (and since ζ 1 = 1) we have (U m-1 , U m ) = (0, 0) for each integer m ≥ 1. Define n by n = ℓ if U ℓ = 0, and n = ℓ -1 if U ℓ = 0. By the above we have U n = 0.

Then, using our estimates of A n , B n and V n , given V ∈ T , we find


Clément Boudaud

Stéphane Hameury

Carole Faye

Laurent Daudeville

EUROPEAN SEISMIC DESIGN OF SHEAR WALLS: EXPERIMENTAL AND NUMERICAL TESTS AND OBSERVATIONS

Keywords: Wood shear wall, cyclic tests, Eurocode 8, finite element modelling

This paper presents series of monotonic and cyclic tests on wood shear walls and on single nail connections used to connect wood based panels to studs of shear walls. Tests results underline difficulties in interpreting the European seismic-code, Eurocode 8 (EC8). Moreover, a finite element model of shear walls has been built using all the tests results as input data. The aim of the work is to gather enough data to propose an update of the EC8, taking into account missing material (OSB panels) and new criteria for the ductility classification of the tested elements.

INTRODUCTION 123

It is generally accepted that timber frame houses display a good behaviour and resistance when facing a seismic event. Nevertheless, the need of codes dealing with earthquake resistance cannot be disregarded. Europe is about to deal with a modern design code, Eurocode 8 [START_REF]EN 1998-1: Design of structures for earthquake resistance. Part1: General rules, seismic actions and rules for buildings[END_REF], which brings up harmonized methods to justify earthquake resistance of structure for limitation of damages and against collapse [START_REF] Toratti | Design guidance on EC8 for practicing engineers for timber structures[END_REF]. According to the future French regulations, the necessity of justify the earthquake resistance of a building depends on the building importance classification and the seismic area. Buildings are classified in 4 importance classes, depending on the consequences of collapse. Seismic areas define a classification of seismic hazards and provide for each region a reference Peak Ground Acceleration a gr .

Different design approaches are given in EC8. Depending on the regularity of the building, the structural model and analysis can be either simplified (plain) or global (spatial). The method mostly used to design timber frame houses is based on lateral forces analysis. It consists on performing an elastic analysis of 1 Clément Boudaud, Stéphane Hameury, Centre Scientifique et technique du Bâtiment -CSTB, 84 av. Jean Jaurès, Champssur-Marne, 77447 Marne-la-Vallée Cedex 2, France. Email : clement.boudaud@cstb.fr, stephane.hameury@cstb.fr 2 Carole Faye, Technological Institue FCBA, Forestry, Cellulose, Timber-construction, Furniture, Allée Boutaut, BP227, 33028 Bordeaux cedex, France. Email : carole.faye@fcba.fr 3 Laurent Daudeville, Université Joseph Fourier, Laboratoire 3S-R, BP53, ENSE 3 , Rue de la Piscine, 38041 Grenoble cedex 9, France. Email : laurent.daudeville@ujf-grenoble.fr.

the structure and introducing a behaviour factor q to take into account the ductility and the energy dissipation of a structure. Eventually, the base shear force equivalent to earthquake (F b ) is divided by the behaviour factor. The more the structure can dissipate strain energy and develop plastic strain through identified dissipative zones, the greater is the behaviour factor value. This value goes from 1.5 for low dissipative structures (Low Ductility Level) to 5 for dissipative structures (classified in Medium Ductility Level for q=2 and q=2.5 and High Ductility Level for higher q values). We can note that French authorities enforced q=3 as a maximum. The obtained F b value is compared to the design capacity of the structure calculated according Eurocode 5 [START_REF]EN 1995-1-1: Design of timber structures. Part1-1: General -Common rules and rules for buildings[END_REF]. One of the critical point of this method is the determination of the behaviour factor. Two approaches are available in EC8 when designing wood structures. The first proposes prescriptions in order to classify a few standard structures in ductility levels with corresponding q value. The second method requires monotonic and cyclic tests, according to EN standards protocols. The results of tests can be used to classify the ductility of the structure according to criteria defined in EC8. As experimental tests on shear walls are long and expensive, tests were also performed on single nail connections. Others reasons are that behaviour of shear walls is mostly dependant of its connections [START_REF] Dolan | Monotonic and cyclic tests of timber shear walls[END_REF][START_REF] Richard | Timber shear walls with large openings: experiment and numerical prediction of structural behaviour[END_REF] and that EC8 states that tests on dissipative zones shall be performed on connections, structures or parts of the structure. The non linear analysis of timber structures under seismic loading was based on this assumption in previous papers of the authors [5 -7].

This paper provides: a comparison between these two approaches, a comparison of shear walls and single nails connections behaviour which led to discuss the criteria used in the EC8 to classify the ductility level, a finite element model of the cyclic behaviour of shear walls using experimental results on connections as input data.

For the first point, tests were performed on particleboards sheathed walls, in order to confront tests results to EC8 prescriptions (these walls are High Ductility Level according to EC8). Moreover, tests on OSB sheathed walls have been performed because prescriptions of EC8 did not deal with them. Shear walls tested were standard configurations commonly used in France. The idea was to realize tests according to the second method of EC8 in order to find adequate prescriptions for OSB panels. Tests on single nail connections were conducted in order to compare behaviour between a shear wall and its constitutive connections.

EXPERIMENTAL INVESTIGATIONS

According to EN 12512 [START_REF]Timber structures -Test methods -Cyclic testing of joints made with mechanical fasteners[END_REF], cyclic test is a sequence of 3 cycles with the same amplitude of displacement. Failure of the component is defined as the loss of 20% of the shear resistance of the wall between the first and the third cycle at a given amplitude. If the component resists three cycles of amplitude 4×Vy (where V y is the yield slip) the component is classified in Medium Ductilty Level (MDL). For the High Ductility Level (HDL), the component has to endure three cycles of amplitude 6×V y without failure. Every failure occurring before or at the amplitude 4×Vy classifies the shear wall in Low Ductility Level (LDL).

TESTS METHOD ON SHEAR WALLS

Tests plan

Experimental tests consist on applying a horizontal load F at the top of the shear wall, as shown in Figure 1. The load is in the plane of the shear wall and can be applied in both directions (left and right). Roof and upper storey masses can be simulate by a vertical dead load P. When used, P is between 5.5 and 6kN on each vertical stud of the shear wall, this corresponds to the load applied on the ground floor of a two storeys building. For each configuration of shear wall, one test was carried out according to EN 594 [START_REF]Timber structures -Test methods -Racking strength and stiffness of timber frame wall panels[END_REF]. This method describes the setting of shear walls on the test machine and the loading protocol for a monotonic test. It has to be noted that EN 594 considers that a vertical dead load does not influence the result of the test. Monotonic tests provide the reference value needed to setup the loading protocol of cyclic test. For each panel configuration, two cyclic tests were performed according to EN 12512 [START_REF]Timber structures -Test methods -Cyclic testing of joints made with mechanical fasteners[END_REF], the reference value used to determine the loading protocol is the yield slip (V y ). Few cyclic tests were performed the ISO 21581 [START_REF]Timber structures -Static and cyclic lateral load test method for shear walls[END_REF], this test method uses ultimate slip (V u ) as reference value to define the loading protocol.

Shear Walls description

Figure 1 shows dimensions of the shear wall and all configurations tested are shown in Table 1, where panels type is given according to EN 12369 [START_REF]Characteristic values for structural design[END_REF] and mechanical class of studs according to EN 338 [START_REF]Structural timber -Strength classes[END_REF]. Moreover, the space between nails -which all have a ring shank -around the perimeter of the panel is set to 150mm. This space is set to 300mm along intermediates studs. At the connections between external studs and inferior plate, angles were settled with nails at the stud and bolts were screwed on anchors to maintain the plate. Angles aim at preventing a premature failure of the stud/plate connection, which would result in stopping the test in early cycles. In addition, one anchor was settled in every space between studs. and a nail. The load-displacement curve is obtained by using a universal testing machine, see Figure 2. In order to evaluate the effect of the variability of materials and connections properties, three monotonic and five cyclic tests were performed for each configuration.

TESTS METHOD ON CONNECTIONS

Tests plan

Configurations tested

Tests were conducted on the same configurations as the shear walls previously tested, as well as 29 others configurations in order to observe the effect of panel thickness and type and nail size, shape and material. These configurations were chosen after an industrialists survey and a look at the provisions of the French code for timber frame houses [START_REF]DTU 31-2 Construction de maisons et de bâtiments à ossature en bois[END_REF]. Both showed a wide variety of solutions. RS: Ring-shank nail, X: "X" shaped cross section nail, SS: Stainless Steel (X5 CrNiMo 18-10), ZP: Zinc-plated c3+ (15µ), Ga: hot-dip galvanisation (50µ) Sections and shank shape of nails tested are shown in Figure 3.

RESULTS FOR SHEAR WALLS

Figure 4 shows the force-displacement curve for a 16mm particleboard sheathed wall and Table 3 shows the resistance weakening in percentage for each amplitude and the viscous damping coefficient υ eq .

For the amplitude corresponding to 4×V y , the resistance weakening is more than 20% -see encircled area -which is the failure criteria used by EC8. A failure at 4×V y means that the shear wall is classified in LDL.

Results show that OSB shear walls with OSB are classified in LDL whatever the thickness tested. Walls sheathed with particleboards and made in accordance with prescriptions of EC8 for HDL class showed the same result.

RESULTS FOR CONNECTIONS

Figure 5 shows a force-displacement hysteretic evolution of a nail connection.

Table 4: Comparison of ductility levels between shear walls and nail connections using the same connectors

Table 4 contains results of the single nail connections corresponding to shear walls tested. Ductility level according to EC8 can be compared and numbers indicate the amplitude of the last cycle before the failure happens e. g. the number 2 in the X×Vy column means that cycles of amplitude 2×Vy were completed and the rupture happened at amplitude 4×Vy.

For each cyclic test, the force-displacement response has been characterized according to the behaviour law described later in §3. Curves plotted in Figure 6 are average responses of five tests for each diameter. For each configuration, it shows the envelop curve of the first loading at every amplitude, the envelop curve of the third loading at every amplitude and the hysteresis curves at the six millimetres amplitude. As tests were symmetrical, only the positive range of amplitudes has been represented. Influence of thickness, see Figure 7, and type of panel was very small. Shape and material coating for nails were not much influential either. Moreover, these tests showed that rupture was often a combination of the phenomenon described in §3.2, e.g. a rupture consecutive to the pull-out from the stud along with a plastically deformed shank in the stud and a slightly pull-through of the connector head in the panel.

NUMERICAL APPROACH

Finite elements method is used to model the mechanical behaviour of shear walls. Force-displacement curves of panels to studs connections are modelled using the behaviour law developed by N. Richard [START_REF] Richard | Timber shear walls with large openings: experiment and numerical prediction of structural behaviour[END_REF], which has been adjusted to fit better to tests data.

MODEL

Studs and shear panels geometry and dimensions correspond to the shear walls tested, see §2. We suppose that all non-linearities are concentrated in connectors. Boundary conditions impose a null displacement in the three directions on inferior plate. The imposed displacement of loading is applied on the node on the left of the top plate. A vertical dead load is also applied along the top plate, this load is equivalent to 6kN on each stud to fit with experimental tests.

BEHAVIOUR LAW FOR CONNECTORS

Pre observations

Failure mode and design calculation of the resistance of a single nail connection is given by the theory of Johansen [START_REF] Johansen | Theory of timber connections[END_REF]. This theory supposes six failure modes (a, b, c, d, e and f). The most dissipative one is the mode f, which consists in a double plastic hinges. Yet, when calculating the resistance of connections in a shear wall, the dominant mode is d, which is a single plastic deformation in the stud. Behaviour description of connectors is a complex matter due to the important number of phenomenon taking place in such a connection when subjected to alternate loads. It is then almost impossible to get an analytical solution of the resistance under a seismic loading taking into account following failure modes: squashing of the stud, panel or both (mode a, b or c), plastic deformation of the connector in the stud, panel or both (mode d, e or f), pull-out of the connector from the stud, pull-through of the connector head in the panel.

Law description

Behaviour of connectors is represented by a set of rules describing the force-displacement evolution, this is given for a unique direction of loading. This law was developed by N. Richard [START_REF] Richard | Timber shear walls with large openings: experiment and numerical prediction of structural behaviour[END_REF].

MATERIALS PROPERTIES

Characteristics values 1 of the materials properties are resumed in Table 5.

CALCULATION

The procedure of calculation is the same as for tests. It first simulates a monotonic test and calculates the yield slip. This value is then used to elaborate the protocol of loading of a cyclic test and this test is simulated. At each amplitude, results are the force-displacement curve and the percentage of resistance weakening between the first and the third cycle and the viscous damping coefficient. The percentage of resistance weakening is directly linked to the ductility levels of EC8.

1 Characteristic value: 5% fractile of the probability function of the random variable modelling the property value

NUMERICAL MODEL

Model is now robust, except in some cases that will need further improvements of the law. A satisfying adequacy between numerical and experimental results has been found for particleboards panels. Figure 9 displays the experimental and numerical curves for the same configuration as in Figures 4 and5. Assessment with OSB panels has still to be conducted.

DISCUSSION

SHEAR WALLS

Experimental results on shear walls for P5 and OSB were classified in LDL. These results were unexpected because prescriptions of EC8 for 16mm thick P5 classify the shear wall in HDL. Moreover, OSB and P5 mechanical properties are not that different. That leads us to think that prescriptions for HDL shear walls sheathed with OSB would not have been very different of prescriptions for P5 sheathed walls. We also have to note that all walls tested showed ductile behaviour and failure (plastic deformation of nails in timber frame and in shear panel), which is not consistent with the LDL classification.

SINGLE NAIL CONNECTIONS

The first aim of these experimental tests was to build a large data base describing single nail connections behaviour and to use it as input data for the finite elements model. The second aim was to classify connections in the ductility levels of EC8 in order to observe the effect of the scale of test (single nail connection or shear wall). Results showed that connections were as much or more ductile than a shear wall. Table 5 and figures 4 and 5 show it quite well.

The last aim was to observe the effects of different parameters. Results show that fastener dimensions have the most significant influence on the resistance. This can be explained by the failure mode commonly observedplastic deformation of nails and pull out of the stud. Resistance to bending depends on the section of the shank (difference in material, stainless steel or coated steel, are not relevant compared to the section evolution for diameters from 2.1 to 3.1mm). Pull out depends on the penetration length of fasteners and the length of a fastener is linked to diameter. Then, the more the dimensions are important, the more nails will resist to dominant failure mode.

TEST PROCEDURE AND CRITERIA

This paragraph deals with the test procedure performed on both shear walls and single nail connections and the ductility classification criteria. A bibliographic study [15 -22] concluded that the test procedure was a good compromise between results relevance and costs. However, some difficulties were pointed out, like the lack of a reliable method to determine the yield slip V y from the load-displacement curve of a monotonic test. As previously said in §1, EC8 allows to perform tests on different scales (connection, part of the structure or structure). Yet, tests methods and even ductility classification criteria are the same whatever the test scale. Tests methods fit to test scale by using a reference value obtained after a first test to realize the second. There is no such thing for classification criteria, hypothesis according to which behaviour of a shear wall can be compared to behaviour of its constitutive connections is certainly right. However, ductility classification criteria still have to be suited to what is really tested, taking into account that additional loads (e.g. roof, upper stories), angles, anchorages and studs connections are also factors in the equation. This leads to consider that testing a single nail connection to determine the ductility level of a shear wall by using the current criteria of EC8 is not accurate.

FINITE ELEMENTS MODEL

When developing the finite elements model on Code_Aster [23], some modifications have been brought to the behaviour law previously described. The first observation was that for very small amplitudes, the law equations led to discontinuities around zero displacement. This is explained by the fact that these formulae are asymptotic to the point of maximal displacement, which seems too close of the zero displacement point in case of small amplitudes. This has been corrected by adding a α coefficient in the exponential term in order to accelerate the convergence.

Others modifications of the law may follow as the numerical model application is still ongoing work.

CONCLUSIONS

This paper focuses on an experimental approach according to the new European code on earthquake resistance design, EC8. Tests performed on timber frame shear walls and on single nail connections underlined that ductility classification criteria should not be the same whatever the scale of test. Moreover, to classify the ductility of a shear wall by only performing tests on single nail connections is not satisfactory. This first observation being made, a finite elements model has been built in order to complete the tests already carried out by a wide numerical tests study. The post-processing of this study is ongoing research. It should confirm -or not -our first observations and add to our understanding of shear walls behaviour. This research recognizes that the scale of test has an influence on the tests results and therefore onto the classification of a shear wall to one of the ductility level according to EC8.

Figure 1 :

 1 Figure 1: Test shear wall dimensions and loading

Figure 2 :

 2 Figure 2: Connection and testing machine

Figure 3 :

 3 Figure 3: Ring-shank nail (left) and "X" shaped nail (right)

Figure 4 :

 4 Figure 4: Force-displacement hysteretic evolution of a16mm particleboard sheathed wall undergoing the load F

Figure 5 :

 5 Figure 5: Force-displacement hysteretic evolution of the single nail connection used in the shear wall of Figure 4

Figure 6 :

 6 Figure 6: Influence of the fastener diameter on the forcedisplacement evolution

Figure 7 :

 7 Figure 7: Influence of the panel thickness on the forcedisplacement evolution

Figure 9 :

 9 Figure 9: Nail connection behaviour -test result versus N. Richard law (Title: Cyclic curve)

Table 1 :

 1 Configurations of shear walls tested

	Panel Thickness	Nail	Stud section	Stud Mechanical class
	OSB 3	9	2.1x45	35x120	C24
	OSB 3	12	2.1x45	35x120	C24
	P5	10	2.5x55	45x120	C24
	P5	16	2.5x50	45x115	C24
	Dimensions are given in mm
	OSB 3: waterproof OSB panel glue for construction outdoor
	P5: waterproof particleboard for construction outdoor

  Table 2 summarizes all tested configurations tested in order to cover this variety.

	Nail Size	Shape	Material Panel type	Panel thick. (mm)
	2.1x38	RS	SS	OSB 3	9, 12, 15
	2.5x60	RS	SS	OSB 3	9, 12, 15
	3.1x85	RS	SS	OSB 3	9, 12, 15
	2.1x45	RS	ZP	OSB 3	9, 12
	2.5x50	RS	ZP	P 5	10, 16
	2.1x55	RS	ZP	OSB 3	9, 15
	2.8x80	RS	ZP	OSB 3	9, 15
	2.3x60	RS	Ga	OSB 3	9, 15
	3.1x90	RS	Ga	OSB 3	9, 15
	2.3x60	X	SS	OSB 3	9, 15
	3.1x85	X	SS	OSB 3	9, 15
	2.3x60	X	Ga	OSB 3	9, 15
	3.1x75	X	Ga	OSB 3	9, 15
	2.3x60	RS	Ga	P 5	10, 16
	3.1x90	RS	Ga	P 5	10, 16

Table 2 :

 2 Configurations of connections tested

Table 3 :

 3 Materials properties

Table 5 :

 5 Materials properties

ACKNOWLEDGEMENT

This testing program was carried out in the Centre Scientifique et Technique du Bâtiment (CSTB) and the Fibre Cellulose, Bois construction et Ameublement institute (FCBA). It was supported by the Syndicat National des Fabricants et Constructeurs de Maisons à Ossatures Bois (SYMOB) and French authorities. Partners of the ANR Risknat SISBAT project are also acknowledged for their advice.


Kishore Kuchibhotla

Brice Bathellier
email: brice.bathellier@unic.cnrs-gif.fr

Neural encoding of sensory and behavioral complexity in the auditory cortex

Converging evidence now supports the idea that auditory cortex is an important step for the emergence of auditory percepts. Recent studies have extended the list of complex, nonlinear sound features coded by cortical neurons. Moreover, we are beginning to uncover general properties of cortical representations, such as invariance and discreteness, which reflect the structure of auditory perception. Complexity, however, emerges not only through nonlinear shaping of auditory information into perceptual bricks. Behavioral context and task-related information strongly influence cortical encoding of sounds via ascending neuromodulation and descending top-down frontal control. These effects appear to be mediated through local inhibitory networks. Thus, auditory cortex can be seen as a hub linking structured sensory representations with behavioral variables.

Introduction

Auditory perception is an essential, complex process through which humans and other animals use pressure waves to interpret and interact with the environment. The cochlea initially decomposes pressure waves into their frequency spectrum and transduces these signals into neural impulses [START_REF] Hudspeth | Integrating the active process of hair cells with cochlear function[END_REF]. This tonotopic organization propagates throughout the auditory system. At the level of perception, however, audition is not merely designed for precise frequency encoding [START_REF] Frégnac | Cortical correlates of low-level perception: from neural circuits to percepts[END_REF], but rather interprets complex acoustic motifs into distinct auditory objects reflecting our experience of the acoustic environment. Thus, one may struggle to identify the absolute pitch of a tone but most people accustomed to western music will recognize if the tone is coming from a piano [START_REF] Cutting | Categories and boundaries in speech and music[END_REF]. How does the auditory system turn the initial spectral decomposition of sounds into the timbre of instruments, the words of a phrase, or the siren of a police car? In vision science, Gestalt theories [START_REF] Wagemans | A century of Gestalt psychology in visual perception: I. Perceptual grouping and figure-ground organization[END_REF][START_REF] Wagemans | A century of Gestalt psychology in visual perception: II. Conceptual and theoretical foundations[END_REF] argue that perception is based on a skeleton of complex representations which does not correspond to a 'one-to-one copy' of the input signals but rather to archetypal building blocks that are innate or experience-dependent and are used to construct perception [START_REF] Frégnac | Cortical correlates of low-level perception: from neural circuits to percepts[END_REF][START_REF] Gregory | Knowledge in perception and illusion[END_REF]. This idea likely extends to audition [START_REF] Nelken | Primary auditory cortex of cats: feature detection or something else?[END_REF][START_REF] Petkov | Illusory sound perception in macaque monkeys[END_REF][START_REF] Bregman | Auditory Scene Analysis: The Perceptual Organization of Sound[END_REF]. A major challenge for understanding auditory perception is to experimentally isolate and mechanistically explain the elements of such complex auditory representations. These sensory percepts, moreover, are bound to be sensitive to the current internal state and near-term goals of the subject while it explores or interacts with the environment. Increasing evidence now shows that, in auditory cortex, an elaborate vocabulary of auditory representations comes along with information about behavioral context. The purpose of this review is to highlight recent discoveries about these two levels of complexity in auditory cortex and discuss their functional roles and potential interactions.

Structure of auditory cortex representations and their link to perception

Whether auditory cortex represents simple acoustic features or building blocks for 'auditory objects' is a longstanding question [START_REF] Nelken | Primary auditory cortex of cats: feature detection or something else?[END_REF][START_REF] Mizrahi | Single neuron and population coding of natural sounds in auditory cortex[END_REF][START_REF] Nelken | Auditory cortical processing in real-world listening: the auditory system going real[END_REF][START_REF] Wang | The harmonic organization of auditory cortex[END_REF][START_REF] Chechik | Auditory abstraction from spectrotemporal features to coding auditory entities[END_REF]. The existence of frequency tuned neurons and their tonotopic mapping in primary auditory subfields [START_REF] Kanold | Local versus global scales of organization in auditory cortex[END_REF][START_REF] Issa | Multiscale optical Ca2+ imaging of tonal organization in mouse auditory cortex[END_REF] suggests a degree of similarity to cochlear representations. Yet this might be the only similarity. The precision of the tonotopic map weakens in secondary areas [START_REF] Issa | Multiscale optical Ca2+ imaging of tonal organization in mouse auditory cortex[END_REF]. Moreover, frequency selectivity is, in general, more broadly tuned in auditory cortex than in subcortical areas [START_REF] Barnstedt | Functional microarchitecture of the mouse dorsal inferior colliculus revealed through in vivo two-photon calcium imaging[END_REF][START_REF] Rauschecker | Auditory and visual cortex of primates: a comparison of two sensory systems[END_REF], and response time constants appear to be longer [START_REF] Nelken | Primary auditory cortex of cats: feature detection or something else?[END_REF] so that fine temporal details present in the input are lost or converted into rate codes [18 • ]. This readily suggests that cortex integrates over a wider range of information than subcortical structures. But most importantly, multiple studies have shown that selectivity of auditory cortex neurons greatly extend pure tone frequency coding [START_REF] Chechik | Auditory abstraction from spectrotemporal features to coding auditory entities[END_REF][START_REF] Nelken | Processing of complex stimuli and natural scenes in the auditory cortex[END_REF][START_REF] Nelken | Responses of auditory-cortex neurons to structural features of natural sounds[END_REF][START_REF] Decharms | Optimizing sound features for cortical neurons[END_REF][START_REF] Chambers | Online stimulus optimization rapidly reveals multidimensional selectivity in auditory cortical neurons[END_REF]. Recently, two-photon calcium imaging and electrophysiology in rodents demonstrate that loosely mapped ensembles of cortical neurons in primary and secondary auditory cortex code for the direction of frequency variations [START_REF] Issa | Multiscale mapping of frequency sweep rate in mouse auditory cortex[END_REF], interaural differences [24 • ] as well as the frequency [START_REF] Gao | A coding transformation for temporally structured sounds within auditory cortical neurons[END_REF], amplitude and direction [26 •• ] of intensity variations (Figure 1). These temporal features are crucial for recognizing particular classes of sounds. For example, recognition of musical instruments greatly depends on the steepness of tone intensity rise and decay [START_REF] Cutting | Categories and boundaries in speech and music[END_REF] which is coded in primary auditory cortex, even in mice [26 •• ]. Frequency modulations are important components of vocalization in most species [START_REF] Shepard | Behavioral relevance helps untangle natural vocal categories in a specific subset of core auditory cortical pyramidal neurons[END_REF][START_REF] Perks | Subthreshold membrane responses underlying sparse spiking to natural vocal signals in auditory cortex[END_REF][START_REF] Miller | Marmosets: a neuroscientific model of human social behavior[END_REF]. For highly vocal animals, the structure of frequency harmonics represent important cues, as in human language [START_REF] Wang | The harmonic organization of auditory cortex[END_REF]. Interestingly, a recent study showed that the core auditory cortex of primates includes cells which specifically detect patterns of frequency harmonics [30 •• ] (Figure 1).

The presence of complex features in auditory cortex, however, does not prove that it codes for auditory objects. To address this issue, two major and somewhat orthogonal properties of object-like representations must be observed: invariance and discreteness. Invariance refers to the stability of representations with respect to small changes in acoustic parameters. Discreteness refers to the categorical and rapid switching of representations between objects as separate entities. Two recent studies in rats have shown that auditory cortex neurons respond to vocalizations or water sounds with a certain degree of robustness against various acoustic modifications [START_REF] Carruthers | Emergence of invariant representation of vocalizations in the auditory cortex[END_REF][START_REF] Blackwell | Stable encoding of sounds over a broad range of statistical parameters in the auditory cortex[END_REF]. Interestingly, invariance for vocalization was tested both in primary and non-primary auditory cortex and was found to be more pronounced in non-primary areas [START_REF] Carruthers | Emergence of invariant representation of vocalizations in the auditory cortex[END_REF] (Figure 1), suggesting that invariance properties progressively emerge along the cortical hierarchy, correlating with the weakening of the tonotopic map [START_REF] Issa | Multiscale optical Ca2+ imaging of tonal organization in mouse auditory cortex[END_REF]. As for discreteness, a two-photon calcium imaging study has shown that local ensemble of neurons in the mouse auditory cortex respond in a step-wise manner to gradual changes in sound mixtures [START_REF] Bathellier | Discrete neocortical dynamics predict behavioral categorization of sounds[END_REF] (Figure 1). The object-like representation in cortex predicted how mice categorized diverse sounds during a behavioral task [START_REF] Bathellier | Discrete neocortical dynamics predict behavioral categorization of sounds[END_REF]. Thus, many of the ingredients necessary to build object-like or categorical representations are present in auditory cortex and evidence exists that cortical representations are close to perceptual space.

Are object-like representations hard-wired into the cortex or are they experience-dependent, dynamically updating based on the ecological needs of a particular animal? In rodents, for example, ultrasonic pup vocalizations elicit robust responses even in low frequency areas [START_REF] Shepard | Contrast enhancement without transient map expansion for species-specific vocalizations in core auditory cortex during learning[END_REF], suggesting the possibility for an object-like representation. Future studies will need to address this on the axes described above, invariance and discreteness, and determine whether these representations are innate or experience-dependent. Moreover, a comparative approach that looks across the phylogenetic tree in mammals might yield further insights into the types of objects encoded by the auditory cortex, including speech in humans.

Computations and circuits underlying auditory cortex representations

Computational models of sound encoding must account for the richness, diversity and complexity of auditory cortex representations. One influential model, linear spectrotemporal receptive fields (STRFs) [START_REF] Aertsen | The spectro-temporal receptive field. A functional characteristic of auditory neurons[END_REF][START_REF] Depireux | Spectro-temporal response field characterization with dynamic ripples in ferret primary auditory cortex[END_REF] attempts to explain the activity of auditory cortex neurons as generic weighted sums of spectral characteristics over multiple time steps (i.e. the neurons computations are approximated by a linear filter). Combined with a simple threshold non-linearity, this model can capture many frequency selectivity characteristics, including selectivity to the direction of frequency variations [START_REF] Patil | Music in our ears: the biological bases of musical timbre perception[END_REF], and can even be used, with significant success, to reconstruct played sound spectrograms based on cortical activity [START_REF] Pasley | Reconstructing speech from human auditory cortex[END_REF]. But while they successfully account of the raw input space, STRFs struggle to capture many of the transformations of the acoustic inputs that are required to structure auditory information into percepts. In general, STRFs can only predict a fraction of cortical neurons' responses [START_REF] Machens | Linearity of cortical receptive fields measured with natural sounds[END_REF][START_REF] Christianson | The consequences of response nonlinearities for interpretation of spectrotemporal receptive fields[END_REF]. The linearity hypothesis prevents them from capturing selectivity to amplitude modulations [26 •• ,41], as well as the invariance [START_REF] Carruthers | Emergence of invariant representation of vocalizations in the auditory cortex[END_REF][START_REF] Blackwell | Stable encoding of sounds over a broad range of statistical parameters in the auditory cortex[END_REF] and discreteness [START_REF] Bathellier | Discrete neocortical dynamics predict behavioral categorization of sounds[END_REF] properties of cortical representations. A generic solution to these limitations might be to consider that the auditory system implements several layers of linear (STRF-like) computations interleaved with nonlinearities (Figure 1) [26 •• ,42], as in artificial deep networks [START_REF] Lecun | Deep learning[END_REF]. The primate visual system has functional analogies with deep networks trained to recognize visual objects [START_REF] Cadieu | Deep neural networks rival the representation of primate IT cortex for core visual object recognition[END_REF] and ongoing work is beginning to address this in the auditory system [START_REF] Mlynarski | Learning midlevel auditory codes from natural sound statistics[END_REF]. Thus, deep networks might represent new tools, complementary to biologically constrained modeling [START_REF] Ahmad | Harmonic training and the formation of pitch representation in a neural network model of the auditory brain[END_REF], to identify the actual features encoded by auditory cortex neurons, particularly where linear filter methods and their weakly non-linear extensions [START_REF] Mcfarland | Inferring nonlinear neuronal computation based on physiologically plausible inputs[END_REF][START_REF] Atencio | Cooperative nonlinearities in auditory cortical neurons[END_REF][START_REF] Schinkel-Bielefeld | Inferring the role of inhibition in auditory processing of complex natural stimuli[END_REF] are not successful.

A major goal will be to link the emerging computational models, identified by deep networks or other approaches, to the underlying neural implementation and circuit mechanisms. A promising avenue is to combine observation of auditory cortical representations with manipulation of local and long-range circuit elements with optogenetics. Recent work demonstrates that interneurons influence representation of frequency in auditory cortex [START_REF] Kato | Network-level control of frequency tuning in auditory cortex[END_REF][START_REF] Aizenberg | Bidirectional regulation of innate and learned behaviors that rely on frequency discrimination by cortical inhibitory neurons[END_REF]. Currently, these studies are in their infancy with the tools now emerging to obtain more precise control of neural circuitry. Going forward, robust circuit dissection will require exploration of more complex sensory stimuli, cell-type specific genetic strategies [START_REF] Guo | A corticothalamic circuit for dynamic switching between feature detection and discrimination[END_REF], patterned and activity-dependent optogenetics [START_REF] Zhu | High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device[END_REF][START_REF] Szabo | Spatially selective holographic photoactivation and functional fluorescence imaging in freely behaving mice with a fiberscope[END_REF][START_REF] Packer | Simultaneous all-optical manipulation and recording of neural circuit activity with cellular resolution in vivo[END_REF], and incorporation of large-scale network modeling. This largely remains to be done, not only for interneuronal circuits but also for important elements of biological circuits such as intracortical layer connections and interareal feedback projections.

Modulations by behavior and context

Another current limitation of the essentially 'sensory-centric' framework proposed by existing encoding models is the absence of account for the context in which auditory perception occurs. However, this is an essential parameter in auditory cortex computations. David Hubel and colleagues demonstrated in the 1950s that the auditory cortex responded to non-sensory features when they found putative excitatory neurons that fired only when a cat would 'pay attention' [START_REF] Hubel | Attention units in the auditory cortex[END_REF]. They remarked, however, that 'attention is an elusive variable that no one has yet been able to quantify' [START_REF] Hubel | Attention units in the auditory cortex[END_REF]. This simple statement continues to resonate fifty years later. Yet beginning with pioneering studies done in the ferret [START_REF] Fritz | Rapid task-related plasticity of spectrotemporal receptive fields in primary auditory cortex[END_REF], several experiments have chipped away at this notion and identified several emerging cortical computations related to behavioral context. First, during locomotion, in which sounds have no apparent valence, multiple circuits work in concert to achieve robust suppression of auditory cortical output during locomotion [57 • , 58,59 • ]. Top-down projections from the motor cortex feed back into the auditory cortex, activate local inhibitory circuits, and suppress excitatory output [57 • ,58]. In parallel, ascending cholinergic projections from the basal forebrain innervate multiple inhibitory neurons to suppress excitatory output [START_REF] Nelson | The basal forebrain and motor cortex provide convergent yet distinct movement-related inputs to the auditory cortex[END_REF]. These signals likely supplement a reduction of the feedforward thalamic drive also induced by locomotion [START_REF] Nelson | The basal forebrain and motor cortex provide convergent yet distinct movement-related inputs to the auditory cortex[END_REF]. This shows that even a simple neural computation, such as blanket suppression when auditory information is less important to the animal, can be accomplished by the coordinated actions of distributed brain regions projecting in the auditory cortex.

Second, in contrast to these suppressive effects, cognitively demanding tasks appear to both suppress and facilitate auditory cortical responses [61 •• ,62,63 • ,64]. Anticipatory top-down inputs from the pre-frontal cortex prepares auditory cortex to receive incoming sensory information based on behavioral conditions [63 • ]. Moreover, rats given the opportunity to voluntarily initiate a given trial exhibit similar anticipatory patterns of activity in the auditory cortex [START_REF] Carcea | Dynamics of auditory cortical activity during behavioural engagement and auditory perception[END_REF] and a reduction of baseline activity with trial initiation is also seen in gerbils [START_REF] Buran | Behaviorally gated reduction of spontaneous discharge can improve detection thresholds in auditory cortex[END_REF]. These signals likely augment ascending cholinergic neuromodulation which alters the output of the auditory cortex during engagement in a cognitively demanding task [61 •• ]. Interestingly, cholinergic modulation works to simultaneously suppress some excitatory neurons (via direct inhibition through parvalbumin and somatotastin-positive interneurons) and facilitate others (via disinhibition through VIP+ interneurons) (Figure 2). Thus, inhibitory networks in auditory cortex are not only enabling stimulus-related sensory computations [START_REF] Aizenberg | Bidirectional regulation of innate and learned behaviors that rely on frequency discrimination by cortical inhibitory neurons[END_REF] but also mediate the complex action of multiple long-range modulatory inputs [61 These data suggest that all contexts are not created equal and, rather, that the auditory cortex integrates the specific features of the surrounding context. These effects may be distinct from and complementary to long-term neural plasticity, including changes in receptive fields and tonotopic maps, which have been observed in associative and perceptual learning [START_REF] Bieszczad | Remodeling the cortex in memory: increased use of a learning strategy increases the representational area of relevant acoustic cues[END_REF][START_REF] Diamond | Physiological plasticity of single neurons in auditory cortex of the cat during acquisition of the pupillary conditioned response: II. Secondary field (AII)[END_REF][START_REF] Reed | Cortical map plasticity improves learning but is not necessary for improved performance[END_REF][START_REF] Caras | Top-down modulation of sensory cortex gates perceptual learning[END_REF]. Why and how does the auditory cortex treat locomotion differently than cognitively demanding tasks? These two different contexts likely reflect the importance of experiencedependent changes in circuitry. One possibility is that locomotion-related suppression reflects the hard-wiring of feedforward, top-down, and ascending projections onto inhibitory neurons that directly suppress excitatory output. Since sounds hold little to no value in these contexts, suppression dominates as the critical computation in auditory cortex. Some contexts, in contrast, may require the processing of behaviorally relevant sounds. In these tasks, the facilitation of a subnetwork of excitatory neurons may be crucial for faithful execution of a given task.

Broad suppression and selective facilitation may therefore be a prudent coding scheme in these cases to minimize metabolically expensive spikes [START_REF] Deneve | Efficient codes and balanced networks[END_REF] while maximizing information transfer. Population analysis further suggests that choice-related information transiently exists in the auditory cortex such that rate increases may be predictive of behavioral choice [START_REF] Runyan | Distinct timescales of population coding across cortex[END_REF]. Moreover, these effects extend to more ethological behaviors such as motherhood. Oxytocin appears to work on auditory cortex through disinhibitory mechanisms to increase the sensitivity of auditory cortex to ultrasonic distress calls of pups [START_REF] Marlin | Oxytocin enables maternal behaviour by balancing cortical inhibition[END_REF][START_REF] Mitre | A distributed network for social cognition enriched for oxytocin receptors[END_REF] likely via multiple interneuron subtypes [START_REF] Cohen | Plasticity during motherhood: changes in excitatory and inhibitory layer 2/3 neurons in auditory cortex[END_REF]. Taken together, the auditory cortex exhibits remarkable context-dependent responses, allowing for simple computations such as blanket suppression during locomotion and more complex computations such as bidirectional modulation during cognitively demanding tasks. In both cases, however, neurons in the auditory cortex negotiate a barrage of inputs including feedforward sensory drive, ascending neuromodulation and descending frontal control. These signals likely converge in the auditory cortex to prime downstream circuits for action.

Conclusions/perspectives

There is now broad evidence that auditory cortex combines high level representations of sounds with different types of behavioral modulations. Future work will need to systematically investigate the related algorithms, computations and neural circuit implementations. First, at the algorithmic level, some questions are unexplored. For example pioneering work has shown that sound guided navigation leads to plasticity of sound amplitude representations [START_REF] Polley | Associative learning shapes the neural code for stimulus magnitude in primary auditory cortex[END_REF]. However it remains unclear which operations allow the brain to guide navigation with sounds and whether this involves processing of contextual inputs by auditory cortex. Second, at the computational level, it is not yet clear what are the circuit principles allowing auditory objects to gain behavioral relevance? One possibility is that there exist hard-wired computations in auditory cortex fostering the detection of behaviorally relevant stimuli. This could concern conspecific communication (e.g. motherpup interactions in rodents, vocal interactions in primates) and/or vocalization-specific features (e.g. frequency modulations, harmonics) which could thereby be boosted against other sensory features. Similarly, sounds that gain behavioral relevance through experience may be detected by similar, but learnt computations operating above the blanket suppression induced by a non-specific active state (i.e. locomotion) to enhance them as particular auditory objects. Third, at the circuit level, what are the local and long-range connections that enable the interplay between sensory and behavioral complexity? Addressing these questions will require probing the activity of precisely identified functional cell types or neural ensembles during parametrically controlled behavioral contexts using more complex auditory objects and representations. Emerging technologies including large-scale optical and electrophysiological recordings, cell-type specific targeting, and longitudinal observation of the same neurons promise to aid in this endeavor. Thus, exciting times lie ahead, in which we may soon get a handle on the sensory and behavioral complexity of audition. Structuring of auditory information via nonlinear computations in the auditory cortex. (a) Throughout the auditory system raw cochlear inputs are structured into biologically relevant percepts (auditory objects). This transformation requires a complex ensemble of non-linear computations, which we here schematize as a multilayer network linking simple nonlinearities (e.g. spike threshold) with an elaborate connection graph that includes feedforward, feedback and lateral connections. Interestingly, appropriately trained multilayer networks (Deep Learning) were recently demonstrated to boost performance on artificial perceptual tasks such as speech recognition. (b) Beyond simple computations such as spatially organized preference for particular frequency ranges (left), the auditory cortex displays a number of non-linear computations. These leads to the emergence of neurons sensitive to specific features, as sketched in the middle column, including sound onsets and offsets of particular amplitude [26 •• ], amplitude modulation (AM) frequencies [START_REF] Gao | A coding transformation for temporally structured sounds within auditory cortical neurons[END_REF], or even harmonicity (at least in primates) [30 •• ]. Also, generic response properties, such as invariance to modification of basic acoustic parameters as sketched in the top right graph [START_REF] Carruthers | Emergence of invariant representation of vocalizations in the auditory cortex[END_REF] and discrete coordinated population response switches (bottom right) (data from [START_REF] Bathellier | Discrete neocortical dynamics predict behavioral categorization of sounds[END_REF]), indicate that these nonlinear computations endow cortical representations with some of the expected properties of auditory object representations. Two computations that govern behavioral context in auditory cortex. During locomotion (left), evidence suggests that top-down inputs from the motor cortex (black solid line) drive the activity of PV+ interneurons which suppress the activity of pyramidal neurons. These same inputs may also activate somatostatin neuron (black dashed line, hypothetical). At the same time, ascending cholinergic neuromodulation (red lines) works on the same cell types but potentially in non-overlapping neurons. During task engagement (right), evidence suggests that all three major interneuron subtypes are co-activated by ascending cholinergic neuromodulation (red lines). Top-down inputs from the frontal cortex are known to modulate PV+ interneurons (black solid line) while potentially also operating on other interneuron subtypes (black dashed line, hypothetical).

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

Curr Opin Neurobiol. Author manuscript; available in PMC 2019 December 20.

Acknowledgements

B.B. acknowledges support by the Agence Nationale pour la Recherche (ANR 'SENSEMAKER'), the Marie Curie FP7 Program (CIG 334581), the Human Brain Project (SP3 -WP5.2), the Fyssen foundation, the DIM 'Region Ile de France', the International Human Frontier Science Program Organization (CDA-0064-2015), the Fondation pour l'Audition (Laboratory grant) and the Paris-Saclay University (Lidex NeuroSaclay, IRS iCode and IRS Brainscopes). We would like to thank Y. Frégnac for comments on the manuscript and Srdjan Ostojic for hosting K.V.K. at Ecole Normale Superieure as a visitor funded through an ANR TERC grant to SO and the NIDCD (DC05014).