2.1 Scientific ground

2.2 Synergies

Outside applications which can clearly be seen as transversal acitivies, our development directions are linked at several levels : shared computable objects, computational strategies and transversal research directions.

Sharing basic algebraic objects As seen above, is the well-known fact that the elimination theory for functional systems is deeply intertwined with the one for polynomial systems so that, topology in small dimension, applications in control theory, signal theory and robotics share naturally a large set of computable objects developped in our project team.

Performing efficient basic arithmetic operations in number fields is also a key ingredient to most of our algorithms, in Number theory as well as in topology in small dimension or , more generally in the use of roots of polynomials systems. In particular, finding good representations of number fields, lead to the same computational problems as working with roots of polynomial systems by means of triangular systems (towers of number fields) or rational parameterizations (unique number field). Making any progress in one direction will probably have direct consequences for almost all the problems we want to tackle.

Elimination theory is also deeply connected to Gröbner bases and rewriting, which are themselves linked to Garside theory and Koszul duality, establishing a continuum with the effective methods studied in algebraic topology.

Symbolic-numeric strategies. Several general low-level tools are also shared such as the use of approximate arithmetic to speed up certified computations. Sometimes these can also lead to improvement for a different purpose (for example computations over the rationals, deeply used in geometry can often be performed in parallel combining computations in finite fields together with fast Chinese remaindering and modular evaluations).

As simple example of this sharing of tools and strategies, the use of approximate arithmetic is common to the work on LLL (used in the evaluation of the security of cryptographic systems), resolutions of real-world algebraic systems (used in our applications in robotics, control theory, and signal theory), computations of signs of trigonometric expressions used in knot theory or to certified evaluations of dilogarithm functions on an algebraic variety for the computation of volumes of representations in our work in topology, numerical integration and computations of $L$-functions.

Transversal research directions. The study of the topology of complex algebraic curves is central in the computation of periods of algebraic curves (number theory) but also in the study of character varieties (topology in small dimension) as well as in control theory (stability criteria). Very few computational tools exists for that purpose and they mostly convert the problem to the one of variety over the reals (we can then recycle our work in computational geometry).

As for real algebraic curves, finding a way to describe the topology (an equivalent to the graph obtained in the real case) or computing certified drawings (in the case of a complex plane curve, a useful drawing is the so called associated amoeba) are central subjects for Ouragan.

As mentioned in the section 3.3 the computation of the Mahler measure of an algebraic implicit curve is either a challenging problem in number theory and a new direction in topology. The basic formula requires the study of points of moduli 1 , as for stability problems in Control Theory (stability problems), and certified numerical evaluations of non algebraic functions at algebraic points as for many computations for $L$-Functions.

3.1 Basic computable objects and algorithms

The development of basic computable objects is somehow on demand and depends on all the other directions. However, some critical computations are already known to be bottlenecks and are sources of constant efforts.

Computations with algebraic numbers appear in almost all our activities: when working with number fields in our work in algorithmic number theory as well as in all the computations that involve the use of solutions of zero-dimensional systems of polynomial equations. Among the identified problems: finding good representations for single number fields (optimizing the size and degree of the defining polynomials), finding good representations for towers or products of number fields (typically working with a tower or finding a unique good extension), efficiently computing in practice with number fields (using certified approximation vs working with the formal description based on polynomial arithmetics). Strong efforts are currently done in the understanding of the various strategies by means of tight theoretical complexity studies 76, 120, 56 and many other efforts will be required to find the right representation for the right problem in practice. For example, for isolating critical points of plane algebraic curves, it is still unclear (at least the theoretical complexity cannot help) that an intermediate formal parameterization is more efficient than a triangular decomposition of the system and it is still unclear that these intermediate computations could be dominated in time by the certified final approximation of the roots.

3.2 Algorithmic Number Theory

Concerning algorithmic number theory, the main problems we will be considering in the coming years are the following:

• Number fields. We will continue working on the problems of class groups and generators. In particular, the existence and accessibility of good defining polynomials for a fixed number field remain very largely open. The impact of better polynomials on the algorithmic performance is a very important parameter, which makes this problem essential.
• Lattice reduction. Despite a great amount of work in the past 35 years on the LLL algorithm and its successors, many open problems remain. We will continue the study of the use of interval arithmetic in this field and the analysis of variants of LLL along the lines of the Potential-LLL which provides improved reduction comparable to BKZ with a small block size but has better performance.
• Elliptic curves and Drinfeld modules. The study of elliptic curves is a very fruitful area of number theory with many applications in crypto and algorithms. Drinfeld modules are “cousins” of elliptic curves which have been less explored in the algorithm context. However, some recent advances  80 have used them to provide some fast sophisticated factoring algorithms. As a consequence, it is natural to include these objects in our research directions.

3.3 Topology in small dimension

3.4 Algebraic analysis of functional systems

We want to further develop our expertise in the computational aspects of algebraic analysis by continuing to develop effective versions of results of module theory, homological algebra, category theory and sheaf theory 142 which play important roles in algebraic analysis 51, 108, 109 and in the algorithmic study of linear functional systems. In particular, we shall focus on linear systems of integro-differential-constant/varying/distributed delay equations 128, 131 which play an important role in mathematical systems theory, control theory, and signal processing 128, 137, 133, 134.

The rings of integro-differential operators are highly more complicated than the purely differential case (i.e. Weyl algebras) 15, due to the existence of zero-divisors, or the fact of having a coherent ring instead of a noetherian ring 48. Therefore, we want to develop an algorithmic study of these rings. Following the direction initiated in 131 for the computation of zero divisors (based on the polynomial null spaces of certain operators), we first want to develop algorithms for the computation of left/right kernels and left/right/generalized inverses of matrices with entries in such rings, and to use these results in module theory (e.g. computation of syzygy modules, (shorter/shortest) free resolutions, split short/long exact sequences). Moreover, Stafford's results 143, algorithmically developed in 15 for rings of partial differential operators (i.e. the Weyl algebras), are known to still hold for rings of integro-differential operators. We shall study their algorithmic extensions. Our corresponding implementation will be extended accordingly.

Finally, within a computer algebra viewpoint, we shall continue to algorithmically study issues on rings of integro-differential-delay operators 128, 133 and their applications to the study of equivalences of differential constant/varying/distributed delay systems (e.g. Artstein's reduction, Fiagbedzi-Pearson's transformation) which play an important role in control theory.

4.1 Security of cryptographic systems

The study of the security of asymmetric cryptographic systems comes as an application of the work carried out in algorithmic number theory and revolves around the development and the use of a small number of general purpose algorithms (lattice reduction, class groups in number fields, discrete logarithms in finite fields, ...). For example, the computation of generators of principal ideals of cyclotomic fields can be seen as one of these applications since these are used in a number of recent public key cryptosystems.

The cryptographic community is currently very actively assessing the threat coming for the development of quantum computers. Indeed, such computers would permit tremendous progress on many number theoretic problems such as factoring or discrete logarithm computations and would put the security of current cryptosystem under a major risk. For this reason, there is a large global research effort dedicated to finding alternative methods of securing data. For example, the US standardization agency called NIST has recently launched a standardization process around this issue. In this context, OURAGAN is part of the competition and has submitted a candidate (which has not been selected) 46. This method is based on number-theoretic ideas involving a new presumably difficult problem concerning the Hamming distance of integers modulo large numbers of Mersenne.

4.2 Robotics

Algebraic computations have tremendously been used in Robotics, especially in kinematics, since the last quarter of the 20th century 99. For example, one can find algebraic proofs for the 40 possible solutions to the direct kinematics problem 123 for steward platforms and companion experiments based on Gröbner basis computations 89. On the one hand, hard general kinematics problems involve too many variables for pure algebraic methods to be used in place of existing numerical or semi-numerical methods everywhere and everytime, and on the other hand, global algebraic studies allow to propose exhaustive classifications that cannot be reached by other methods,for some quite large classes.

Robotics is a long-standing collaborative work with LS2N (Laboratory of Numerical Sciences of Nantes). Work has recently focused on the offline study of mechanisms, mostly parallel, their singularities or at least some types of singularities (cuspidals robots 147).

For most parallel or serial manipulators, pose variables and joints variables are linked by algebraic equations and thus lie an algebraic variety. The two-kinematics problems (the direct kinematics problem - DKP- and the inverse kinematics problem - IKP) consist in studying the preimage of the projection of this algebraic variety onto a subset of unknowns. Solving the DKP remains to computing the possible positions for a given set of joint variables values while solving the IKP remains to computing the possible joints variables values for a given position. Algebraic methods have been deeply used in several situations for studying parallel and serial mechanisms, but finally their use stays quite confidential in the design process. Cylindrical Algebraic Decomposition coupled with variable's eliminations by means of Gröbner based computations can be used to model the workspace, the joint space and the computation of singularities. On the one hand, such methods suffer immediately when increasing the number of parameters or when working with imprecise data. On the other hand, when the problem can be handled, they might provide full and exhaustive classifications. The tools we use in that context 66, 65, 101, 103, 102 depend mainly on the resolution of parameter-based systems and therefore of study-dependent curves or flat algebraic surfaces (2 or 3 parameters), thus joining our thematic Computational Geometry.

4.3 Control theory

Certain problems studied in mathematical systems theory and control theory can be better understood and finely studied by means of algebraic structures and methods. Hence, the rich interplay between algebra, computer algebra, and control theory has a long history.

For instance, the first main paper on Gröbner bases written by their creators, Buchberger, was published in Bose's book 52 on control theory of multidimensional systems. Moreover, the differential algebra approach to nonlinear control theory (see 79, 78 and the references therein) was a major motivation for the algorithmic study of differential algebra 53, 82. Finally, the behaviour approach to linear systems theory 148, 125 advocates for an algorithmic study of algebraic analysis (see Section 2.1.4). More generally, control theory is porous to computer algebra since one finds algebraic criteria of all kinds in the literature even if the control theory community has a very few knowledge in computer algebra.

OURAGAN has a strong interest in the computer algebra aspects of mathematical systems theory and control theory related to both functional and polynomial systems, particularly in the direction of robust stability analysis and robust stabilization problems for multidimensional systems 52, 125 and infinite-dimensional systems 72 (such as, e.g., differential time-delay systems).

Let us shortly state a few points of our recent interests in this direction.

In control theory, stability analysis of linear time-invariant control systems is based on the famous Routh-Hurwitz criterion (late 19th century) and its relation with Sturm sequences and Cauchy index. Thus, stability tests were only involving tools for univariate polynomials 107. While extending those tests to multidimensional systems or differential time-delay systems, one had to tackle multivariate problems recursively with respect to the variables 52. Recent works use a mix of symbolic/numeric strategies, Linear Matrix Inequalities (LMI), sums of squares, etc. But still very few practical experiments are currently involving certified algebraic computations based on general solvers for polynomial equations. We have recently started to study certified stability tests for multidimensional systems or differential time-delay systems with an important observation: with a correct modelization, some recent algebraic methods $-$ derived from our work in algorithmic geometry and shared with applications in robotics $-$ can now handle previously impossible computations and lead to a better understanding of the problems to be solved 58, 59, 61. The previous approaches seem to be blocked on a recursive use of one-variable methods, whereas our approach involves the direct processing of the problem for a larger number of variables.

The structural stability of $n$-D discrete linear systems (with $n\ge 2$) is a good source of problems of several kinds ranging from solving univariate polynomials to studying algebraic systems depending on parameters. For instance, we show 60, 59, 61 that the standard characterization of the structural stability of a multivariate rational transfer function (namely, the denominator of the transfer function does not have solutions in the unit polydisc of ${ℂ}^n$) is equivalent to deciding whether or not a certain system of polynomial equations has real solutions. The use state-of-the-art computer algebra algorithms to check this last condition, and thus the structural stability of multidimensional systems has been validated in several situations from toy examples with parameters to state-of-the-art examples involving, e.g., the resolution of bivariate systems 57, 56.

The rich interplay between control theory, algebra, and computer algebra is also well illustrated with our recent work on robust stabilization problems for multidimensional and finite/infinite-dimensional systems 54, 130, 135, 138, 136, 137.

4.4 Signal processing

Due to numerous applications (e.g. sensor network, mobile robots), sources and sensors localization has intensively been studied in the literature of signal processing. The anchor position self calibration problem is a well-known problem which consists in estimating the positions of both the moving sources and a set of fixed sensors (anchors) when only the distance information between the points from the different sets is available. The position self-calibration problem is a particular case of the Multidimensional Unfolding (MDU) problem for the Euclidean space of dimension 3. In the signal processing literature, this problem is attacked by means of optimization problems (see 71 and the references therein). Based on computer algebra methods for polynomial systems, we have recently developed a new approach for the MDU problem which yields closed-form solutions and a very efficient algorithm for the estimation of the positions 73 based only on linear algebra techniques. This first result, done in collaboration with Dagher (Inria Chile) and Zheng (DEFROST, Inria Lille), yielded a recent patent 74. This result advocates for the study of other localization problems based on the computational polynomial techniques developed in OURAGAN.

In collaboration with Safran Tech (Barau, Hubert) and Dagher (Inria Chile), a symbolic-numeric study of the new multi-carrier demodulation method98 has recently been initiated. Gear fault diagnosis is an important issue in aeronautics industry since a damage in a gearbox, which is not detected in time, can have dramatic effects on the safety of a plane. Since the vibrations of a spur gear can be modeled as a product of two periodic functions related to the gearbox kinematic, it is proposed to recover each function from the global signal by means of an optimal reconstruction problem which, based on Fourier analysis, can be rewritten as ${\mathrm{argmin}}_{u\in {ℂ}^n,v_1,v_2\in {ℂ}^m}\parallel M-u{v}_1^{☆}-Du{v}_2^{☆}{\parallel }_F$, where $M\in {ℂ}^{n\times m}$ (resp. $D\in {ℂ}^{n\times n}$) is a given matrix with a special shape (resp. diagonal matrix), $\parallel ·{\parallel }_F$ is the Frobenius norm, and $v^{☆}$ is the Hermitian transpose of $v$. We have recently obtained closed-form solutions for the exact problem, i.e., $M=u{v}_1^{☆}+Du{v}_2^{☆}$, which is a polynomial system with parameters. This first result gives interesting new insides for the study of the non-exact case, i.e. for the above optimization problem.

Our expertise on algebraic parameter estimation problem, developed in the former Non-A project-team (Inria Lille), will be further developed. Following this work 90, the problem consists in estimating a set $\theta$ of parameters of a signal $x(\theta ,t)$$-$ which satisfies a certain dynamics $-$ when the signal $y\left(t\right)=x(\theta ,t)+\gamma \left(t\right)+\varpi \left(t\right)$ is observed, where $\gamma$ denotes a structured perturbation and $\varpi$ a noise. It has been shown that $\theta$ can sometimes be explicitly determined by means of closed-form expressions using iterated integrals of $y$. These integrals are used to filter the noise $\varpi$. Based on a combination of algebraic analysis techniques (rings of differential operators), differential elimination theory (Gröbner basis techniques for Weyl algebras), and operational calculus (Laplace transform, convolution), an algorithmic approach to algebraic parameter estimation problem has been initiated in 133 for a particular type of structured perturbations (i.e. bias) and was implemented in the Maple prototype NonA. The case of a general structured perturbation is still lacking.

6.1 Limit sets

Limit sets in the 3-dimensional sphere are a general generalization of the well-known fractal sets in the plane given by the complex dynamic, a.k.a Julia and Mandelbrot sets. The efficient computation and rendering of the latter has been a powerful drive for their theoretical and algorithmic study and can now be considered as achieved (real time rendering, arbitrary precision...). However, passing to dimension 3 is a challenge and is yet at its first steps. The website limit-sets.imj-prg.fr presents the first result of an effort leaded by R. Alexandre and A. Guilloux to compute and render such limit sets. Though still experimental, they are already used for improving the theoretical understanding of these limit sets. Moreover, one can hope that their visual attractiveness helps attract interest for this fields of research.

6.2 Hecke Grossencharacters

A collaboration between Pascal Molin and Aurel Page (Inria Bordeaux, Lfant team) recently added to the Pari/GP number theory software the support of Hecke Grossencharacters. These objects have been mathematically understood from the 1950s and establish explicit links between remote parts of number theory (algebraic curves, galois representations, L functions). Despite their immense computational interest, there has been no computer implementation until a partial Magma package in 2015. This is due to the fact that these characters are defined on infinite spaces which cannot be represented on a computer. The algorithm introduced by Molin and Page is the first to give access to the full family of Hecke characters, with routine interfaces to Pari/GP.

6.3 Parallel mechanisms

In 2021, we did sign a contract with Safran Technologies and Defense and the CNRS Laboratory LS2B (Nantes) for working on the design of parallel manipulators. After few months it has been decided that this contrat will be extended 3 years more at least and a co-dupervised PhD will start on the subject in January 2022.

7.1 New software

7.2 New platforms

No new platforms.

8.1 Number Theory

8.2 Computer Algebra

8.3 Algebraic Analysis

8.4 Geometry

8.5 Control Theory

8.6 Algebraic aspects of a rank factorization problem arising in vibration analysis

The article 21 continues the study of a rank factorization problem arising in gear fault surveillance. The structure of a class of solutions – important in practice – of the rank factorization problem is studied. We show that these solutions can be parametrized. Using module theory and computer algebra methods, the parameter space P is explicitly characterized and is shown to be the complementary of an algebraic set. Finally, a finite open cover of P is obtained and for each basic open subset of the cover of P, a closed-form solution is characterized.

9.1 Bilateral contracts with industry

• The objective of our Agrement with WATERLOO MAPLE INC. is to promote software developments to which we actively contribute.

  On the one hand, WMI provides man power, software licenses, technical support (development, documentation and testing) for an inclusion of our developments in their commercial products. On the other hand, OURAGAN offers perpetual licenses for the use of the concerned source code.

  As past results of this agreement one can cite our C-Library RS for the computations of the real solutions zero-dimensional systems or also our collaborative development around the Maple package DV for solving parametric systems of equations.

  For this term, the agreement covers algorithms developed in areas including but not limited to: 1) solving of systems of polynomial equations, 2) validated numerical polynomial root finding, 3) computational geometry, 4) curves and surfaces topology, 5) parametric algebraic systems, 6) cylindrical algebraic decompositions, 7) robotics applications.

  In particular, it covers our collaborative work with some of our partners, especially the Gamble Project-Team - Inria Nancy Grand Est.

• A research contract was signed with the company Safran Electronics & Defense on the study of parallel mecanisms.

10.1 International initiatives

10.2 National initiatives

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

11.3 Popularization

12.1 Major publications

• 1 articleY.Yacine Bouzidi, S.Sylvain Lazard, G.Guillaume Moroz, M.Marc Pouget, F.Fabrice Rouillier and M.Michael Sagraloff. Solving bivariate systems using Rational Univariate Representations.Journal of Complexity372016, 34--75
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• 2 articleE.Erwan Brugallé, P.-V.Pierre-Vincent Koseleff and D.Daniel Pecker. On the lexicographic degree of two-bridge knots.Journal Of Knot Theory And Its Ramifications (JKTR)25714p., 21 figsJune 2016
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• 3 articleE.Erwan Brugallé, P.-V.Pierre-Vincent Koseleff and D.Daniel Pecker. Untangling trigonal diagrams.Journal Of Knot Theory And Its Ramifications (JKTR)25710p., 24 figsJune 2016
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• 4 articleF.Frédéric Chyzak, A.Alban Quadrat and D.Daniel Robertz. Effective algorithms for parametrizing linear control systems over Ore algebras.Applicable Algebra in Engineering, Communications and Computing162005, 319--376
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• 5 articleT.Thomas Cluzeau and A.Alban Quadrat. Factoring and decomposing a class of linear functional systems.Linear Algebra and Its Applications4282008, 324--381
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• 6 articleE.Elisha Falbel and A.Antonin Guilloux. Dimension of character varieties for 3-manifolds.Proceedings of the American Mathematical Society2016
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• 7 articleE.Elisha Falbel, A.Antonin Guilloux, P.-V.Pierre-Vincent Koseleff, F.Fabrice Rouillier and M.Morwen Thistlethwaite. Character Varieties For SL(3,C): The Figure Eight Knot.Experimental Mathematics2522016, 17
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• 8 articleE.Elisha Falbel and J.Jieyan Wang. Branched spherical CR structures on the complement of the figure-eight knot.Michigan Mathematical Journal632014, 635-667
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• 9 articleS.Stéphane Gaussent, Y.Yves Guiraud and P.Philippe Malbos. Coherent presentations of Artin monoids.Compositio Mathematica15152015, 957-998
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• 10 articleY.Yves Guiraud, E.Eric Hoffbeck and P.Philippe Malbos. Convergent presentations and polygraphic resolutions of associative algebras.Mathematische Zeitschrift2931-22019, 113-179
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• 11 articleY.Yves Guiraud and P.Philippe Malbos. Higher-dimensional normalisation strategies for acyclicity.Advances in Mathematics2313-42012, 2294-2351
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• 12 articleA.Antoine Joux. A one round protocol for tripartite Diffie-Hellman.J. Cryptology1742004, 263--276
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• 13 articleA.Antoine Joux and R.Reynald Lercier. Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the gaussian integer method.Math. Comput.722422003, 953-967
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• 14 articleD.Daniel Lazard and F.Fabrice Rouillier. Solving Parametric Polynomial Systems.Journal of Symbolic Computation42June 2007, 636-667
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• 15 articleA.Alban Quadrat and D.Daniel Robertz. Computation of bases of free modules over the Weyl algebras.Journal of Symbolic Computation422007, 1113--1141
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• 16 articleF.Fabrice Rouillier. Solving zero-dimensional systems through the rational univariate representation.Journal of Applicable Algebra in Engineering, Communication and Computing951999, 433--461
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• 17 articleF.Fabrice Rouillier and P.Paul Zimmermann. Efficient Isolation of Polynomial Real Roots.Journal of Computational and Applied Mathematics16212003, 33--50
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12.2 Publications of the year

12.3 Other

12.4 Cited publications

• 46 inproceedingsD.Divesh Aggarwal, A.Antoine Joux, A.Anupam Prakash and M.Miklos Santha. A New Public-Key Cryptosystem via Mersenne Numbers.Advances in Cryptology - CRYPTO 2018 - 38th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 19-23, 2018, Proceedings, Part III2018, 459--482URL: https://doi.org/10.1007/978-3-319-96878-0_16
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• 47 bookS.Saugata Basu, R.Richard Pollack and M.-F.Marie-Françoise Roy. Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics).Berlin, HeidelbergSpringer-Verlag2006
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• 48 articleV.Vladimir Bavula. The algebra of integro-differential operators on an affine line and its modules.J. Pure Appl. Algebra2172013, 495--529
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• 49 articleN.Nicolas Bergeron, E.Elisha Falbel and A.Antonin Guilloux. Tetrahedra of flags, volume and homology of SL(3).Geometry & Topology Monographs182014
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• 50 inproceedingsJ.-F.Jean-François Biasse, T.Thomas Espitau, P.-A.Pierre-Alain Fouque, A.Alexandre Gélin and P.Paul Kirchner. Computing generator in cyclotomic integer rings.36th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2017)10210Lecture Notes in Computer ScienceParis, FranceApril 2017, 60-88
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• 51 bookA.A. Borel. Algebraic D-modules.Perspectives in mathematicsAcademic Press1987
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• 52 bookN.N.K. Bose. Multidimensional Systems Theory: Progress, Directions and Open Problems in Multidimensional Systems.Mathematics and Its ApplicationsSpringer Netherlands2001
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• 53 articleF.François Boulier, D.Daniel Lazard, F.François Ollivier and M.Michel Petitot. Computing representations for radicals of finitely generated differential ideals.Applicable Algebra in Engineering, Communication and Computing202009, 73--121
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• 54 inproceedingsY.Yacine Bouzidi, T.Thomas Cluzeau, G.Guillaume Moroz and A.Alban Quadrat. Computing effectively stabilizing controllers for a class of $n$D systems.The 20th World Congress of the International Federation of Automatic Control501Toulouse, FranceJuly 2017, 1847 -- 1852
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• 55 inproceedingsY.Yacine Bouzidi, S.Sylvain Lazard, G.Guillaume Moroz, M.Marc Pouget and F.Fabrice Rouillier. Improved algorithm for computing separating linear forms for bivariate systems.ISSAC - 39th International Symposium on Symbolic and Algebraic ComputationKobe, JapanJuly 2014
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• 56 articleY.Yacine Bouzidi, S.Sylvain Lazard, G.Guillaume Moroz, M.Marc Pouget, F.Fabrice Rouillier and M.Michael Sagraloff. Solving bivariate systems using Rational Univariate Representations.Journal of Complexity372016, 34--75
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• 57 articleY.Yacine Bouzidi, S.Sylvain Lazard, M.Marc Pouget and F.Fabrice Rouillier. Separating linear forms and Rational Univariate Representations of bivariate systems.Journal of Symbolic Computation680May 2015, 84-119
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• 58 incollectionY.Yacine Bouzidi, A.Adrien Poteaux and A.Alban Quadrat. A symbolic computation approach to the asymptotic stability analysis of differential systems with commensurate delays.Delays and Interconnections: Methodology, Algorithms and ApplicationsAdvances on Delays and Dynamics at SpringerSpringer VerlagMarch 2017
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• 59 unpublishedY.Yacine Bouzidi, A.Alban Quadrat and F.Fabrice Rouillier. Certified Non-conservative Tests for the Structural Stability of Multidimensional Systems.August 2017, To appear in Multidimensional Systems and Signal Processing, https://link.springer.com/article/10.1007/s11045-018-0596-y
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• 60 inproceedingsY.Y. Bouzidi, A.Alban Quadrat and F.Fabrice Rouillier. Computer algebra methods for testing the structural stability of multidimensional systems. IEEE 9th International Workshop on Multidimensional (nD) Systems (IEEE nDS 2015)Proceedings of the IEEE 9th International Workshop on Multidimensional (nD) Systems (IEEE nDS 2015)Vila Real, PortugalSeptember 2015
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• 61 inproceedingsY.Yacine Bouzidi and F.Fabrice Rouillier. Certified Algorithms for proving the structural stability of two dimensional systems possibly with parameters.MNTS 2016 - 22nd International Symposium on Mathematical Theory of Networks and SystemsProceedings of the 22nd International Symposium on Mathematical Theory of Networks and SystemsMinneapolis, United StatesJuly 2016
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• 62 articleE.Erwan Brugallé, P.-V.Pierre-Vincent Koseleff and D.Daniel Pecker. On the lexicographic degree of two-bridge knots.Journal Of Knot Theory And Its Ramifications (JKTR)25714p., 21 figsJune 2016
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• 63 articleE.Erwan Brugallé, P.-V.Pierre-Vincent Koseleff and D.Daniel Pecker. The lexicographic degree of the first two-bridge knots.Annales de la Faculté des Sciences de Toulouse. Mathématiques.294December 2020, 761-793
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• 64 articleE.Erwan Brugallé, P.-V.Pierre-Vincent Koseleff and D.Daniel Pecker. Untangling trigonal diagrams.Journal Of Knot Theory And Its Ramifications (JKTR)25710p., 24 figsJune 2016
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• 65 inproceedingsD.Damien Chablat, R.Ranjan Jha, F.Fabrice Rouillier and G.Guillaume Moroz. Non-singular assembly mode changing trajectories in the workspace for the 3-RPS parallel robot.14th International Symposium on Advances in Robot KinematicsLjubljana, SloveniaJune 2014, 149 -- 159
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• 66 inproceedingsD.Damien Chablat, R.Ranjan Jha, F.Fabrice Rouillier and G.Guillaume Moroz. Workspace and joint space analysis of the 3-RPS parallel robot.ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering ConferenceVolume 5ABuffalo, United StatesAugust 2014, 1-10
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• 67 articleF.Frédéric Chyzak, A.Alban Quadrat and D.Daniel Robertz. Effective algorithms for parametrizing linear control systems over Ore algebras.Applicable Algebra in Engineering, Communications and Computing162005, 319--376
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• 68 articleF.Frédéric Chyzak and B.Bruno Salvy. Non-commutative elimination in Ore algebras proves multivariate identities.Journal of Symbolic Computation2621998, 187--227
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• 69 inproceedingsG. E.George E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decompostion.Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern, May 20--23, 1975Berlin, HeidelbergSpringer Berlin Heidelberg1975, 134--183
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• 70 bookD. A.David A. Cox, J.John Little and D.Donal O'Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics).Berlin, HeidelbergSpringer-Verlag2007
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• 71 articleM.Marco Crocco, A.Alessio Del Bue and V.Vittorio Murino. A bilinear approach to the position self-calibration of multiple sensors.IEEE Transactions on Signal Processing6022012, 660--673
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• 72 bookR.R.F. Curtain and H.H. Zwart. An Introduction to Infinite-Dimensional Linear Systems Theory.Texts in Applied MathematicsSpringer New York2012
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• 73 articleR.Roudy Dagher, A.Alban Quadrat and G.Gang Zheng. Algebraic solutions to the metric multidimensional unfolding. Application to the position self-calibration problem.in preparation2019
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• 74 articleR.Roudy Dagher, A.Alban Quadrat and G.Gang Zheng. Auto-localisation par mesure de distances.Pattern n. FR18535532018
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