Yoan Miche

Patrick Bas

Amaury Lendasse

Christian Jutten

Olli Simula

Advantages of Using Feature Selection Techniques on Steganalysis Schemes

Steganalysis consists in classifying documents as steganographied or genuine. This paper presents a methodology for steganalysis based on a set of 193 features with two main goals: determine a sufficient number of images for effective training of a classifier in the obtained high-dimensional space, and use feature selection to select most relevant features for the desired classification. Dimensionality reduction is performed using a forward selection and reduces the original 193 features set by a factor of 13, with overall same performance.

Introduction

Steganography has been known and used for a very long time, as a way to exchange information in an unnoticeable manner between parties, by embedding it in another, apparently innocuous, document. For example, during the 80's, Margaret Thatcher decided to have each word processor of the government's administration members changed with an unique word spacing for each, giving a sort of "invisible signature" [START_REF] Maxemchuck | Electronic document distribution[END_REF] to documents. This was done to prevent the continuation of sensitive government information leaks.

Nowadays steganographic techniques are also used on digital contents. The online newspaper, Wired News, reported in one of its articles [START_REF] Mccullagh | Secret messages come in .wavs[END_REF] on steganography that several steganographic contents have been found on websites with very large image database such as eBay.

Most of the time research about steganography is not as much to hide information, but more to detect that there is hidden information. This "reverse" part of the steganography is called steganalysis and is specifically aimed at making the difference between genuine documents, and steganographied -called stego -ones. Consequently, steganalysis can be seen as a classification problem where the goal is to build a classifier able to distinguish these two sorts of documents.

During the steganographic process, a message is embedded in an image so that it is as undetectable as possible. Basically, it uses several heuristics in order to guarantee that the statistics of the stego content are as close as possible to the statistics of the original one. Afterwards, steganalysis techniques classically use features extracted from the analysed image and an appropriately trained classifier to decide whether the image is genuine or not.

In this paper the Outguess algorithm proposed by Niels Provos in [START_REF] Provos | Defending against statistical steganalysis[END_REF] is analysed. This algorithm, according to its author, is supposed to resist especially well to statistical attacks by work on Least Significant Bits of quantized DCT coefficients of JPEG compressed images. In practice, it is often used as a reference steganographic algorithm for performance comparison, eventhough it is highly detectable, as shown for example in [START_REF] Fridrich | Feature-based steganalysis for jpeg images and its implications for future design of steganographic schemes[END_REF][START_REF] Miche | A feature selection methodology for steganalysis[END_REF].

In this work, the 193 image features proposed by Pevni and Fridrich in [START_REF] Pevny | Merging markov and dct features for multi-class jpeg steganalysis[END_REF] have been used. Theses features consider statistics of JPEG compressed images such as histograms of DCT coefficients for different frequencies, histograms of DCT coefficients for different values, global histograms, blockiness measures and co-occurrence measures. They are an extension of an original set of 23 features [START_REF] Fridrich | Feature-based steganalysis for jpeg images and its implications for future design of steganographic schemes[END_REF].

Fridrich proposes afterwards to train a classifier according to the extracted features. Consequently a set of 193 features for each image of the database is obtained, giving an especially high dimensionality space for classifiers to work on. Earlier research about these high dimensionality spaces has shown that a lot of issues come out when the number of features is as high as the one we use.

Drawbacks of Performing Steganalysis in

High-Dimensional Spaces with a Constrained Data-Set

The common term "curse of dimensionality" [START_REF] Bellman | Adaptive control processes: a guided tour[END_REF] refers to a wide range of problems related to a high number of features. Some details are given below about three inherent issues that occur in the framework of steganalysis, namely the need for data points (images), the increase of complexity and the lack of interpretability.

The need for data points: In the general case, in order for any tool to be able to analyze and find an underlying structure within the data, the number of needed points is growing exponentially with the dimension. Indeed, consider a d-dimensional unit side hypercube, the number of points needed to fill the Cartesian grid of step inside of it, is growing as O((1/ ) d ). Thus, using a common grid of step 1/10 and a dimension of 10, it requires 10 10 points to fill the grid. In practice, steganalysis work makes often use of at least 10 to 20 dimensions, implying a "needed" number of images impossible to achieve. As a consequence, the feature space may be not correctly filled with data points, which can give wrong models when building classifiers, having to extrapolate for the missing images: thus, an estimation of the required minimum number of images has to be obtained, in order to have a reliable training of the selected model.

Increase of complexity:

Computational time is another main reason. Nearest neighbours methods are usually implemented with a O(d) dimension relationship, as for SVMs. Clearly, reducing the dimensionality by a significant order of magnitude gives much more achievable computational times. As a consequence, one can use more images and lower this "missing images" effect. Meanwhile, the finally chosen number of images should still remain within a reasonable range defined by these computational times. The best compromise between the number of images and the future reliability of our model has to be found.

Lack of interpretability:

Eventhough the nearest neighbours classifiers keep good performance in high dimensions [START_REF] François | High-dimensional data analysis: optimal metrics and feature selection[END_REF][START_REF] Verleysen | The curse of dimensionality in data mining and time series prediction[END_REF], other obvious problems of high dimensionality motivate the idea of feature selection. The interpretability is an important one: high performance can indeed be reached using the whole 193 features set for classification. Meanwhile, if looking for the weaknesses and reasons why these features react vividly to a specific algorithm, it seems rather impossible on this important set. Reducing the required number of features to a small amount through feature selection enables to understand better why a steganographic model is weak on these particular details, highlighted by the selected features. Indeed, steganalysis can tend to make the whole process aimed only at performance without possible interpretations, while having knowledge about the selected features gives interesting insights on the steganographic algorithm.

Methodology and Techniques Used

Classifiers and Appropriate Number of Images

The "appropriate" number of images (or at least the minimum required for the dimensionality of our data) should be determined. For this matter, a KNN classifier is used with a Monte-Carlo technique [START_REF] Christian | Monte Carlo statistical methods[END_REF]. This enables to estimate the noise and give a confidence interval for our results. We randomly draw (without repetitions) a subset of the whole data set, and use the obtained classifier on it.

For our experiments, two different types of classifiers have mainly been used: the first one, KNN, for its overall good performance even in high dimensional spaces, but most of all, because it is computationally very fast. SVM was also chosen because it is among the classifiers giving the best results. Major drawback is of course the computational time. KNN classifier is a supervised distance-based classifier, proposed by Devijver and Kittler in [START_REF] Devijver | Pattern recognition: a statistical approach[END_REF], usually using the euclidean metric. It is based on a majority vote among the k nearest neighbours classes to assign the class of the new considered point. The SVM has been created by Vapnik [START_REF] Vapnik | Statistical Learning Theory[END_REF] in 1963 and then improved more recently (1992,1995) by Boser, Guyon and Vapnik [START_REF] Boser | A training algorithm for optimal margin classifiers[END_REF]. The original idea was to separate data using a hyperplane: this was a linear classifier. The extension of this method adds a non-linear part by the use of kernel functions.

Feature Selection Technique: Forward Selection

The forward selection algorithm is a greedy algorithm proposed in [START_REF] Whitney | A direct method of nonparametric measurement selection[END_REF]; the algorithm selects one by one the dimensions, trying to find the one that combines best with the already selected ones. Even if its capacity to isolate efficient features is obvious, the forward technique has some drawbacks: in the case where two features would have a high dependency and be "unefficient" when alone but very good when put together, forward might not take these into account soon enough in the selection process. Nevertheless, the feature selection using forward has been showing very good results and seems to perform well on our feature set; this is presented in the next section.

Our Methodology

The three main points of the proposed methodology are detailed in the following. number of images to use for training with the prepared database. Using a Monte-Carlo method on low numbers of images with both SVM and KNN, averaged plots are obtained. From it, a correct idea of a sufficient number of images for the later study can be obtained, as shown in the following experiments.

Since KNN is the fastest classifier between the two presented, it is used for the next step with forward technique. This produces a ranking of the features showing how much each new feature contributes to the correct classification rate. The best features combination is selected. A SVM is finally used on this combination, to improve the performance and obtain the final best classification rate achieved.

Experiments, Results and Analysis

Our image base was constituted of 13 000 images of natural scenes, coming from 5 different digital cameras. Images are then all reduced to a size of 800 × 600 (multiples of 8) to avoid some possible block effects and artifacts due to JPEG recompression on another grid. At the same time, they are changed from their original colorspace to grayscale colorspace (256 gray levels).

A cropping operation to 512 × 512 follows, since our implementation of the extractor of Fridrich's 193 features works on 512 × 512 image blocks (powers of 2). In the end, the whole set of images is separated into two equal parts: one is kept genuine while the other one is steganographied with the Outguess algorithm at an embedding rate of 25%. This choice of half steganographied and half genuine can be discussed as it does not reflects a real world situation. Meanwhile, the whole steganalysis process presented is designed to be used on one image at a time, determining whether it is genuine or not. Furthermore, this choice has been done to be able to compare performances with the steganalysis community current research.

For classification and test purposes, the training set has been made with at most 8000 images. Test set is composed of the remaining, that is 5000 images. The 193 features proposed by Fridrich are used as in [START_REF] Pevny | Merging markov and dct features for multi-class jpeg steganalysis[END_REF].

Determination of Sufficient Number of Images

Presented first is the result of the evaluation of a sufficient number of images, as explained in the methodology. The Monte-Carlo is used on randomly taken subsets of 200 up to 2000 images with 10 iterations. Each model built -using KNN and SVM -is also evaluated on the test set of 5000 images.

A single point is evaluated with a randomly chosen set of 4000 images, since computational time becomes very important with such number of images. In practice, on Fig. 2 presenting the results of this study, the two cross-validation results (SVM and KNN) should not be strictly compared since they do not use the same number of images to validate the model: SVM uses a 10-fold crossvalidation, while a Leave-One-Out (LOO) method is used for KNN. Test results are, on the other hand, comparable. One can really see on these plots that an apparently sufficient number of images is over 2000, since the classification rate seems to increase exponentially slowly over this value. For the experiments, a bigger set of 4000 images has been chosen.

Forward Selection and Optimisation by SVM

Here, the results of the forward selection are presented shortly. As can be seen from Fig. 3, the whole process of forward selection is not fully achieved -for computational time reasons -since we do not go over 21 features. Meanwhile, as will be more detailled in the analysis part of these results, good performance is already performed before this value. Since the goal is to have the smallest possible feature set, while keeping average same performances, the forward selection could be stopped at this point.

Test and 10-fold cross-validation remain in a much thinner interval than for our only-KNN tryouts. Moreover, the performance gain with SVM is significative as expected and reaches up to 2% in the frame of these plots. Table 1 presents the main values obtained using 193 features set. Our feature selection gives interesting results on this set. Indeed, using as few as 14 features, we are less than 1.9% behind the value obtained with all features for 10-fold cross-validation. Test values are following the exact same pattern.

Analysis

From machine learning point of view, a major achievement was obtained: reducing the dimensionality by more than 13 and keeping roughly the same performance, in the variance interval. This result is interesting for different reasons: First, the computational time is drastically reduced, since the classifiers complexity relationships to dimensionality are linear. Second, because computational time is decreased by 11, it allows future new analyses and experiments previously not possible.

From steganalysis point of view, the obtained results are of course behind the actual best values, obtained for the Outguess algorithm in [START_REF] Pevny | Merging markov and dct features for multi-class jpeg steganalysis[END_REF]. Nevertheless, the two advantages coming out of these results -namely the decrease of computational time and the gain in interpretability -can counterbalance this opinion. In the end, this set of features describes in a more precise way the functionning and problems of the Outguess algorithm. Taking these into account might help improve the steganographic scheme and make it less detectable: the first (and thus most "efficient") features selected by the forward algorithm show that the Outguess algorithm is especially weak when the analysis is made on features using -1 and -2 DCT coefficients, leading to already more than 90% of correct classification with the SVM classifier.

Conclusions and Future Work

This paper has presented a new methodology for dimensionality reduction by feature selection in the framework of steganalysis.

The issues of dimensionality have been adressed and the first step of our methodology proves that the theoretically required number of images for correct training is far from being needed. By the use of a Monte-Carlo technique on up to 4000 images, it has been shown that such numbers of images are sufficient for stable results. A set of 193 features extracted from all images serves the classification process, preceded by the dimensionality reduction step. This part of our methodology is achieved using a forward selection with a KNN classifier. It enables to reduce the number of required features to 14, while keeping roughly the same classification results. Computational time is thus greatly improved, divided by about 11. Further analysis becomes again possible with this low number of features: conclusions and precisions about the steganographic scheme can be infered from the obtained feature set. The last step using SVM for improvement over the previous KNN results achieves high classification results for so small a feature set, proving that many features among the full 193 set might not be relevant enough to be kept for classification purposes.

A comparison between the obtained reduced sets of features for various steganographic algorithms might reveal some common sensitive features. An analysis of these common points could help design a more generic steganalysis method using a "low" number of features.

Fig. 1 Fig. 1 .

 11 Fig.1. Schematic view of the proposed methodology: (1) An appropriate number of data points to work with is determined using a Monte-Carlo method for statistical stability; (2) The forward selection is performed using a KNN classifier; (3) A good feature set is selected and performance is improved using SVM.

Fig. 2 .

 2 Fig. 2. Correct classification rate for SVM (circles, top curve) and KNN (squares, bottom curve) with associated variance

Fig. 3 .

 3 Fig. 3. Plot of the correct classification rate for SVM: 10-fold cross-validation (circles, top curve) and test (squares, bottom curve). Crosses are for performance for random sets of 14 features with a KNN classifier.

Table 1 .

 1 Results of the different classifiers for cross-validation and test

		LOO / 10-fold Test Comp. time
	KNN 193	86.65%	85.89% 4.5min
	KNN 193→14	93.20%	89.02%	60s
	SVM 193	96.92%	96.76%	49h
	SVM 193→14	95.08%	94.86%	4.5h

Acknowledgements

Part of the work of Yoan Miche, Patrick Bas and Christian Jutten is supported by the French national funding under the project RIAM Estivale (ANR-05-RIAM-O1903). Part of the work of Yoan Miche, Olli Simula and Amaury Lendasse is supported by the project of New Information Processing Principles, 44886, of the Academy of Finland.


Philippe Aubry

AND Jérémy Marrez

Annick Valibouze

Computing real solutions of polynomial fuzzy systems

come

INTRODUCTION

Modeling problems with uncertain data has important applications in engineering, economics and social sciences [START_REF]Fuzzy Systems in Economy and Engineering[END_REF]. In this context, many researchers have been interested in searching for real solutions of a polynomial equation with fuzzy coefficients. This issue is crucial for the interpolation of fuzzy functions [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF]. The methods of resolution were initially based on local techniques [START_REF] Abbasbandy | Numerical solution of fuzzy polynomials by fuzzy neural network[END_REF][START_REF] Abbasbandy | Numerical solution of a system of fuzzy polynomials by fuzzy neural network[END_REF][START_REF] Rahman | Multi-solutions of fuzzy polynomials: a review[END_REF][START_REF] Rouhparvar | Solving fuzzy polynomial equation by ranking method[END_REF]. Recently, a global approach using classical algebraic techniques has been developed [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF][START_REF] Molai | Resolution of a system of fuzzy polynomial equations using the Gröbner basis[END_REF]. Indeed, despite a name that may be confusing, fuzzy numbers benefit from a perfectly formal definition.

We revisit this approach and strengthen it. In the past, both local and global approaches focused on so-called triangular fuzzy numbers,that is, with linear spread functions. The results presented here apply more generally to fuzzy numbers with finite support and bijective spread functions (such as, for example, quadratic fuzzy numbers, that have quadratic spread functions).

In a fuzzy algebraic system, the fuzzy coefficients (coming from the experiments) are generally given under a representation called "tuple". Although the tuple representation is formal, it cannot be handle by usual algebraic methods (Gröbner bases [START_REF] Becker | Gröbner bases: a computational approach to commutative algebra[END_REF], triangular decomposition [START_REF] Aubry | Triangular sets for solving polynomial systems: a comparative implementation of four methods[END_REF], rational univariate representation [START_REF] Rouillier | Solving zero dimensional systems through the Rational Univariate Representation[END_REF], . . . ) to solve the system. Nevertheless, for any fuzzy number with finite support and of bijective spread functions, this tuple representation is transformable into another representation called "parametric", where the coefficients are no longer fuzzy but real. We give the expression of the parametric representation as a function of the inverse of its spread functions (see Proposition 2.9).

We show how, in the context of fuzzy numbers with finite support and bijective spread functions, the resolution of a system (S) of s equations with k variables and with fuzzy coefficients reduces to computing positive solutions of 2 k systems, each of them formed by 3 s equations of k variables with real coefficients (Theorem 3.9 and main Theorem 4.1). This new algebraic real system denoted by T (S) is called the real transform of (S). We extend this result Date: December 18, 2017. 0 AMS Subject Classification 2000: . Keywords: Fuzzy numbers, polynomial systems.

to so-called trapezoidal fuzzy numbers to get 4 s equations instead of 3 s (Section 3.6). After describing a basic algorithm requiring the resolution of 2 k systems, we propose an optimized algorithm called SolveFuzzySystem which reduces the number of systems to solve.

In Section 2, we introduce fuzzy numbers, their different representations, and the transition from one to the other in the case of a fuzzy number with finite support. In Section 3, we define the real transform and the induced fuzzy equations of a given fuzzy polynomial equation. We show that, in the triangular fuzzy case, the real transform is a system equivalent to the collected crisp form calculated by previous methods. Finally, the study is extended to the trapezoidal case. Section 4 applies the results of previous section to a fuzzy polynomial system. It establishes main Theorem 4.1 which leads to a basic algorithm called BA-SFS. In Section 5, this latter is optimized in the algorithm SolveFuzzySystem, or SFS. A discussion about the parallelization of SFS follows. The implementation of Algorithm SFS in the Fuzzy package in SageMath [START_REF]SageMath, the Sage Mathematics Software System[END_REF] is briefly presented in Section 6 together with examples illustrating the sequential algorithm and the parallel algorithm.

FUZZY NUMBERS

This section presents basic generalities on the theory of fuzzy sets and fuzzy numbers, and the two classical representations of fuzzy numbers respectively called "tuple" and "parametric". To go further the reader may be interested in [START_REF] Dubois | Fuzzy Interval Analysis[END_REF]. We give some formulas which express the parametric representation of a fuzzy number in function of its tuple representation in case where the number has a finite support and bijective spread functions (Proposition 2.9). These formulas are the key of our algebraic method to solve in R the algebraic fuzzy systems, where R is the set of reals.

Generalities.

Fuzzy numbers are particular fuzzy sets, that we first define. In classical set theory, a subset E of the universal set X is defined by its Dirichlet's function, called also its characteristic function, defined from X to {0, 1} such that the image of x ∈ X is 1 if x ∈ E and 0 otherwise. The concept of fuzzy sets generalizes this classical concept. As depicted in Zadeh's seminal paper [START_REF] Zadeh | Fuzzy sets[END_REF], a fuzzy set E is a class of objects with a continuum of grades of membership. An element x belongs to E with a validity degree which is represented by a function with values between 0 and 1.

Definition 2.1.

A fuzzy set E is a pair formed by a set E of X and its membership function µ E : X → [0, 1]. For each element x of X, µ E (x) is called the grade of membership of x to E.

The support Supp( f ) of a function f defined on a set A is the set of a ∈ A such that f (a) = 0. The support of E is the support of its membership function: Supp( E) := Supp(µ E ). Fuzzy numbers are defined from fuzzy sets by means of the notion of r -cut, which is also involved in the parametric representation of Section 2.3. Definition 2.2. Let E be a fuzzy set and r be a real number in ]0, 1]. The r -cut of E is the following set

E r = {x ∈ X | µ E (x) ≥ r } .
The 0-cut E 0 is the closure of Supp( E). A fuzzy set is said convex if each of its r -cuts is convex for all r in [0, [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF].

Some examples of r -cuts for finite and infinite supports appear in Figures 1 to 3. They represent specific fuzzy sets of R, defined below: Definition 2.3. Let n be a real number. A fuzzy number n is a convex fuzzy set whose membership function µ n from R to the real interval [0, 1] is continuous and satisfies µ -1 n ({1}) = {n}; i.e. n is the only real with grade of membership 1. The value n is called the mean value of the fuzzy number n.

According to the definition of a fuzzy number n, the grade of membership of a real x increases when x approaches the mean value n. Two fuzzy numbers are equal if they have the same membership functions [START_REF] Zadeh | Fuzzy sets[END_REF].

Remark 2.4. In literature there are more general definitions than the one given above. They include the so-called trapezoidal fuzzy numbers, for which the grade of membership equals 1 over an interval of R containing the mean value. Definition 2.3 excludes them for the sake of clarity in our study. Note that most applications make use of non-trapezoidal numbers. However, we will show in Section 3.6 that our analysis, once established, simply extends to the trapezoidal case.

Let n be a fuzzy number and µ n its membership function. The left restriction (resp. right restriction) µ n -(resp. µ n + ) is the restriction of µ n to the left (resp. right) of the mean value n. Some families of fuzzy numbers can be characterized from the functions µ n -and µ n + . Their denomination derives directly from the nature of these restrictions. The most popular family contains the triangular fuzzy numbers for which the functions µ n -and µ n + are linear on Supp( n). Similarly one speaks of fuzzy numbers which are gaussian, quadratic,. . . Remark that both restrictions functions are not necessarily of the same type. Then, for example, a triangular-quadratic fuzzy number denotes a fuzzy number n with a left restriction µ n -linear on Supp( n)∩] -∞, n] and a right restriction µ n + quadratic on Supp( n) ∩ [n, +∞[.

Tuple representation.

The tuple representation of a fuzzy number with finite support was proposed in 1978 by D. Dubois and H. Prade in [START_REF] Dubois | Operations on fuzzy numbers[END_REF]. In this representation the arithmetic operations have very simple expressions as soon as they are performed within a family described in Definition 2.5, based on spread functions.

A function H defined on the real interval [0, +∞[ with values in the real interval ] -∞, 1] is called a spread function if H(0) = 1, H(1) = 0, H is continue and decreasing on its domain. Definition 2.5. Let L and R be two spread functions. A fuzzy number n with a finite support is said of type L-R if there exist two positive real numbers α and β such that the membership function µ n of n is given as follows: Inside a given family F(L,R), a tuple (n, α, β) represents an unique element n. The addition is an internal law of F(L,R) defined by:

µ n (x) =          L n-x α for n -α ≤ x < n when α = 0 1 for x = n R x-n β for n < x ≤ n + β when β = 0 0 for x ∈] -∞, n -α[ ∪ ]n + β,
(n, α, β) + (n , α , β ) = (n + n , α + α , β + β ) .
The family (F(L, R), +) is an abelian monoïd with the tuple 0 = (0, 0, 0) as identity element. There exists an approximate product "•" such that (F(L, R), •) is an abelian monoïd with the tuple 1 = (1, 0, 0) as identity element. The product is distributive with respect to addition. As we study equations with fuzzy equations with real indeterminates, we consider only products of the form q • n, where q ∈ R. In this case, the product described by Dubois and Prade becomes exact and is defined by:

q • (n, α, β) = (q n, q α, q β) if q ≥ 0 (q n, -q β, -q α) if q ≤ 0 . (1)
Note that the inversion of the spreads that keeps them positive when q < 0 : -q β and -q α are respectively the left spread and the right spread of q • (n, α, β). In particular, we have

-n = -1 • (n, α, β) = (-n, β, α) . (2)

Parametric representation.

The parametric representation introduced in 1986 by R. Goetschel et W. Voxman [START_REF] Goetschel | Elementary fuzzy calculus[END_REF] allows them to embed all the trapezoidal fuzzy numbers into a topological vector space. The following definition is an adaptation for non-trapezoidal fuzzy numbers: Definition 2.7. The parametric form of a fuzzy number n is an ordered pair [n, n] of functions from the real interval [0, 1] to R which satisfy the following conditions:

(i) n is a bounded left continuous non-increasing function on [0, 1], (ii) n is a bounded left continuous non-decreasing function on [0, 1], (iii) n(1) = n(1) = n.
Fuzzy number n defined by functions n and n has membership function

µ n : R → [0, 1] such that µ(x) = sup{r | n(r ) ≤ x ≤ n(r )}.
A fuzzy arithmetic, described in the following lemma, operates on parametric representation. It is coherent with those of the tuple representation given in Section 2.2. Lemma 2.8. [START_REF] Stefanini | Fuzzy arithmetic with parametric lr fuzzy numbers[END_REF] Let q ∈ R and m = [m, m] and n = [n, n] two fuzzy numbers. Then

(2) m + n = [m + n, m + n], (3) q • n = [q • n, q • n] if q ≥ 0, [q • n, q • n] if q ≤ 0
where, for any function f from R to R, the product g = q • f represents the function defined as g (r ) = q f (r ) for each r ∈ R.

From tuple to parametric representation.

In this paper, we consider polynomials equations with coefficients that are fuzzy numbers of a same family F(L,R) satisfying the sufficient requirement that the spread functions L and R are bijective. Our solving method is based on the change of representation given by formulas of Proposition 2.9 below. It is fundamental because we have to rewrite algebraically each

fuzzy coefficient n ∈ F(L,R) from tuple representation (n, α, β) into parametric representa- tion.
The parametric representation of n is strongly related to its r -cuts n r since functions n and n defined by

n(r ) = inf r ∈[0,1] n r and n(r ) = sup r ∈[0,1] n r
satisfy the requirements of Definition 2.7. This relation appears graphically in Figure 4 where x 1 = n(1/2) and x 2 = n(1/2) for a triangular fuzzy number 3 = (3, 2, 3).

The graph of functions n and n is obtained by a plane rotation of the graph of the membership function followed by a vertical symmetry. This transformation is illustrated in Figure 4 and in Figure 5 for a quadratic fuzzy number 3 = (3, 2, 3). Formally, it is given by the formulas below.

3) n(r ) = n -α L -1 (r ) n(r ) = n + β R -1 (r ) .
In particular, when the fuzzy number n is triangular, we have: In the triangular case, we obtain formula (4) because L = R = F where F(x) = 1x is bijective with F -1 = F

In the rest of the paper the different fuzzy numbers appearing in an equation or a system are assumed to belong to a same family F(L,R).

THE REAL TRANSFORM OF A FUZZY EQUATION

We will use the following terminology: a variable is said real if its represents any real number; a real variable is said positive if it represents any positive real number, i.e. belonging to

R + ; a k-uplet (b 1 , . . . , b k ) of real variables or real numbers is said positive if each component b i is positive.
In this section we consider an algebraic equation (E) with fuzzy coefficients and k real variables x 1 , . . . , x k also called the indeterminates.

The problem when computing with real variables and fuzzy numbers comes from the fuzzy numbers expressed as a product q • n because the spreads depend on the sign of q ∈ R (see Lemma 2.8). In the fuzzy equation (E) each monomial m = x

d 1 1 • • • x d k k (d i ∈ N)
whose sign is generally a priori unknown, plays the role of q; note that if each exponent d i is even then m is positive. When the k-uplet x = (x 1 , . . . , x k ) is positive, the monomial m is necessary positive.

To avoid this sign problem, Section 3.2 seeks to obtain the real solutions of (E) in R k from positive solutions in R k of 2 k auxiliary fuzzy equations E(I), where I ∈ {-1, 1} k . We called the latter equations the induced equations of E.

Considering only positive real variables, in Section 3.3 a crisp form of (E) is constructed in order to deduce a collected crisp form of (E) ; in other words, an algebraic system of equations with real coefficients whose positive solutions are those of (E). In the literature this collected crisp form is formed by four equations obtained from (E) by an algorithm and only valid for equations whose coefficients are triangular fuzzy numbers. In this paper, the coefficients are only required to have a finite support and bijective spread functions.

Moreover Section 3.4 establishes a formula that provides a particular collected crisp form of (E) formed by only three equations. We call it the real transform of (E) and denote it by T (E).

To obtain the real solutions of (E), it is necessary and sufficient to collect the positive real solutions of the 2 k real transforms T (E(I)), where E(I) is an induced fuzzy equation of (E). In practice, the equations E(I) are not pairwise distinct and it is not necessary to solve each of the 2 k systems T (E(I)). This subjet will be the center of the discussion of Section 4.

Section 3.5 compares the real transform T (E) to the usual collected crisp form given in literature for the triangular case. Section 3.6 finally generalizes the results to trapezoidal fuzzy numbers.

Preliminaries.

Let d = (d 1 , . . . , d k ) ∈ N k and x = (x 1 , . . . , x k ). The monomial of multidegree d in the variables x 1 , . . . , x k is the product x d = x d 1 1 • • • x d k k .
In the same way, for a = (a 1 , . . . , a k ) ∈ R k , we denote by a d the product

a d 1 1 • • • a d k k .
We denote by x × y the classical product (x 1 y 1 , . . . , x k y k ) where y = (y 1 , . . . , y k ). Note that (x × y) d = x d y d .

In this section, we consider the polynomial equation (E) below whith coefficients n d and m belonging to a same family F(L,R) whose spread functions L and R are bijective. These numbers are assumed to be known under their respective tuple representation m = (m, α, β) and

n d = (n d , α d , β d ). Let (5) (E) : d∈Supp(E) n d x d = m ,
where Supp(E) is the support of (E), i.e. the finite set of d ∈ N k such that n d = (0, 0, 0) (we do not take into account the right hand side of (E) which may be zero or not).

We denote by Sol(E) the set of solutions of (E) in R k :

Sol(E) = { a ∈ R k | d∈Supp(E) n d a d = m } .
The set Sol + (E) of positive solutions of (E) is defined by:

Sol + (E) = {(a 1 , . . . , a k ) ∈ Sol(E) | ∀i ∈ [[1, k]] a i ≥ 0} = Sol(E) ∩ R + k .
We search for Sol(E) by using the r -cuts in order to obtain an algebraic system with real coefficients that can be solved with classical computer algebra methods.

However, according to Identity (3) of Lemma 2.8, the r -cut of n d x d depends on the sign of x d while this sign is as unknown as those of the indeterminates x 1 , . . . , x n . This is why we need to consider the case where the indeterminates x 1 , . . . , x n are both real and positive. But we want all real zeros of (E), not only positive zeros. This is the discussion of Section 3.2. This section aims to construct Sol(E) from the 2 k sets of positive real zeros Sol + (E(I)) of the induced fuzzy equations E(I), where I runs throughout the k-uplets of {-1, 1} k .

After Section 3.2, it will remain to know how to calculate the 2 k sets Sol + (E(I)). For this perspective, Section 3.3 and 3.4 suppose that the variables are both real and positive. In this context, we seek the formula of the real transform T (E) of (E). The positive solutions of the real algebraic system T (E) are exactly the positive solutions of the fuzzy algebraic equation (E). In these two sections the fuzzy equation (E) plays the role of each of its induced equation E(I).

Solutions of (E) in function of positive solutions of its induced equations.

Let a = (a 1 , . . . , a k ) ∈ R k . We put |a| = (|a 1 |, . . . , |a k |) where |b| denotes the absolue value of b ∈ R. Note that |a d | = |a| d . The k-uplet of signs of the components of a is ε(a) = ( sign(a 1 ), . . . , sign(a k ) ) ∈ {-1, 1} k . The value ε(a) d = k i =1 sign(a i ) d i is
[ n d (r ) |a| d , n d (r ) |a| d ] ,
and if this sign is -1 then the r -cut is

[ -n d (r ) |a| d , -n d (r ) |a| d ] .
It is therefore impossible to transform n d x d in its parametric representation without knowing the sign of x d . Our idea to work around this problem of unknown sign of x d , is to, introduce an artificial k-uplet I ∈ {-1, 1} k for the signs of the indeterminates. Then the real I d can be mixed in with the coefficient n d and x can be supposed to be positive. It leads to 2 k induced fuzzy equations E(I) defined below. Definition 3.1. Let (E) be fuzzy algebraic equation with k variables given by [START_REF] Aubry | Triangular sets for solving polynomial systems: a comparative implementation of four methods[END_REF]. The induced equations of (E) are the following equations E(I) :

d∈Supp(E) I d n d x d = m where I ∈ {-1, 1} k . ( 6
)
Note that (E) = E((1, 1, . . . , 1)). In E(I), the k-uplet I plays the role of the k-uplet ε(a) of the signs of a possible solution a of (E). For each term I d n d x d in the left side of E(I), I d ∈ {-1, 1} plays the role of the sign ε(a) d of a d while the sign of x d itself is considered as positive. In E(I), there is no problem to express the r -cuts of I d n d x d with x positive whereas it is impossible in (E) for n d x d with x not necessarily positive.

To describe relationships between the solutions of equations (E) and E(I), for a k-uplet of real numbers q = (q 1 , . . . , q k ) and any subset S of R k we introduce the following notation: Proof. As I × I = (1, . . . , 1), it is sufficient to prove first equality. For b ∈ R k a solution of E(I), we have

q ⊗ S = { q × a = (q 1 a 1 , . . . , q k a k ) | a = (a 1 , . . . , a k ) ∈ S } .
m = d I d n d b d = d n d (I × b) d . Hence I × b is a solution of (E). Conversely, let a ∈ Sol(E). Then b = I × a is a solution of E(I) since m = d n d (I × I × a) d = d I d n d (I × a) d
For each a ∈ R k , there exists I ∈ {-1, 1} k such that ε(a) = I and for which the evaluation of E(I) in |a| is identical to the evaluation of (E) in a. The question of finding all solutions of (E) from the positive solutions of its induced fuzzy equations E(I) is solved by the following fundamental theorem:

(E) : 3 x 2 1 x 2 + 1 x 4 3 = 6 .
There are 8 = 2 3 3-uplets in {-1, 1} 3 :

I 1 = (1, 1, 1), I 2 = (1, 1, -1), I 3 = (-1, 1, 1), I 4 = (-1, 1, -1), I 5 = (1, -1, 1), I 6 = (1, -1, -1), I 7 = (-1, -1, 1
) and I 8 = (-1, -1, -1). There are only two distinct induced equations E(I j ). Indeed, since Supp(E) = {d 1 = (2, 1, 0), d 2 = (0, 0, 4)}, we have I d 2 j = 1 for all j ∈ [ [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF][START_REF] Chou | Ritt-Wu's decomposition algorithm and geometry theorem proving[END_REF]] ; thus the eight I j split up into the two following groups: that of j ∈ [ [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF][START_REF]Fuzzy Systems in Economy and Engineering[END_REF]] with I d 1 j = 1 and that of j ∈ [ [START_REF] Aubry | Triangular sets for solving polynomial systems: a comparative implementation of four methods[END_REF][START_REF] Chou | Ritt-Wu's decomposition algorithm and geometry theorem proving[END_REF]] with I d 1 j = -1. Hence E(I 1 ) = E(I j ) for all j ∈ [ [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF][START_REF]Fuzzy Systems in Economy and Engineering[END_REF]] and E(I 5 ) = E(I j ) for all j ∈ [ [START_REF] Aubry | Triangular sets for solving polynomial systems: a comparative implementation of four methods[END_REF][START_REF] Chou | Ritt-Wu's decomposition algorithm and geometry theorem proving[END_REF]]. According to Theorem 3.3, we have

Sol(E) = 4 j =1 I j ⊗ Sol + (E(I 1 )) 8 j =5 I j ⊗ Sol + (E(I 5 )) .
We have E(I j ) = (E) for j ∈ [ [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF][START_REF]Fuzzy Systems in Economy and Engineering[END_REF]] and the tuple representation of the coefficients of E(I j ) for j ∈ [ [START_REF] Aubry | Triangular sets for solving polynomial systems: a comparative implementation of four methods[END_REF][START_REF] Chou | Ritt-Wu's decomposition algorithm and geometry theorem proving[END_REF]] is determined with the help of Remark 3.4. With 3 = (3, α 1 , β 1 ), 1 = (1, α 2 , β 2 ) and 6 = (6, α, β), we have -3 = (-3, β 1 , α 1 ) and [START_REF] Aubry | Triangular sets for solving polynomial systems: a comparative implementation of four methods[END_REF][START_REF] Chou | Ritt-Wu's decomposition algorithm and geometry theorem proving[END_REF]]. The following sections are devoted to finding the positive solutions of (E) that we will apply to each equation E(I) taking into account Remark 3.4 illustrated in previous example.

E(I j ) : (-3, β 1 , α 1 )x 2 1 x 2 + (1, α 2 , β 2 )x 4 3 = (6, α, β) for j ∈ [[

Crisp form of (E) to find Sol + (E).

Algebraic solving of fuzzy equation (E) is usually based on the passage of the L-R representation of fuzzy numbers to their parametric representation. In the presentation below we significally strengthen this classical method for triangular fuzzy coefficients by applying it to a generic system and by extending it to more general fuzzy coefficients.

Following Lemma 2.8, equation (E) rewrites into two equalities on the r -cuts of the left and right members of (E) if all the x d , d ∈ Supp(E), are supposed to represent reals of the same sign. Indeed, according to Rule (3) of this lemma, the multiplication of a fuzzy number by a scalar q splits into two cases: q ≤ 0 and q ≥ 0. Thus we search only for the solutions a ∈ R + k since the real q := a d is then positive for each d ∈ N k . We will apply the results to every induced equation E(I).

For a ∈ R + k , according to Lemma 2.8, the following equivalence applies:

a ∈ Sol + (E) ⇐⇒ d∈Supp(E) n d (r ) a d , d∈Supp(E) n d (r ) a d = m(r ), m(r ) . (7)
This equivalence leads us to consider C (E), the following system of two equations with real coefficients and k + 1 variables x 1 , . . . , x k , r , called the crisp form of (E) :

C (E) :      d∈Supp(E) n d (r ) x d = m(r ) d∈Supp(E) n d (r ) x d = m(r ) . Let F be a set of equations in R[x 1 , . . . , x k , r ]. We put Sol + k (F) = {a ∈ R + k | ∀r ∈ [0, 1] (a 1 , . . . , a k , r ) ∈ Sol(F)}
where Sol(F) is the set of the solutions of F in R k+1 . Take a = (a 1 , . . . , a k ) ∈ R + k . According to Equivalence [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF], the k-uplet a is a solution of (E) if and only if for all real r ∈ [0, 1] the (k + 1)uplet (a 1 , . . . , a k , r ) is a solution of the crisp form C (E). In other words, Sol + k (C (E)) is the set of the positive solutions of the fuzzy equation (E):

(8) Sol + (E) = Sol + k (C (E)
) . Example 3.6. Let us take again the fuzzy equation of Example 3.5 with triangular fuzzy numbers as coefficients. We recall that their r -cuts are given by the formulas (4) and that the triplets of the common support to E(I 1 ) = (E) and E(I 5 ) are d 1 = (2, 1, 0) and d 2 = (0, 0, 4). The parametric representations of the coefficients of the equation are then written as follows:

• for both E(I 1 ) and E(I 5 ), we have n

d 2 = 1 = [α 2 r + 1 -α 2 , -β 2 r + 1 -β 2 ] and 6 = [αr + 6 - α, -βr + 6 + β]; • for E(I 1 ), we have n d 1 = 3 = [α 1 r + 3 -α 1 , -β 1 r + 3 -β 1 ]; • for E(I 5 ), we have n d 1 = -3 = [β 1 r -3 -β 1 , -α 1 r -3 -α 1 ].
Therefore the respective crisp forms of the two distincts induced equation of (E) which are necessary and sufficient to obtain the real solutions of (E) are:

C (E(I 1 )) : (α 1 r + 3 -α 1 ) x 2 1 x 2 + (α 2 r + 1 -α 2 ) x 4 3 = αr + 6 -α (-β 1 r + 3 -β 1 )x 2 1 x 2 + (-β 2 r + 1 -β 2 ) x 4 3 = -βr + 6 + β and C (E(I 5 )) : (β 1 r -3 -β 1 ) x 2 1 x 2 + (α 2 r + 1 -α 2 ) x 4 3 = αr + 6 -α (-α 1 r -3 -α 1 ) x 2 1 x 2 + (-β 2 r + 1 -β 2 ) x 4 3 = -βr + 6 + β .
In the particular triangular case, illustrated in Example 3.6, the crisp form has two equations with linear expressions w.r.t. the variable r in each side. It is a consequence of formulas (4). The triangular case is easy because the spread functions are equal to F : x → 1x with F -1 = F. The general case, when the spread functions are simply bijective, requires using inversion formulas [START_REF] Abbasbandy | Numerical solution of a system of fuzzy polynomials by fuzzy neural network[END_REF] with two indeterminates instead of only one. This leads to the crisp form with two parameters in the following theorem: C (E) :

   d n d x d -m + (α - d α d x d ) u = 0 d n d x d -m + (-β + d β d x d ) v = 0 , where u = L -1 (r ) and v = R -1 (r ) for all r ∈ [0, 1]. For a ∈ R + k , we have a ∈ Sol + (E) if and only if, for all r ∈ [0, 1], system (9) is satisfied by the (k + 2)-uplet (a 1 , . . . , a k , L -1 (r ), R -1 (r )).
Proof. By definition, a spread function H sends [0, 1] to itself and if moreover H is bijective then its inverse H -1 is continue and decreasing with H -1 (1) = 0 and H -1 (0) = 1. Suppose that the spread functions L and R are bijective. As each r ∈ [0, 1], we can put u = L -1 (r ) and v = R -1 (r ). When r runs throughout [0, 1] in the increasing sens, the parameters u and v run throughout the same interval [0, 1] in the decreasing sens. According to formulas (3), the parametric form of the coefficients of the equation are given by ( 10)

n d (r ) = n d -α d u , n d (r ) = n d + β d v for d ∈ Supp(E) m(r ) = m -α u , m(r ) = m + β v .
Then the crisp form C (E) of (E) given in ( 8) is written as a system of two equations with k + 2 variables x 1 , . . . , x k , u, v, where u and v are dependent on each other:

C (E) :    d n d x d -α d u x d = m -α u d n d x d + β d v x d = m + β v
By collecting the respective factors of u and v as well as the constant factor into both u and v, the crisp form is thus expressed as we want in the form (9) of the theorem.

Last assertion in the theorem about Sol + (E) follows directly from equality [START_REF] Chou | Ritt-Wu's decomposition algorithm and geometry theorem proving[END_REF] Note that Theorem 3.7 only requires that the restrictions on [0, 1] of the two spread functions L and R are bijective.

Our approach allows at the same time to improve and to generalize the methods known so far. For instance, results in [START_REF] Molai | Resolution of a system of fuzzy polynomial equations using the Gröbner basis[END_REF] and [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF] are restricted to triangular fuzzy numbers. Indeed, the crisp form of (E) with two parameters u = L -1 (r ) and v = R -1 (r ) given in Identity ( 9) is a generalization of the crisp form known in triangular case with only one parameter r where r ∈ [0, 1].

In the aforementioned articles, for each problem to be solved, the algorithm computes the system C (E) in variables x 1 , . . . , x k , r , which is linear w.r.t. r . Then it is rewritten into an equivalent system of four algebraic equations in x 1 , . . . , x k with real coefficients called collected crisp form of (E). In next section 3.4, we will show how to get a particular collected crisp form reduced to three equations, for any family F(L,R) such that the spread functions L and R are bijective. This is the real transform of (E). In addition, we explicitly give its formulation from (E).

The real transform and the positive real solutions of (E).

We define here the real transform of a fuzzy equation (E) and show that its positive real solutions are also those of (E).

n d = (n d , α d , β d ) (d ∈ Supp(E)) and m = (m, α, β).
The real transform T (E) of (E) is the following polynomial system over R: [START_REF] Goetschel | Elementary fuzzy calculus[END_REF] T (E) :

           d∈Supp(E) n d x d = m d∈Supp(E) α d x d = α d∈Supp(E) β d x d = β .
This definition naturally extends to a system (S) of fuzzy equations such (E). We denote by T (S) its real transform, i.e. the system formed by the real transforms of the equations in (S).

Theorem 3.9. Under the assumptions of Definition 3.8, if the two spread functions L and R are bijective then the set of positive real solutions of (E) equals the one of its real transform; in other words:

Sol + (E) = Sol + (T (E)) .
Proof. As the spread functions L and R are bijective, we can apply Theorem 3.7. Let be a ∈ R + k . According to this theorem, we know that a ∈ Sol + (E) if and only if, for all r ∈ [0, 1], the crisp form of (E) expressed in ( 9) is satisfied by the (k + 2)-uplet (a 1 , . . . , a k , L -1 (r ), R -1 (r )).

With r = 1, we have

u = L -1 (1) = 0. Then a ∈ Sol + (E) satisfies the equation d n d x d = m.
Note that when r = 1 we have v = 0 too because R(0) = 1, and we find the same equation and not a second one. This is why we obtain three equations instead of four. Then, by taking r = 0

we have u = L -1 (0) = 1 and v = R -1 (0) = 1. By replacing in (9) the expression d n d x dm by 0 and each variable u and v by 1, we deduce that a positive solution of (E) is also a positive solution of the real transform T (E) of (E).

For the inverse inclusion, consider the crisp form C (E) as a polynomial system in the variables x and with coefficients in the ring

R[u, v]. Any solution (a 1 , . . . , a k ) ∈ R k of T (E)
is also a solution of C (E) in R k whatever the parameters u and v may be in the interval [0, 1]. Obviously it remains true when they are furthermore connected by the constraint

G -1 (u) = D -1 (v) ∈ [0, 1]
. Hence any positive real solution of the real transform T (E) is also a positive real solution of the fuzzy equation (E) Theorem 3.9 ensures that finding the positive real roots of (E) amounts to finding the positive real roots of its real transform T (E). Therefore it is no use to develop intermediate computations on parametric representation like the previous methods did in the specific triangular case.

From the expression of the real transform T (E) of E, we can now derive those of its induced equations E(I). With reference to Remark 3.4 we immediately obtain the following corollary: Corollary 3.10. Under the assumptions of Definition 3.8, let I ∈ {-1, 1} k and E(I) the induced equation of E defined in [START_REF] Becker | Gröbner bases: a computational approach to commutative algebra[END_REF]. Then the real transform of E(I) is given by: [START_REF] Marrez | Bibliothèque Fuzzy en SageMath, Documentation[END_REF] T (E(I)) :

           d|I d >0 n d x d - d|I d <0 n d x d = m d|I d >0 α d x d + d|I d <0 β d x d = α d|I d >0 β d x d + d|I d <0 α d x d = β .
Moreover, if the spread functions L and R are both bijective then Sol + (E(I)) = Sol + (T (E(I))).

Comparison with previous methods in the triangular case.

Consider a system (S) formed by s polynomial equations with fuzzy coefficients. In the specific case of triangular fuzzy numbers as coefficients, the authors of [START_REF] Molai | Resolution of a system of fuzzy polynomial equations using the Gröbner basis[END_REF] and [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF] compute a collected crisp form of (S) formed by 4 s real algebraic equations. In this part we are interested in the relationship between their collected crisp form with 4 s equations and our particular collected crisp system with 3 s equations, that is the real transform of (S). For both systems their positive real solutions are the same than those of (S). It is the principle of any collected crisp form of (S).

Consider below the system F 1 of Section 6 in [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF]:

F 1 :
(2, 1, 1)x y + (3, 1, 1)x 2 y 2 + (2, 1, 1)x 3 y 3 = (7, 3, 3) (5, 1, 1)x y + (2, 3, 1)x 2 y 2 + (2, 2, 1)x 3 y 3 = (9, 6, 3) .

Applied to first equation, the algorithm proposed in [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF] produces the following collected crisp form:

       x y + x 3 y 3 -3 + x 2 y 2 = 0, x y + 2 x 2 y 2 -4 + x 3 y 3 = 0 -x y -x 3 y 3 + 3 -x 2 y 2 = 0 3 x y + 4 x 2 y 2 -10 + 3 x 3 y 3 = 0 ;
and it produces the following collected crisp form of the second equation of F 1 :

       x y + 3 x 2 y 2 + 2 x 3 y 3 -6 = 0, 4 x y -x 2 y 2 -3 = 0, -x y -x 3 y 3 + 3 -x 2 y 2 = 0, 6 x y + 3 x 2 y 2 -12 + 3 x 3 y 3 = 0 .
Call T 1 the system formed of the eight preceding equations. Besides, by applying to F 1 our formula [START_REF] Goetschel | Elementary fuzzy calculus[END_REF] defining the real transform, we get T (F 1 ), the following system of six equations:

T (F 1 ) :                  2x y + 3x 2 y 2 + 2x 3 y 3 = 7 x y + x 2 y 2 + x 3 y 3 = 3 x y + x 2 y 2 + x 3 y 3 = 3 5x y + 2x 2 y 2 + 2x 3 y 3 = 9 x y + 3x 2 y 2 + 2x 3 y 3 = 6 x y + x 2 y 2 + x 3 y 3 = 3 .
An easy computation shows the equivalence of the systems T 1 and T (F 1 ), whose set of solutions is {(x, y) ∈ R | x y = 1}. This phenomenon of equivalence between both approaches may be explained in a very general way as we show below by considering the classical computation of the collected crisp form obtained by an application of the algorithm of [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF] on the generic equation ( 5) of (E).

Let m = (n, α, β) and n d = (n d , α d , β d ) (d ∈ Supp(E)} be the respective tuple representations of the fuzzy coefficients of (E) that are assumed to be triangular. According to the formulas (4), in the triangular case the r -cuts are given by

n d (r ) = [α d r + n d -α d , -β d r + n d + β d ] for d ∈ Supp(E), and
m(r ) = [αr + m -α , -βr + m + β] .
Unlike our treatment of the general case, the transformation does not require the use of u = L -1 (r ) and v = R -1 (r ) where the spread functions L and R are bijective but not determined. For a triangular fuzzy number, L = R = F, where F(x) = F -1 (x) = 1x, being known, the previous methods replace directly L -1 (r ) and R -1 (r ) by their expression in the variable r in the equations. That's how they end up in the crisp form of (E) below expressed as two polynomials in the variable r :

C (E) :    ( d α d x d -α)r + d (n d -α d ) x d -m + α = 0 (β - d β d x d )r + d (n d + β d ) x d -m -β = 0 . A k-uplet (x 1 , . . . , x k ) ∈ R k is a solution of C (E)
for all r ∈ [0, 1] if and only if each coefficient w.r.t. the variable r of these independent equations is zero. The collected crisp form of (E) is therefore written

                 d α d x d = α d (n d -α d ) x d = m -α d β d x d = β d (n d + β d ) x d = m + β .
By applying this transformation to each equation in the system F 1 , we find the collected crisp form T 1 of our example. For the generic equation (E), by injecting the first equation into the second one and by noting that the last equation is the sum of the three other ones, we obtain the real transform T (E) with three equations defined in [START_REF] Goetschel | Elementary fuzzy calculus[END_REF].

Case of trapezoidal fuzzy numbers.

As mentioned in Remark 2.4, our results adapt to trapezoidal fuzzy numbers with finite support. The latter numbers extend Definition 2.3 that we give about fuzzy numbers by allowing µ -1 n ({1}) to be an interval [a, b]. In this context, a fuzzy number n with finite support is of type L-R if its membership function µ n has the following form:

µ n (x) =          L a-x α for a -α ≤ x < a when α = 0 1 for x ∈ [a, b] R x-n β for b < x ≤ b + β when β = 0 0 for x ∈] -∞, a -α[ ∪ ]b + β, +∞[ .
Then the tuple representation of the fuzzy number n is the quadruplet (a, b, α, β).

The expression of the parametric representation given in Proposition 2.9 takes the following form for a trapezoidal number of type L -R whose spread functions L and R are bijective:

(13) n(r ) = a -α L -1 (r ) n(r ) = b + β R -1 (r ) .
n d (r ) = a d -α d u , n d (r ) = b d + β d v for d ∈ Supp(E) ; m(r ) = a -α u , m(r ) = b + β v .
Applying the argument of Section 3.4 (here L(1) = L(1) = 0 and R(0) = R(0) = 1), we obtain in the same way a real transform of (E), but this time with four equations: [START_REF] Rahman | Multi-solutions of fuzzy polynomials: a review[END_REF] T (E) :

                 d a d x d = a d b d x d = b d α d x d = α d β d x d = β .
Consequently the use of the real transform presented further for solving polynomial fuzzy systems will directly transpose to systems with trapezoidal fuzzy coefficients.

The algorithms proposed in this article will remain valid for trapezoidal fuzzy numbers. Only the algorithmic function RealTransform(S), which returns the real transform of a fuzzy (S) system of s equations, will have to be adapted in order to return the real transform T (S) with 4 s instead of 3 s equations by slightly applying formula [START_REF] Rahman | Multi-solutions of fuzzy polynomials: a review[END_REF] to each equation of (S).

REAL SOLVING OF FUZZY POLYNOMIAL SYSTEMS

The resolution of a fuzzy system (S) of s equations follows directly from the results for a single equation. In Section 3, we deduce from equation (E) 2 k fuzzy induced equations E(I) where I runs throughout the 2 k k -uplets of {-1, 1} k . Following Theorem 3.3 the solutions of (E) are deduced from the positive solutions of every fuzzy induced equation E(I). And following Corollary 3.10, the positive solutions of each E(I) are the positive solutions of the real transform T (E(I)). In the same way, we will consider 2 k real transforms T (S(I)) of induced systems S(I), with 3 s equations, where I runs throughout the 2 k k -uplets of {-1, 1} k . This section establishes the main Theorem 4.1 that expresses the real solutions of a system from the positive solutions of the 2 k real transforms of these induced systems. A first algorithm results from a direct application of this theorem. Then it is discussed how to reduce the number of computational branches by deriving some of the 2 k sets of real solutions of (S) from the positive real solutions of some system T (S(I)) previously processed by the algorithm.These elements will lead to an optimized algorithm SolveFuzzySystem presented in next section.

Foundations.

Let (E 1 ), . . . , (E s ) be the s fuzzy equations of the system (S). For each I ∈ {-1, 1} k , we denote by S(I) the induced fuzzy system formed by the s induced fuzzy equations E 1 (I), . . . , E s (I) according to the notation of Section 3.2. For each I ∈ {-1, 1} k the real transform T (S(I)) of the induced sytem S(I) is the system formed by the 3 s equations coming from the real transforms of the induced equations E 1 (I), . . . , E s (I).

The following main theorem is a direct consequence of Theorem 3.

Sol(S) = I∈{-1,1} k { I × b | b ∈ Sol + (T (S(I))) } = I∈{-1,1} k I ⊗ Sol + (

T (S(I))).

A first algorithm for the real solving of fuzzy systems, called BA-SFS, derives naturally from Theorem 4.1 and the results above. It is given below. Its input is a polynomial system with fuzzy coefficients given in tuple representation and supposed to belong to a same family F(L,R) whose spread functions L and R are bijective. It is based on the following functions:

• The function Multisign( j ) is the natural bijection between the interval [[0, 2 k -1]] and {-1, 1} k : the image of the integer j

= 2 k -1 i =0 b i 2 i ∈ [[0, 2 k -1]] is the k-uplet I = (c o , . . . , c 2 k -1 ) ∈ {-1, 1} k with c i = 2b i -1;
• the function SolPos(SR) returns the positive real solutions of a system SR of polynomial equations whith coefficients in R. It can be based on any known computer algebra method. In the implementation described in section 6 it is Wu's method of decomposition into triangular polynomial systems that is used [START_REF] Wu | A zero structure theorem for polynomial equations solving[END_REF]; • the function RealTransform(S, I) applies formula [START_REF] Marrez | Bibliothèque Fuzzy en SageMath, Documentation[END_REF] to return the 3 s equations of the real transform of the fuzzy system S(I) with s equations (it is 4 s equations when the fuzzy numbers are trapezoidal).

Algorithm 1 BA-SFS, a basic algorithm for solving fuzzy systems Require: S, a polynomial system with fuzzy coefficients under tuple representation k, the dimension, i.e. the number of variables Ensure: sol , the set of real solutions of (S) By extension of the notation used for one equation, we denote Supp(S) the support of (S), that is to say the set of d in N k such that x d appears in the left hand side of some equation of (S) with a nonzero coefficient (the constant in right side can be possibly 0 = (0, 0, 0)).

In the for loop of Algorithm BA-SFS, the system S(I) may be identical to one of a previous step. In this case, a new calculation of PR, the positive real solutions of S(I), is redundant and it is desirable to avoid it.

The optimized algorithm called SolveFuzzySystem, or SFS, proposed in Section 5 provides every real solution of (S) avoiding unnecessary calculations.

Thus we first seek to identify, from (S), induced systems S(I) which are identical. Example 3.5 dealing with a single equation shows how this situation can occur from the eight fuzzy equations that reduces to only two distinct ones. When, for example, all the components of each k -uplet d of the support of (S) are even, the 2 k induced systems S(I) are identical. Our goal is to automize the recognition of identical induced systems. , according to the definition of the induced equations given in [START_REF] Becker | Gröbner bases: a computational approach to commutative algebra[END_REF].

The number of distinct systems S(I) among the 2 k induced by (S) is thus at most equal to the number of distinct columns in M(S). The matrix of signs avoids solving many induced systems. Other rapid detections of identical induced systems can be implemented. Indeed, two systems S(I 1 ) and S(I 2 ) can be identical up to a permutation of the equations whereas C(I 1 ) = C(I 2 ). We can also find equations of the form "a = 0" and "-a = 0" since they have the same solutions.

ALGORITHM SolveFuzzySystem

This section proposes an optimized algorithm SolveFuzzySystem, or SFS in contracted form, for computing the real solutions of a polynomial fuzzy systems. The coefficients of the systems are fuzzy numbers given under tuple representation and are supposed to belong to a same family F(L,R) where the spread functions L and R are bijective. After the description of sequential algorithm SFS in Section 5.1, we discuss its parallelization in Section 5.2.

The sequential algorithm SolveFuzzySystem.

Like the first algorithm BA-SFS, algorithm SolveFuzzySystem (or SFS) consists in iteratively going through the columns of the matrix of signs M(S) described in Section 4.2. For each column C(I) indexed by a k -uplet I of signs, it looks for real solutions of the system (S) from S(I) avoiding unnecessary computations if one of the previous columns allows it.

For this purpose, it uses the following three vectors, indexed from 0 to 2 k -1 which is empty at the beginning of the algorithm:

• Di st i nct Col umns will contain the distinct columns of the matrix of signs; • Di st i nct Sy st ems will contain the distinct fuzzy systems whose positive solutions have been calculated; Its indexes are related to the indexes of Di st i nc t Col umns; • l b will contain positive real solutions of S(I) such that C(I) is in Di st i nc t Col umns; so it will be naturally indexed by following Di st i nc t Col umns.

As explained in the section 4.2, two cases can avoid redundant calculations when the current column is I:

(1) If C(I) is identical to a previous column C(J) in M(S) then S(I) = S(J). In this case, the algorithm does not add the column C(I) in the vector Di st i nc t Col umns. It uses the positive real solutions of S(J) already been calculated and stored in l b during a previous step. This is the first step of the algorithm (see the 4: statement). (2) If S(I) is identical to a system S(J) where C(J) is a column preceding C(I) but being distinct from it, then the algorithm adds to the following index c pt the column C(I) to the vector Di st i nct Col umns and the positive solutions of S(J) already calculated in l b. As it is not relevant to add S(I) to vector Di st i nc t Sy st ems, the algorithm does not add to it at index cpt and its place remains empty. We can then apply tests ( 1) and ( 2) to the following columns of M(S) (see the 7: statement).

Outside of situations ( 1) and ( 2), the positive solutions of the fuzzy system S(I) are computed, as in the basic algorithm BA-SFS, with the functions RealTransform and SolPos. The system S(I) is added to the Di st i nc t Sy st ems vector at the c pt index, which contains the fuzzy (distinct) systems whose positive solutions were calculated and stored in a l b vector at the same corresponding index. The column C(I) in Di st i nc t Col umns is kept at index c pt also; it will be possible to apply tests ( 1) and ( 2) to the following columns of M(S) (see the 9: statements).

In each step of the for loop, in other words, for each k-uplet I of signs Theorem 4.1 is applied to obtain the real solutions of (S) associated with the positive real solutions of the induced system S(I).

Functions used by the the algorithm SolveFuzzySystem are those of the basic algorithm BA-SFS completed by the following ones:

• the function SignColumn( j , S) returns the ( j + 1)-th column of the matrix M(S) of signs ( j starts to 0 not 1); • the function IsIn( e , Di st i nc t ) returns -1 if e is not in the vector Di st i nc t , otherwise it returns the index of the first occurence of e in Di st i nc t . This function is called indifferently on the signs columns of M(S) and on the polynomial systems. For an efficient search on a polynomial system, we order polynomials according to a total order on the monomials and by assigning the sign + to the dominant monomial (the monomials in a same polynomial are ordered and the equations in the system are also ordered).

A parallel version of the algorithm SolveFuzzySystem,

The positive solutions of each induced system S(I) are computed independently of those of the other fuzzy induced systems (with successively the functions RealTransform and SolPos).

This part being the most expensive of the algorithm SFS, its possible parallelization is welcome. It requires to modify the algorithm in order to identify the distinct induced systems S(I) to be solved, but without performing the resolution, and simultaneously store the useful informations for further application of Theorem 4.1.

To achieve this aim, we modify the role of vector lb. It will be indexed from 0 to 2 k -1, like the columns of M(S), and l b[ j ] will contain the index in Di st i nc t Sy st ems of the system corresponding to column SignColumn( j ). When the resolution of a system would have been The general parallel algorithm takes place in three steps, as follows.

(1) Detect efficiently the distinct systems to solve in parallel. Moreover, for each column I of M(S) (and equivalently for each j from 0 to 2 k -1), the index of the first previous system equals to the real transform of S(I) is affected to l b[ j ] (it can be j itself if S(I) is new). The distinct systems are stored in the vector Di st i nc t Sy st ems.

This is realized within the framework of the sequential algorithm SFS modified in the A detailed example of this parallel version is given in Section 6.2.

IMPLEMENTATION OF THE ALGORITHM SFS AND EXAMPLES

We already have a package Fuzzy under SageMath written by J. Marrez [START_REF] Marrez | Bibliothèque Fuzzy en SageMath, Documentation[END_REF]. It contains a function resolution_reelle_systemes_flous that implements algorithm SFS of Section 5. Recall that fuzzy numbers must have bounded support and must lies in a family F(L,R)

where L and R are bijective.

In Section 6.1 we describe this function resolution_reelle_systemes_flous. Then two complete examples are given in Section 6.2. The first one details intermediate computations of the function resolution_reelle_systemes_flous. The second one presents on another fuzzy system the computations performed by the parallel version of the algorithm.

Representations and main functions implemented in Fuzzy.

The external representation of data in Fuzzy package is described below:

• a spread function H has a representation rep(H); for example "Quad" if H is quadratic; For example, with quadratic fuzzy numbers, the system

F : x + (-1, 1, 1) = (-2, 1, 1)y 2 , x + (3, 1, 1) = (2, 1, 1)y 2
is represented by the variable System defined as follows:

LeftSide1 = [(NombreFlouRed(1,0,0,"Quad","Quad"),x), (NombreFlouRed(-1,1,1,"Quad","Quad"),1)] RightSide1 = [(NombreFlouRed(-2,1,1,"Quad","Quad"), y**2) ] LeftSide2 = [(NombreFlouRed(1,0,0,"Quad","Quad"),x), (NombreFlouRed(3,1,1,"Quad","Quad"),1)] RightSide2 = [(NombreFlouRed(2,1,1,"Quad","Quad"),y**2) ] System = [ (LeftSide1, RightSide1 ), (LeftSide2, RightSide2)] Remark 6.1. The fuzzy equations of system F does not have exactly the same form than the generic equation (E) studied in our paper. The right side of (E) is not restricted to a fuzzy number m. However our results clearly extend to such a form of equations.

The function resolution_reelle_systemes_flous(S,k) implements the sequential algorithm SFS of this paper. It takes as first parameter the representation of a fuzzy system (S) with s equations. The second parameter k is the number of variables. It returns the set of k-uplets real solutions of (S). For this, it mainly uses the functions transformee_reelle() and SolPos() described below.

The function transformee_reelle(S,rep(I)), where rep(I)) is a list representing a kuplet of signs I ∈ {-1, 1} k , returns the real transform of the induced fuzzy system S(I), that is the system of 3 s equations with real coefficients obtained by transforming each of its equations E(I) as in Corollary 3.10. It implements our function RealTransform of the algorithms in Sections 4 and 5.

The function SolPos(Sr) returns the set of positive real solutions of a polynomial system Sr with real coefficients. The representation of Sr is a list [p 1 ., . . . , p r ] of polynomials such that p i = 0. It first calls the function Wu(Sr) which implements Wu's algorithm [START_REF] Wu | A zero structure theorem for polynomial equations solving[END_REF]. This one returns a set Z of polynomial sets called characteristics sets such that the variety V(Sr) of zeroes of Sr admits a decomposition into triangular polynomial systems of the form V(Sr) = B∈Z V(B) \ V(I B ) où I B = b∈B i ni t (b). One can find the definition of the initial i ni t (b) of a polynomial b in ( [START_REF] Chou | Ritt-Wu's decomposition algorithm and geometry theorem proving[END_REF]) or in [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF], a paper where Wu's method is described in order to solve polynomial systems with triangular fuzzy numbers as coefficients. From the set Z, a function get_zeros(Z) returns the elements in R + k of the variety V(Sr).

Examples.

Example 1 : As in [7] (example 6.1), the call resolution_reelle_systemes_flous(F,2) returns the variety V(F) = {(x = -1, y ± 1)}, solution of the system F given above. Here we describe the intermediate computations.

With k = 2 variables x, y, there are the 2 2 = 4 following multisigns:

I 0 = [-1, -1], I 1 = [-1, 1], I 2 = [1, -1], I 3 = [1, 1]
. By taking them in this order corresponding respectively to j = 0, 1, 2, 3 in the algorithm SFS and by ordering the monomials of the support as follows: x, 1, y 2 , the matrix of signs M(F) is

I 0 I 1 I 2 I 3 x 1 y 2   -1 -1 1 1 1 1 1 1 1 1 1 1  
Note that in the performance of the algorithm below, a column is represented by a line vector.

The variable Sr is the real transform of the induced fuzzy system F(I) of s = 2 equations, with I ∈ {I 1 , . . . , I 4 }; in theory Sr is formed by 3s = 6 equations. But in practice, some are identical. For example, when I = [-1, -1], the real transform of F(I) is formed by the six equations 2y 2x -1 = 0, -y 2 +1 = 0, -y 2 +1 = 0, -2y 2x +3 = 0, -y 2 +1 = 0, -y 2 +1 = 0, that reduces to Sr= [-y 2 + 1, 2y 2x -1, -2y 2x + 3] by removing duplicates polynomials. It is the first real polynomial system to solve in the program. For obvious practical reasons, this system is the one registered in the variable DistinctSystems, and not the fuzzy system F(I) as mentioned in the algorithm SFS. I 0 I 1 I 2 I 3

x 3 y 3 x 2 y 2 x y 1

    1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1    
We see that it is sufficient to solve only the systems corresponding to the first two columns ; that is, we have the following identities on fuzzy induced systems: F 1 (I 0 ) = F 1 (I 3 ) and F 1 (I 1 ) = F 1 (I 2 ). In the performance of the algorithm, this information will be found in the vector lb.

Thus, the number of systems to solve is reduced by half.

In the first step of the parallel algorithm, after the loops j= 0 and j=1, we have: After the loops j=2 and j=3, only the vector lb is modified, as follows:

lb=[0,1,1,0]. As lb [START_REF] Abbasbandy | Numerical solution of fuzzy polynomials by fuzzy neural network[END_REF]=lb [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF] the positive real solutions of the third induced system F 1 (I 2 ) corresponding to lb [START_REF] Abbasbandy | Numerical solution of fuzzy polynomials by fuzzy neural network[END_REF] are those of the second induced system F 1 (I 1 ) corresponding to lb [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF]; after second step, this positive real solutions will be in SPos [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF]. The same occurs with lb [START_REF] Abbasbandy | Numerical solution of a system of fuzzy polynomials by fuzzy neural network[END_REF]=lb [0].

In second step, function SolPos is called in parallel on each of both systems stored in DistinctSystems in order to compute their respective positive real solutions. These solutions are recovered and stored in the vector SPos. We then have:

SPos=[set([(1/y, 'R+')]),set([])]. Last step computes in parallel for each j = 0,1,2,3 the products I ⊗ SPos[lb [j]] corresponding to the last column "returned solutions" below. The four results are added in the variable sol containing the real solutions of the fuzzy system F 1 . Concretely, this gives: As in the sequential version, only two distinct systems are solved. As announced, we find the variety V(F 1 ) = {(x = 1 y , y) | y ∈ R\{0})}.

CONCLUSION

Up to now, given a fuzzy system (S) of s equations and k indeterminates, the existing algebraic methods have performed computations with the parametric representation of the coefficients to obtain the collected crisp form of (S) formed by 4s real equations. We show that these computations are superfluous and exhibit a formula that defines an equivalent system with 3s real equations. We call it the real transform T (S) of the system (S). As a main property, it has the same positive solutions as (S) (Theorem 3.9).

Unlike the previous methods that were restricted to triangular fuzzy numbers, our results apply to any family F(L,R) where the spread functions L and R are bijective. Moreover there is no use to compute the inverse of the spread functions since the real transform is a universal formula independent from L and R.

For solving equations with fuzzy coefficients, one must face the issue of the sign of solutions. It is intrinsic to fuzzy numbers, since the product by a real scalar is expressed differently depending on the sign of this scalar. Our strategy has been to only focus on positive solutions by putting back the issue on the fuzzy coefficients. Theorem 4.1 made it possible since it expresses the real solutions of (S) from the positive solutions of at most 2 k real systems. From this theorem we devise a first algorithm that automatizes the research of solutions by avoiding the studies of signs needed in previous methods. Our approach is independent of the choice of the method to calculate the positive solutions of a system of polynomial with real coefficients.

Among the 2 k induced systems of (S), some of them are identical. Our examples show that it is not rare to substantially reduce the number of induced systems to solve. We describe a strategy to avoid redundant branches of computations that leads to an optimized algorithm, SolvingFuzzySystem, that is implemented in the package Fuzzy of the computer algebra system SageMath.

The most costly part of this algorithm lies in finding the positive solutions of the real transform of the distinct induced systems of (S). We suggest in Section 5.2 a parallelization of the algorithm SolvingFuzzySystem that executes in parallel these independent computations, and we illustrate its performance with the second example of Section 6.

Figure 1

 1 Figure 1 represents a gaussian fuzzy number 3, i.e. of mean value n = 3. Its support is infinite and the 0-cut E 0 of E = n is the whole set R.

FIGURE 1 .

 1 FIGURE 1. Infinite support: a gaussian fuzzy number 3; R is the 0-cut.

Example 2 . 6 .

 26 +∞[ In particular, real numbern can be identified to fuzzy number n with α = β = 0. The triplet (n, α, β) is called the tuple representation of fuzzy number n. Real numbers α and β are respectively called the left spread and the right spread of n. Functions L et R are respectively called the left spread function and the right spread function of the family of fuzzy numbers of type L-R, denoted by F(L,R), and by extension the spread functions of n itself. The support of n (or equivalently of µ n ) is thus ]n -α, n +β[ when α = 0 and β = 0, [n, n +β[ when α = 0 and β = 0, ]n -α, n] when α = 0 and β = 0 and the singleton {n} when α = β = 0. The triangular family, formed by the triangular fuzzy numbers, is the family F(L,R) where the spread functions L and R are both linear. In this case, as L(0) = R(0) = 1 and L(1) = R(1) = 0, we have L = R and this spread function equals the function F defined by F(x) = -x + 1. Let 3 be a triangular fuzzy number with tuple representation (3, 2, 2). Figures 2 and 3 describe respectively the 1/2-cut E 1/2 = [2, 4] and the 0-cut E 0 = [3-α, 3+β] = [1, 5] where E = 3.

FIGURE 2 .

 2 FIGURE 2. Finite support ]1, 5[: 1/2-cut E 1/2 = [2, 4] of the triangular fuzzy number E = 3 = (3, 2, 2).

FIGURE 3 .

 3 FIGURE 3. Finite support ]1, 5[: 0-cut E 0 = [1, 5] of the triangular fuzzy number E = 3 = (3, 2, 2).

FIGURE 4 .

 4 FIGURE 4. Graph of functions 3 and 3 from the graph of a linear membership function

FIGURE 5 . 2 . 9 .

 529 FIGURE 5. Functions 3 and 3 from the graph of a quadratic membership function

(

( 4 )

 4 n(r ) = α r + n -α et n(r ) = -β r + n + β . Proof. For the real numberr ∈ [0, 1], Definition 2.5 implies r = L n-n(r ) α = R n(r )-n β . As L and R are bijective, n(r ) and n(r ) satisfy Identity (3) of the proposition.

  the sign 1 or -1 of the real a d and we have a d = ε(a) d |a d | = ε(a) d |a| d . The evaluation in the value a of a term n d x d consists in replacing the monomial x d by its value a d = ε(a) d |a| d . According to Identity (3) of Lemma 2.8, when the sign ε(a) d of a d is 1, the r -cut of the fuzzy number n d a d is

Lemma 3 . 2 .

 32 Let I ∈ {-1, 1} k and E(I) be an induced equation of equation (E). Then Sol(E) = I ⊗ Sol(E(I)) and Sol(E(I)) = I ⊗ Sol(E) .

Theorem 3 . 3 .Remark 3 . 4 .

 3334 Consider a fuzzy algebraic equation (E). Then the set of real solutions of (E) is formed by the I × b where I runs throughout the set of k-uplets {-1, 1} k and b runs throughout the set of positive solutions of E(I). In other words, Sol(E) = I∈{-1,1} k I ⊗ Sol + (E(I)) . Proof. It follows from Lemma 3.2 that each I ⊗ Sol + (E(I)) ⊂ Sol(E). Conversely, if a ∈ Sol(E) then a d = ε(a) d |a d | = ε(a) d |a| d . It follows that m = d a d n d = d ε(a) d |a| d n d . Putting I = ε(a) and b = |a| ∈ R + k , we obtain I × b = ε(a) × (ε(a) × a) = a where b is a solution of E(I) The coefficients in (E) being given by their tuple representation, the equations E(I) actually derive directly from E. For every d in Supp(E) the coefficient I d n d is equal to n d itself when I d = 1, otherwise it is equal to (-n d , β d , α d ) when I d = -1, following Equality (2). This ensures that for finding the whole set of real solutions of an equation (E) it is sufficient to be able to compute positives real solutions of any fuzzy equation. In practice, the 2 k fuzzy equations E(I) are not always pairwise distinct. In particular, in case of each component d i of any d = (d 1 , . . . , d k ) ∈ Supp(E) is even, all induced equations E(I) are identical to equation (E). Example 3.5. Take k = 3 variables x 1 , x 2 , x 3 and consider the equation

Theorem 3 . 7 .

 37 Let L and R be two spread functions and (E) : d∈Supp(E) n d x d = m , be a fuzzy equation with coefficients in the family F(L,R) given by their tuple representations as follows: m = (m, α, β) and n d = (n d , α d , β d ) for d ∈ Supp(E). If the spread functions L and Rare bijective then the crisp form of (E) is given by:[START_REF] Dubois | Fuzzy Interval Analysis[END_REF]

Definition 3 . 8 .

 38 Let L and R be two spread functions and (E) :d∈Supp(E) n d x d = m ,a fuzzy equation with coefficients in the family F(L,R) given by their representations in tuple as follows:

  When the equation (E) : d∈Supp(E) n d x d = m has trapezoidal fuzzy coefficients of type L-R, where n d = (a d , b d , α d , β d ) and m = (a, b, α, β) are the tuple representations of the coefficients, the parametric forms (10) given in the proof of Theorem 3.7 become

3 and Corollary 3. 10 : 4 . 1 .

 1041 Theorem Let (S) be a fuzzy system with coefficients in F(L,R). If the spread functions L and R are bijective then the set of solution of (S) is the union of the I×b for all k-uplets b which are positives real solutions of the real transform T (S(I)) where I runs throughout {-1, 1} k . In other words,

4 . 2 .

 42 sol := { } for j := 0 to 2 k -1 do I := MultiSign( j ) TRSI := RealTransform(S, I) PR := SolPos(TRSI) sol := sol I ⊗ PR end for return sol Reduction of the number of algebraic systems to solve.

  For this, let us construct the matrix M(S) of signs where the columns are indexed by the 2 k k -uplets I of {-1, 1} k and the lines by the k -uplets d of Supp(S). The element at line d and column I is I d ∈ {-1, 1} which replaces the sign of x d in S(Let I 1 , I 2 ∈ {-1, 1} k two distinct k -uplets and consider C(I 1 ) and C(I 2 ) the two columns of M(S) respectively indexed by I 1 and I 2 . If C(I 1 ) = C(I 2 ) then S(I 1 ) = S(I 2 ). Indeed, for every equation (E) of the system (S), the coefficient I d 1 n d of each monomial x d appearing in the left hand side of each equation E(I 1 ) is identical to the coefficient I d 2 n d in E(I 2 )

Algorithm 2

 2 SolveFuzzySystem or SFS, an optimized algorithm for solving fuzzy systems Require: S, a polynomial system with fuzzy coefficients under tuple representation k, the dimension, i.e. the number of variables Data: c pt := -1, the counter (up to a unit) of the number of distinct columns of M(S) Di st i nct Col umns := [], the distinct columns of M(S) already scanned Di st i nct Sy st ems := [], the distinct systems S(I) already met l b := [], l b[i ] will be the set Sol + (S(I)) when C(I) will be Di st i nc t Col umns[i ] sol := { } Ensure: sol , the set of real solutions of (S) # the index j runs throughout the columns of M(S) which are indexed by MultiSign( j ) 1: for j := 0 to 2 k -1 do 2: I := MultiSign( j ) C := SignColumn( j ) # C is the current column in M(S) # test if C is equal to a previous column of M(S) i := IsIn( C , Di st i nct Col umns) [i ] contains the positive real solutions of S(I): apply Theorem4.1 sol := sol I ⊗ l b[i ] # move to the next column in M(S) and c pt is not incremented go C is a new column c pt := cpt + 1 Di st i nct Col umns[cpt ] := C # test if S(I) is a new system i := IsIn( S(I) , Di st i nct Sy st ems ) i -th system equals S(I). Its positive real solutions in l b[i ] are copied in l b[c pt ] because C is a "new" column l b[cpt ] := l b[i ] new system at cpt index, calculate its positive real solutions and store them in l b[cpt ] Di st i nct Sy st ems[cpt ] = S(I) TRSI := RealTransform(S(I)) l b[cpt ] := SolPos(TRSI) sol := sol I ⊗ l b[cpt ] 11: end for 12: return sol terminated, its positive solutions will be stored in a new vector SPos, with same indexes as Di st i nct Sy st ems.

following way: • in 4 :Remark 5 . 1 .

 451 replace the statement sol := sol I ⊗ l b[i ] by l b[ j ] := l b[i ]. • in 7: replace the statement l b[c pt ] := l b[i ] by l b[ j ] := l b[i ]. • in 9: replace both statements TRSI := RealTransform(S(I)) and l b[c pt ] := SPos (TRSI) by l b[ j ] := cpt . (2) In parallel, solves every distinct system of the vector Di st i nc t Sy st ems. For each system SI := Di st i nct Sy st ems[c pt ], if it exists (see Remark 5.1), we apply the following statements: TRSI := RealTransform(SI) SPos[cpt ] := SolPos(TRSI) (3) Cross over the vector l b to build all the solutions of the system (S) from the results of previous step ; It consists mainly in applying Theorem 4.1. This last step can also be performed in parallel. Each l b[ j ] owns the value of the c pt index of positive real solutions of S(I) in the vector SPos where I = MultiSign( j ). This step is realized by the following loop: for j := 0 to 2 k -1 do sol := sol MultiSign( j ) ⊗ SPos[l b[ j ]] end for As in algorithm SFS, when C = C(I) is a new column butS(I) is not a new system, nothing is stored at the corresponding index in Di st i nc t Sy st ems.

•

  a fuzzy number n = (n, a, b) with spread functions L and R is represented by :rep( n) = NombreFlouRed((n, a, b,rep(L), rep(R)); • as usually, a term x d is represented by rep(x d ) = x 1 * * d 1 • • • x k * * d k ; • a monomial M = n x d isrepresented by the pair rep(M) = (rep( n), rep(x d ) ); • a polynomial p with fuzzy coefficients is represented by rep(p), the list of the rep(M) where M ∈ Supp(P) ; i.e. it is a sparse representation; • a polynomial equation (E) : p = q is represented by the pair rep((E))=(rep(p), rep(q)) ; • a system (S) formed by s equations (E 1 ), . . . , (E s ) is represented by the list rep((S))=[rep((E 1 )), . . . , rep((E 2 ))].

  Now let us present the trace of the function resolution_reelle_systemes_flous(F,2) by linking the real parameters to formal parameters: S=F et k=2. At the beginning, we have: cpt = -1 , DistinctColumns = [ ] and DistinctSystems = [ ]. j=0 : I=I_{0}=[-1,-1], C=[-1,1,1] As the column C is new cpt=cpt+1 ; i.e. cpt =0 DistinctColumns[0] = C Sr = transformee_reelle(S,I) gives Sr=[-y^2+1,2*y^2-x-1,-2*y^2-x+3] DistinctSystems[0] = Sr SolPos(Sr) gives lb[0]=set([(1, 1)]), the positives real solutions of Sr lb[0]= set([(1, 1)]) the product of I with (1, 1) in lb[0] is added to sol: sol = set([(-1, -1)]) DistinctColumn=[[1,-1,1]] DistinctSystems=[[y^2-1,2*y^2-x-1,2*y^2+x-3]] j=1 : I=I_{1}=[-1,1], C=[-1,1,1] As C equals DistinctColumns[0]

  DistinctColumns = [[1, 1, 1, 1], [-1, 1, -1, 1]] DistinctSystems= [[2*x^3*y^3+2*x^2*y^2+5*x*y-9,2*x^3*y^3+3*x^2*y^2+2*x*y-7, 2*x^3*y^3+3*x^2*y^2+x*y-6, x^3*y^3+x^2*y^2+x*y-3], [2*x^3*y^3-3*x^2*y^2+2*x*y+7,2*x^3*y^3-3*x^2*y^2+x*y+6, x^3*y^3-x^2*y^2+x*y+3, 2*x^3*y^3-2*x^2*y^2+5*x*y+9]] lb = [0,1]

  y, 'R+')]) sol = set([(1/y, 'R+'), (1/y, 'R-')])

Acknowlegment We warmly thank Didier Dubois and Henri Prade for clarifying some aspects of the fuzzy numbers. We also thank Binh-Minh Bui-Xuan for his remarks following his careful reading of the paper.

the product of I with [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF][START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF] in lb[0] is added to sol: sol = set([(-1, 1), (-1, -1)]) j=2 : I=I_{2}= [1,-1], C= [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF][START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF][START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF] As the column C is new cpt=cpt+1 ; i.e. cpt =1 DistinctColumns [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF] = C Sr = transformee_reelle(S,I) gives Sr=[2*y^2+x-1,-2*y^2+x+3,-y^2+1] DistinctSystems [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF] = Sr SolPos(Sr) gives lb [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF]= set([]), the positive real solutions of Sr any product of elements of lb [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF] with I is added in sol: sol = set([(-1, 1), (-1, -1)])

We have

DistinctColumn=

any product of elements of lb [START_REF] Abbasbandy | Numerical approximation of fuzzy functions by fuzzy polynomials[END_REF] with I is added in sol: sol = set([(-1, 1), (-1, -1)])

Therefore we find the variety V(F) = {(x = -1, y ± 1)}. Example 2 : Detailed example of the parallel version of the algorithm SFS on the following fuzzy system F 1 :

(2, 1, 1)x y + (3, 1, 1)x 2 y 2 + (2, 1, 1)x 3 y 3 = (7, 3, 3),

(5, 1, 1)x y + (2, 3, 1)x 2 y 2 + (2, 2, 1)x 3 y 3 = (9, 6, 3)

The solution returned by the function resolution_reelle_systemes_flous(F 1 ,2) of the Fuzzy package is the variety V(F 1 ) = {(x = 1 y , y) | y ∈ R\{0})}, like in [START_REF] Boroujeni | Finding solutions of fuzzy polynomial equations systems by an algebraic method[END_REF]. We notice that in the first equation (E), the left spread of each coefficient equals the right spread. Thus, in the real transform of (E), both equations

are equal to x 3 y 3 + x 2 y 2 + x y = 3. We find again this equation with the right spreads of the second equation of F 1 . This is why we will have 4 and not 6 = 3s equations in the real transform of any induced fuzzy equation F 1 (I). During an implementation, it is possible to take this into account in order to identify the identical equations once and for all on the initial fuzzy system and then use the result on each of its induced systems.

We will find V(F 1 ) by applying the parallel version of our algorithm. The multi-signs are the same as for the F system of the first example and the support of F 1 is {x 3 y 3 , x 2 y 2 , x y, 1}. The matrix of signs of F 1 is the following: