2024Activity reportProject-TeamOURAGAN

2.1.1 Research axis 1: Computable objects

An important axis in our activities consists in identifying and (efficiently) describing the different computational objects that emanate from the other research directions of the group. Henceforth, we develop a mathematical and computational framework to manipulate computational objects, relying mainly on algebraic and geometric tools. This framework considers objects, algorithms, and theoretical and practical developments that we already know that are omnipresent or the bottleneck of various problems: for example advanced techniques for computing with (real) algebraic numbers, efficient algorithms for solving structured polynomial systems, certification of roots of polynomials and polynomial systems. We also focus on theoretical developments that are required to study algebraic and geometric algorithms, for precise bit complexity bounds real solving and resultant computations, separation and zero bounds for polynomials and polynomial systems, worst and average bounds for the condition numbers for non-linear problems. Lastly, we develop efficient general-purpose software for polynomial (systems) solving and geometric computations that supports our theoretical developments. However, our framework could be easily adapted to handle the various problems at hand, for example dedicated state-of-the-art algorithms for solving polynomial systems in two or three variables, new efficient techniques for algebraic elimination based on structure and sparsity, dedicated state-of-the-art algebraic and geometric algorithms for curve manipulation in small and higher dimensions, and dedicated algorithms for classical problems of non-linear computational geometry, like arrangement, sampling, and convex hull computations.

Our overall goal is to provide the best theoretical guarantees but also efficient implementations that solve practical problems.

2.1.2 Research axis 2: Algebraic Analysis, Algebraic Topology and Group Theory

Algebraic analysis is a mathematical theory that studies linear systems of partial differential equations by means of rings of differential operators, algebraic geometry, module theory, sheaf theory, homological theory, etc. It nowadays plays an important role in different branches of mathematics.

Motivated by applications of algebraic analysis to engineering sciences such as mathematical systems theory, control theory, and signal processing, as well as to mathematical physics, the OURAGAN project-team has been continuing to develop its expertise towards the development of effective algebraic analysis methods, extend them to other classes of linear functional systems (e.g., differential time-varying delay systems, integro-differential systems), develop dedicated computer algebra packages, and applications to the above-mentioned fields of applications.

More generally, the OURAGAN project-team works on algorithmic questions on algebraic structures from group theory (mainly braid monoids and some generalisations), algebraic topology (mainly associative algebras), rings of functional operators, module theory, and homological algebra. The rewriting methods and the construction of explicit resolutions of these objects are at the core of the approach developed within the OURAGAN project-team. In particular, the same methods are used to study questions arising from fundamental mathematics to engineering sciences.

Methods coming from Garside theory (originating in combinatorial group theory), mixed with rewriting, are developed to achieve results on more complex algebraic structures: generalisations and variants of braid monoids, and operads and their algebras. Algorithmic elimination theories are investigated such as an instrinsic differential elimination theory based on Spencer's theory of partial differential equations, and an ordinary integro-differential elimination theory based on the coherence property of rings of ordinary integro-differential operators.

2.1.3 Research axis 3: Algorithmic Number Theory

Algorithms and number theory have a long common history, as illustrated by Henri Cohen's book "A Course in Computational Algebraic Number Theory". Our work fits into this context and can be implemented in recognized tools such as Magma. On the other hand, it is also linked to fields such as cryptography. To give an overview of the topics we cover, we describe below the current links between cryptography and number theory.

The frontiers between computable objects, algorithms (above section), computational number theory and applications to security of cryptographic systems are very porous. This union of research fields is mainly driven by the algorithmic improvement to solve presumably hard problems relevant to cryptography, such as computation of discrete logarithms, resolution of hard subset-sum problems, decoding of random binary codes and search for close and short vectors in lattices. While factorization and discrete logarithm problems have a long history in cryptography, the recent post-quantum cryptosystems introduce a new variety of presumably hard problems/objects/algorithms with cryptographic relevance: the shortest vector problem (SVP), the closest vector problem (CVP) or the computation of isogenies between elliptic curves, especially in the supersingular case.

2.1.4 Research axis 4: Geometry and Topology in low dimension

A structuring axis for our team revolves around applications in geometry and topology in low dimension. The aim of this axis is to leverage the shared computable object to obtain effective topological and geometric description of objects of mathematical interest. Following the application-oriented spirit of the team, we try and adapt the shared tools to contribute to the research around deep and interesting objects: general algebraic curves, knots, character varieties and geometric structures.

For a general algebraic curve, or more generally an algebraic variety, a very fundamental question is the description of the topology of its points: are there singularities? when trying to project the curve on a surface, what are the singularities of the projection? The answer to these questions then allows for certified approximated computations of the smooth part and a good understanding of the geometry of the curve. Several objects help answering these questions: amoebas for complex curves and varieties, discriminant subvarieties, construction of graphs isotopic to a curve, construction of meshes for algebraic varieties. A significant part of our work revolves around these fundamental objects. Applications of this work ranges from Robotics (see Research Axis 5) to computation of Mahler measures (see Research Axis 3). Among other cases, a specific attention has been given to polynomial knots, i.e. knots in ${𝐑}^3$ defined by the image of a polynomial embedding of $ℝ$.

Another long-standing field of work for our team is the computational study of character varieties and construction of geometric structures. The notion of a geometry carried by a manifold goes back almost two centuries. For example, it is known that a surface carries either the geometry of the sphere (the sphere itself) or of the plane (the torus) or of the hyperbolic plane (for higher genus surfaces). The modern notion of geometric structure has two faces. One is algebraic, through a representation of a surface group, the other one is geometric: the construction of the geometric structure compatible to the representation.

There is an existing and thriving international field of computational topology and hyperbolic geometry of 3-manifolds, with celebrated softwares as Regina and Snappy. The general approach to understanding the geometry carried by a 3-manifold consists in triangulating the manifold by tetrahedra; parametrize the algebraic object called the character variety; and for points in this character variety try to compatibly geometrize the triangulation (i.e. give shapes to the tetrahedra that glue together nicely).

In a continued effort for more than 10 years, members of our team contribute to the expansion of this field beyond the usual case of real hyperbolic geometry. It involves computational geometric tools for triangulations, algebraic tools such as those developed in our set of computable objects for describing the character variety, and theoretical geometric tools for the last step. Further study of the character varieties, such as the volume function defined on it, necessitates other tools shared by our team: certified numerical computations for example.

2.1.5 Research axis 5: Applications

We develop effective algebraic, and symbolic-numeric methods dedicated to problems studied in cryptography, robotics, control theory and signal processing. Our main keyword is certification : the methods must be conceptually infallible (able to solve the problem without unverifiable assumptions or returns a clear message) and able to keep track on uncertainty on the input (for example manufacturing errors).

In cryptography, applications are part of the theoretical problems to be studied and their description can directly be found in section 2.1.3.

In robotics, we follow some of the directions proposed by Jean-Pierre Merlet, in particular the use of interval analysis, and we combine them with pure algebraic objects such as discriminant varieties.

At the design level, we focus on parallel manipulators, which includes the study of direct and inverse kinematics problems, path planning, with and without parameters, with or without error considerations on the design parameters. At a second level, the study and use of such mechanisms for dedicated tasks meet our work on control theory.

For control theory and signal processing, we combine methods of algebraic analysis, algebraic geometry, and computer algebra to study analysis and synthesis problems such as the effective study of structural properties of linear functional systems, equivalence problems, symbolic-numeric methods for stability analysis and robust stabilization problems for multidimensional systems and infinite-dimensional systems (e.g., differential time-delay systems, partial differential systems), as well as for parameter estimation, demodulation problems, and geo-localization problem. The kernel of the methods used for this axis is the same as the one for robotics problems.

7.1.1 A NewDsc

• Name:
  A New Descartes
• Keyword:
  Scientific computing
• Functional Description:
  Computations of the real roots of univariate polynomials with rational coefficients.
• URL:
  https://­anewdsc.­mpi-inf.­mpg.­de
• Contact:
  Fabrice Rouillier
• Partner:
  Max Planck Institute for Software Systems

7.1.2 Catex

• Keywords:
  LaTeX, String diagram, Algebra
• Functional Description:
  Catex is a Latex package and an external tool to typeset string diagrams easily from their algebraic expression. Catex works similarly to Bibtex.
• URL:
  https://­plmlab.­math.­cnrs.­fr/­guiraud/­catex
• Contact:
  Yves Guiraud
• Participant:
  Yves Guiraud

7.1.3 Cox

• Keywords:
  Computer algebra system (CAS), Rewriting systems, Algebra
• Functional Description:
  Cox is a Python library for the computation of coherent presentations of Artin monoids, with experimental features to compute the lower dimensions of the Salvetti complex.
• URL:
  https://­plmlab.­math.­cnrs.­fr/­guiraud/­cox
• Publications:
  hal-00682233, hal-00818253
• Contact:
  Yves Guiraud
• Participant:
  Yves Guiraud

7.1.4 dCat

• Keywords:
  Rewriting, Algebra, Termination, Complexity
• Functional Description:
  dCat is a prototype for the automatic research of complexity bounds of polygraphic programs. It relies on the "termination by derivation" technique introduced in Termination orders for 3-dimensional rewriting and adapted to complexity analysis in Polygraphic programs and polynomial-time functions.
• URL:
  https://­plmlab.­math.­cnrs.­fr/­guiraud/­dcat
• Publications:
  tel-00006863, hal-00092196, hal-00092204, inria-00129391, inria-00122932
• Contact:
  Yves Guiraud
• Participants:
  Yves Guiraud, Frederic Blanqui

7.1.5 Garside.jl

• Keywords:
  Algebra, Garside, Computer algebra
• Functional Description:
  Garside.jl is a Julia library for the explicit computation of a minimal resolution (resolution by lcms) of Garside monoids, including the classical and dual braid monoids in spherical type, and dual monoids of some complex reflection groups.
• URL:
  https://­plmlab.­math.­cnrs.­fr/­guiraud/­garside.­jl
• Contact:
  Yves Guiraud
• Participant:
  Yves Guiraud

7.1.6 ISOTOP

• Name:
  Topology and geometry of planar algebraic curves
• Keywords:
  Topology, Curve plotting, Geometric computing
• Functional Description:
  Isotop is a Maple software for computing the topology of an algebraic plane curve, that is, for computing an arrangement of polylines isotopic to the input curve. This problem is a necessary key step for computing arrangements of algebraic curves and has also applications for curve plotting. This software has been developed since 2007 in collaboration with F. Rouillier from Inria Paris - Rocquencourt.
• URL:
  https://­isotop.­gamble.­loria.­fr/
• Publications:
  hal-00809430, hal-00809425, inria-00329754, inria-00580431, hal-00992634, hal-01342211, inria-00425383, inria-00517175, hal-01468796, hal-00977671
• Contact:
  Marc Pouget
• Participants:
  Luis Penaranda, Marc Pouget, Sylvain Lazard

7.1.7 MPFI

• Name:
  Multiple Precision Floating-point Interval
• Keyword:
  Arithmetic
• Functional Description:
  MPFI is a C library based on MPFR and GMP for arbitrary precision interval arithmetic.
• Release Contributions:
  Updated for the autoconf installation. New functions added: rev_sqrt, exp10, exp2m1, exp10m1, log2p1, log10p1.
• URL:
  https://­gitlab.­inria.­fr/­mpfi/­mpfi
• Contact:
  Nathalie Revol

7.1.8 OreAlgebraicAnalysis

• Keywords:
  Algebra, Computer algebra, Gröbner bases, Linear system, Ordinary differential equations, Differential algebraic equations, Partial differential equation, Equations algebraic partial derivatives, Polynomial equations, Automatic control
• Functional Description:
  OreAlgebraicAnalysis is a Mathematica implementation of algorithms available in the OreModules and the OreMorphisms packages (developed in Maple). OreAlgebraicAnalysis is based on the implementation of Gröbner bases over Ore algebras available in the Mathematica HolonomicFunctions package developed by Christoph Koutschan (RICAM). OreAlgebraicAnalysis can handle larger classes of Ore algebras than the ones accessible in Maple, and thus we can study larger classes of linear functional systems. Finally, Mathematica internal design allows us to consider classes of systems which could not easily be considered in Maple such as generic linearizations of nonlinear functional systems defined by explicit nonlinear equations and systems containing transcendental functions (e.g., trigonometric functions, special functions). This package has been developed within the PHC Parrot project CASCAC.
• URL:
  https://­who.­rocq.­inria.­fr/­Alban.­Quadrat/­OreAlgebraicAnalysis/­index.­html
• Contact:
  Alban Quadrat
• Participants:
  Alban Quadrat, Thomas Cluzeau

7.1.9 OreMorphisms

• Keywords:
  Algebra, Computer algebra, Gröbner bases, Linear system, Ordinary differential equations, Partial differential equation, Differential algebraic equations, Equations algebraic partial derivatives, Polynomial equations, Automatic control
• Functional Description:
  The OreMorphisms package, based on OreModules, is dedicated to the implementation of homological algebra methods such as the computation of homomorphisms between two finitely presented modules over certain noncommutative polynomial algebras (Ore algebras), of kernel, coimage, image and cokernel of homomorphisms, Galois transformations of linear multidimensional systems and idempotents of the endomorphism ring. Using the packages Stafford and Quillen-Suslin, the factorization, reduction and decomposition problems can be effectively studied for different classes of linear multidimensional systems. Many linear functional systems studied in engineering sciences, mathematical physics and control theory have been factorized, reduced and decomposed thanks to the OreMorphisms package.
• URL:
  https://­who.­rocq.­inria.­fr/­Alban.­Quadrat/­OreMorphisms/­index.­html
• Contact:
  Alban Quadrat
• Participants:
  Alban Quadrat, Thomas Cluzeau

7.1.10 OreModules

• Keywords:
  Algebra, Computer algebra, Gröbner bases, Linear system, Ordinary differential equations, Differential algebraic equations, Partial differential equation, Equations algebraic partial derivatives, Polynomial equations, Automatic control
• Functional Description:
  OreModules is a Maple package dedicated to module theory and homological algebra for finitely presented modules defined over an Ore algebra of functional operators (e.g., ordinary or partial differential operators, shift operators, time-delay operators, difference operators) available in the Maple package Ore_algebra, and to their applications in mathematical systems theory and mathematical physics.
• URL:
  https://­who.­rocq.­inria.­fr/­Alban.­Quadrat/­OreModules/­index.­html
• Contact:
  Alban Quadrat

7.1.11 PTOPO

• Name:
  Topology of Parametric Curves
• Keywords:
  Parametric curve, 2D, 3D, Visualization, Computer algebra, Curve plotting, Topology
• Functional Description:
  PTOPO computes (exactly) the topology and visualize parametric curves in 2D and in 3D.
• URL:
  https://­webusers.­imj-prg.­fr/­~christina.­katsamaki/­ptopo/
• Contact:
  Elias Tsigaridas

7.1.12 PurityFiltration

• Keywords:
  Symbolic computation, Partial differential equation
• Functional Description:
  The PurityFiltration package, built upon the OreModules package, is an implementation of a new effective algorithm which computes the purity/grade filtration of linear functional systems (e.g., partial differential systems, differential time-delay systems, difference systems) and equivalent block-triangular matrices. This package is used to compute closed form solutions of over/underdetermined linear partial differential systems which cannot be integrated by the standard computer algebra systems such as Maple and Mathematica.
• URL:
  https://­who.­rocq.­inria.­fr/­Alban.­Quadrat/­PurityFiltration.­html
• Contact:
  Alban Quadrat

7.1.13 Rewr

• Name:
  Rewriting methods in algebra
• Keywords:
  Computer algebra system (CAS), Rewriting systems, Algebra
• Functional Description:
  Rewr is a prototype of computer algebra system, using rewriting methods to compute resolutions and homotopical invariants of monoids. The library implements various classical constructions of rewriting theory (such as completion), improved by experimental features coming from Garside theory, and allows homotopical algebra computations based on Squier theory. Specific functionalities have been developed for usual classes of monoids, such as Artin monoids and plactic monoids.
• URL:
  https://­plmlab.­math.­cnrs.­fr/­guiraud/­rewr
• Publications:
  hal-00326974, hal-00531242, hal-00682233, hal-00818253, hal-00932845, hal-01141226
• Contact:
  Yves Guiraud
• Participants:
  Yves Guiraud, Samuel Mimram

7.1.14 RS

• Functional Description:
  Real Roots isolation for algebraic systems with rational coefficients with a finite number of Complex Roots
• URL:
  https://­team.­inria.­fr/­ouragan/­software/
• Contact:
  Fabrice Rouillier
• Participant:
  Fabrice Rouillier

7.1.15 SIROPA

• Keywords:
  Robotics, Kinematics
• Functional Description:
  Library of functions for certified computations of the properties of articulated mechanisms, particularly the study of their singularities
• URL:
  http://­siropa.­gforge.­inria.­fr/
• Contact:
  Guillaume Moroz
• Partner:
  LS2N

7.1.16 SLV

• Keywords:
  Univariate polynomial, Real solving
• Functional Description:
  SLV is a software package in C that provides routines for isolating (and subsequently refine) the real roots of univariate polynomials with integer or rational coefficients based on subdivision algorithms and on the continued fraction expansion of real numbers. Special attention is given so that the package can handle polynomials that have degree several thousands and size of coefficients hundrends of Megabytes. Currently the code consists of approx. 5000 lines.
• URL:
  https://­who.­paris.­inria.­fr/­Elias.­Tsigaridas/­soft.­html
• Contact:
  Elias Tsigaridas

7.1.17 Stafford

• Keywords:
  Symbolic computation, Partial differential equation
• Functional Description:
  The Stafford package of OreModules contains an implementation of two constructive versions of Stafford's famous but difficult theorem [96] stating that every ideal over the Weyl algebra An(k) (resp., Bn(k)) of partial differential operators with polynomial (resp., rational) coefficients over a field k of characteristic 0 (e.g., k=Q,R) can be generated by two generators. Based on this implementation and algorithmic results developed by the authors of the package, two algorithms which compute bases of free modules over the Weyl algebras An(Q) and Bn(Q) have been implemented. The rest of Stafford's results developed in [96] have recently been made constructive (e.g., computation of unimodular elements, decomposition of modules, Serre's splitting-off theorem, Stafford's reduction, Bass' cancellation theorem, minimal number of generators) and implemented in the Stafford package. The development of the Stafford package was motivated by applications to linear systems of partial differential equations with polynomial or rational coefficients (e.g., computation of injective parametrization, Monge problem, differential flatness, the reduction and decomposition problems and Serre's reduction problem). To our knowledge, the Stafford package is the only implementation of Stafford's theorems nowadays available.
• URL:
  https://­who.­rocq.­inria.­fr/­Alban.­Quadrat/­OreModules/­stafford.­html
• Contact:
  Alban Quadrat
• Participants:
  Alban Quadrat, Daniel Robertz

On the Complexity of Chow and Hurwitz Forms.

In 23, we consider the bit complexity of computing Chow forms of projective varieties defined over integers and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model; thus, our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space and we explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.

Segre-driven radicality testing.

In 25, we present a probabilistic algorithm to test if a homogeneous polynomial ideal I defining a scheme $X$ in ${\mathbb{P}}^n$ is radical using Segre classes and other geometric notions from intersection theory. Its worst case complexity depends on the geometry of $X$. If the scheme $X$ has reduced isolated primary components and no embedded components supported the singular locus of $X_{red}=𝕍\left(\sqrt{I}\right)$, then the worst case complexity is doubly exponential in $n$; in all the other cases the complexity is singly exponential. The realm of the ideals for which our radical testing procedure requires only single exponential time includes examples which are often considered pathological, such as the ones drawn from the famous Mayr-Meyer set of ideals which exhibit doubly exponential complexity for the ideal membership problem.

Probabilistic bounds on best rank-one approximation ratio.

In 27, we provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric tensors our upper bound unveils that the ratio of norms has the same order of magnitude as the trivial lower bound $1/\sqrt{n^{d-1}}$, when the order of a tensor $d$ is fixed and the dimension of the underlying vector space $n$ tends to infinity. However, when $n$ is fixed and $d$ tends to infinity, our lower bound is better than $1/\sqrt{n^{d-1}}$.

Condition-based Low-Degree Approximation of Real Polynomial Systems I: The Zero-Dimensional Case.

In 34, we provide new bounds on the number of real zeros of a (zero-dimensional) polynomial system in terms of the condition number of the system. In the probabilistic setting, this translates into new estimates on all the moments of the number of the real zeros of random polynomial systems, including Kac and KSS polynomial systems. Moreover, the provided bounds are robust: they do not require the Gaussian assumption.

Effective characterization of evaluation ideals of the ring of integro-differential operators

The article 31 provides a step forward to developing an algorithmic study of linear systems of polynomial ordinary integro-differential equations over a field $𝕜$ of characteristic zero. Such a study can be achieved by first obtaining a constructive proof of the coherence property of the ring $𝕀\left(𝕜\right)$ of linear ordinary integro-differential operators with coefficients in $𝕜\left[t\right]$. To do that, the finiteness of the intersection of two finitely generated ideals has to be algorithmically studied. Three cases must be considered: first when evaluation operators generate the two ideals; second, when only one ideal is generated by evaluation operators; and third, when none is generated by evaluation operators. In this paper, we first explicitly characterize the intersection of two finitely generated ideals defined by evaluation operators. As for the second case, a key result is that the ideals generated by evaluations are semisimple $𝕀\left(𝕜\right)$-modules. We develop an algorithmic proof of this result. In particular, we show how a finite set of generators, defined by “simple” evaluations, can be obtained, that characterizes the class of finitely generated evaluation ideals of $𝕀\left(𝕜\right)$ as finitely generated $𝕜\left[t\right]$-modules. Due to lack of space, the second and third cases will be developed in other publications.

On Polynomial Modular Number Systems over Z/pZ.

Since their introduction in 2004, Polynomial Modular Number Systems (PMNS) have become a very interesting tool for implementing cryptosystems relying on modular arithmetic in a secure and efficient way. However, while their implementation is simple, their parameterization is not trivial and relies on a suitable choice of the polynomial on which the PMNS operates. The initial proposals were based on particular binomials and trinomials. But these polynomials do not always provide systems with interesting characteristics such as small digits, fast reduction, etc. In 18, we study a larger family of polynomials that can be exploited to design a safe and efficient PMNS. To do so, we first state a complete existence theorem for PMNS which provides bounds on the size of the digits for a generic polynomial, significantly improving previous bounds. Then, we present classes of suitable polynomials which provide numerous PMNS for safe and efficient arithmetic.

Limits of Mahler measures in multiple variables.

In 19, we prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.

Reversible primes.

For a $n$-bit positive integer $a$ written in binary as $a={\sum }_{j=0}^{n-1}{ϵ}_j\left(a\right)2^j$, where, ${ϵ}_j\left(a\right)\in \{0,1\},j\in \{0,...,n-1\},{ϵ}_{n-1}\left(a\right)=1$, let us define $\overleftarrow{a}={\sum }_{j=0}^{n-1}{ϵ}_j\left(a\right))2^{n-1-j}$ the digital reversal of $a$. Also let ${ℬ}_n=\{2n-1\le a\le 2n:aodd\}$. In 21, with a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of $p\in {ℬ}_n$ such that $p$ and $\overleftarrow{p}$ are prime. We also prove that for sufficiently large $n$, $\left|\right\{a\in {ℬ}_n:max\{\Omega \left(a\right),\Omega \left(\overleftarrow{a}\right)\le 8\}|\ge c\frac{2^n}{n^2}$,where $\Omega \left(n\right)$ denotes the number of prime factors counted with multiplicity of $n$ and $c>0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of $n$-bit integers $a$ such that $a$ and $\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum ${\sum }_{a\in ℬ}exp(2\pi i(\alpha a+\delta \overleftarrow{a})$$(\alpha ,\delta \in ℝ)$.

Mirror stabilizers for lattice complex hyperbolic triangle groups.

For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizer of a reprentative of each group orbit of mirrors of complex reflections. In 22, we give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some of the triangle groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.

Slim curves, limit sets and spherical CR uniformisations.

In 24, we consider here the 3-sphere $S^3$ seen as the boundary at infinity of the complex hyperbolic plane $H_C^2$. It comes equipped with a contact structure and two classes of special curves. First R-circles are boundaries at infinity of totally real totally geodesic subspaces and are tangent to the contact distribution. Second, C-circles, which are boundaries of complex totally geodesic subspaces and are transverse to the contact distribution. We define a quantitative notion, called slimness, that measures to what extent a continuous path in the sphere $S^3$ is near to be an R-circle. We analyze the classical foliation of the complement of an R-circle by arcs of C-circles. Next, we consider deformations of this situation where the R-circle becomes a slim curve. We apply these concepts to the particular case where the slim curve is the limit set of a quasi-Fuchsian subgroup of PU(2,1). As a consequence, we describe a class of spherical CR uniformizations of certain cusped 3-manifolds.

Semidefinite games.

In 26, we introduce and study the class of semidefinite games, which generalizes bimatrix games and finite N-person games, by replacing the simplex of the mixed strategies for each player by a slice of the positive semidefinite cone in the space of real symmetric matrices. For semidefinite two-player zero-sum games, we show that the optimal strategies can be computed by semidefinite programming. Furthermore, we show that two-player semidefinite zero-sum games are almost equivalent to semidefinite programming, generalizing Dantzig's result on the almost equivalence of bimatrix games and linear programming. For general two-player semidefinite games, we prove a spectrahedral characterization of the Nash equilibria. Moreover, we give constructions of semidefinite games with many Nash equilibria. In particular, we give a construction of semidefinite games whose number of connected components of Nash equilibria exceeds the long standing best known construction for many Nash equilibria in bimatrix games, which was presented by von Stengel in 1999.

dingo: a Python package for metabolic flux sampling.

In 20, we present dingo, a Python package that supports a variety of methods to sample from the flux space of metabolic models, based on state-of-the-art random walks and rounding methods. For uniform sampling dingo's implementation of the Multiphase Monte Carlo Sampling algorithm, provides a significant speed-up and outperforms existing software. Indicatively, dingo can sample from the flux space of the largest metabolic model up to now (Recon3D) in less than 30 hours using a personal computer, under several statistical guarantees; this computation is out of reach for other similar software. In addition, supports common analysis methods, such as Flux Balance Analysis (FBA) and Flux Variability Analysis (FVA), and visualization components. dingo contributes to the arsenal of tools in metabolic modeling by enabling flux sampling in high dimensions (in the order of thousands).

Certified Kinematic Tools for the Design and Control of Parallel Robots.

The article 28 presents a methodology for the design and control of Parallel Kinematic Robots (PKRs). First, one focuses on the problematics of design. In particular, given a parallel mechanism defined by its design parameters and its kinematic modeling as well as its prescribed workspace, the idea is to certify the absence of any numerical instabilities (computational and physical singularities) that may jeopardize the integrity of the robot. This is achieved through two complementary approaches: a global method using symbolic computation and a local one based on continuation techniques and interval calculus, accounting for uncertainties in the design. The methodology is then applied to real PKR examples. Secondly, the paper proposes a control strategy that limits the active joint velocities to ensure the robot remains within its certified workspace. It will be applied to a special class of parallel robots: Spherical Parallel Manipulators (SPM) with coaxial input shafts (CoSPM).

$L_{\infty }$ -norm computation for linear time-invariant systems depending on parameters.

The article 29 focuses on representing the $L_{\infty }$-norm of finite-dimensional linear time-invariant systems with parameter-dependent coefficients. Previous studies tackled the problem in a non-parametric scenario by simplifying it to finding the maximum y-projection of real solutions (x,y) of a system of the form $\Sigma =\{P=0,\partial P/\partial x=0\}$, where $p\in \mathbb{Z}[x,y]$. To solve this problem, standard computer algebra methods were employed and analyzed. In this paper, we extend our approach to address the parametric case. We aim to represent the “maximal" y-projection of real solutions of $\Sigma$ as a function of the given parameters. To accomplish this, we utilize cylindrical algebraic decomposition. This method allows us to determine the desired value as a function of the parameters within specific regions of parameter space.

Inertial line-of-sight stabilization using a 3-dof spherical parallel manipulator with coaxial input shafts.

The article 32 dives into the use of a 3-RRR Spherical Parallel Manipulator (SPM) for the purpose of inertial Line Of Sight (LOS) stabilization. Such a parallel robot provides three Degrees of Freedom (DOF) in orientation and is studied from the kinematic point of view. In particular, one guarantees that the singular loci (with the resulting numerical instabilitiesand inappropriate behavior of the mechanism) are far away from the prescribed workspace. Once the kinematics of the device is certified, a control strategy needs to be implemented in order to stabilize the LOS through the upper platform of the mechanism. Such a work is done with MATLAB Simulink® using a SimMechanics™ model of our robot.

On a Software Joint Velocity Limitation of a Spherical Parallel Manipulator with Coaxial Input Shafts.

The article 35 discusses the implementation of a software joint velocity limitation dedicated to a Spherical Parallel Manipulator (SPM) with coaxial input shafts (CoSPM) using a speed control loop. Such an algorithm takes as input the current joint positions as well as the joint reference velocities computed by the speed controller and limit the latter in order to avoid any known singular configuration. This limitation takes into account the workspace properties of the mechanism and the physical characteristics of its actuators. In particular, one takes advantage of the coaxiality of the input shafts of the CoSPM and the resulting unlimited bearing.

On a general robust stability test based on the homological perturbation lemma.

Within the lattice approach to synthesis problems, the article 46 shows how a general unstructured robust stability test can be obtained directly by applying the homological perturbation lemma, a standard method developed in algebraic topology and homological algebra. This robust stability test generalizes and unifies various results from the robust control literature.

Towards the Computation of Stabilizing Controllers of Multidimensional Systems.

In 30, we further study the effective computation of stabilizing controllers of multidimensional systems. Within the algebraic analysis approach, the stabilization problem can be characterized by the fact that a certain finitely presented $A$-module $M$, naturally associated with the multidimensional system, is projective, where $A$ denotes the ring of multivariate rational functions without poles in the closed complex unit polydisc $D^n$ . This condition can be reduced to the existence of an element $s$ of a polynomial ideal $I$ which has no zero in $D^n$.

According to the Polydisc Nullstellensatz, the latter condition is equivalent to the fact that no complex zero of the elements of $I$ belongs to $D^n$. If this condition is satisfied, using cyclic resultants and linear programming, we then propose a method to compute such a polynomial $s$.

Finally, using computer algebra methods for effectively handling basic operations on $R\left[s^{(-1)}\right]$ modules, where $R$ is a polynomial ring, we show how to compute stabilizing controllers.

10.1.1 Visits of international scientists

• Alexander Demin from July 1st to September 30th
• Jeremie Kaminski from December 2024 1st to March 2025 31th

10.2.1 ANR

• ANR JCJC SHoCoS (Structure and Homotopy of Configuration Spaces)

  Coordinator: Najib Idrissi (Univ. Paris Cité, IMJ-PRG)

  Participant: Yves Guiraud

  Duration: 2022 – 2026

  This is a project of fundamental research in mathematics, specifically algebraic topology, homotopical algebra, and quantum algebra. It is concerned with configuration spaces, which consist in finite sequences of pairwise distinct points in a manifold. Over the past couple of decades, strides have been made in the study and computation of the homotopy types of configuration spaces, i.e., their shape up to continuous deformation. These advances were possible thanks to the rich structure of configuration spaces, which comes from the theory of operads. Moreover, a new theory, factorization homology, allowed the use of configuration spaces to compute topological field theories, topological invariants of manifolds inspired by physics. Our purpose is to exploit the full operadic structure of configuration spaces to obtain new kinds of stabilizations in the homotopy types of configuration spaces, and to use this stability to effectively compute topological field theories from deformation quantization.

• ANR HilbertxField (Géométries de Hilbert sur les corps valués)

  Coordinator Antonin Guilloux (OURAGAN)

  Participant : Antonin Guilloux

  Duration: Sept 2023 - Aug 2027

  A Hilbert geometry is defined on any convex body in a real affine space. This notion is the source of numerous examples of metric spaces and has had many applications in various fields since its definition in 1895 by Hilbert. The participants in this project contribute to different generalizations of this notion and these applications in contexts where the basic body is no longer the body of realities.

  This project has three main objectives: - develop a unified approach to these generalizations: unified definitions, common generalization of the results of Benzécri and the notion of volume; - explore the interactions between the different generalization contexts, using numerous families of examples; - obtain important applications in each case study.

  These applications are expected in different projects including:

  • the study of minimum entropy metrics on symmetric spaces;
  • the geometric study of degenerations of convex projective structures on surfaces;
  • the study of the boundary of Anosov representations, especially in the context of complex hyperbolic geometry;
  • the development of new linear programming algorithms, with Smale's 9th problem in focus.

10.2.2 Inria Exploratory actions

• LOCUS (non‐Linear geOmetriC compUting at Scale) Inria Exploratory Action

  Coordinator: Elias Tsigaridas

  Duration 2022 - 2025

  Summary : LOCUS shapes a novel theoretical, algorithmic, and computational framework at the intersection of computational algebra, high dimensional geometric and statistical computing, and optimization. It focuses on sampling and integrating in convex bodies, algorithms for convex optimization, and applications in structural biology. It aims to deliver effective theoretical algorithms and efficient open source software for the problems of interest.

• Réal (Réécriture algébrique) Inria Exploratory Action

  Coordinator : Yves Guiraud

  Duration : 2022-2025

  Summary : Rewriting is a branch of computer algebra consisting in transforming mathematical expressions according to admissible rules. Examples range from elementary situations, such as a remarkable identity ${(a+b)}^2=a^2+2ab+b^2$ in a ring, to calculations in complex algebraic structures, such as the Jacobi relation [[x,y],z] = [x,[y,z]] - [[x,z],y] in a Lie algebra.

  The Réal project proposes to explore the connections between rewriting and algebra. The aim is to understand the algebraic foundations of rewriting, to integrate similar calculation mechanisms known in algebra, and to develop new calculation tools with a view to applications in three areas of mathematics: combinatorial and higher algebra, theory groups and representations, study of algebraic systems and varieties.

10.2.3 Technological Development Actions

• PACE (2023 - ) Coordinated by F. Rouillier. With the help of a research engineer from Inria who is working full time for OURAGAN, pass from prototypes that did serve to validate some computational strategies related to our collaborations with Safran (Control Theory, Robotics, etc.) to full solutions with an interface directly usable by specialized Engineers.

11.1.1 Scientific events: organisation

• J.-C. Bajard organized the conference NAC'2024 in Paris, Feb 2024.
• Elias Tsigaridas, with Matías Bender, organized a session on Polynomials, Optimization, and Sampling during the PGMO days 2024.

11.1.2 Scientific events: selection

11.1.3 Journal

11.1.4 Invited talks

• Jean-Claude Bajard was invited to the "Colloquium Jacques Morgenstern" in Nice, Mai 2024.
• Yves Guiraud was invited to the Séminaire “Groupes, Algèbre, Topologie” in Amiens, April 2024.
• Alban Quadrat was invited to the MAX team seminar (LIX Laboratory & Ecole Polytechnique), on 16/12/2024.
• Fabrice Rouillier was invited to séminaire GAE in Rennes, September 2024.
• Cathy Swaenepoel was invited to :
  • 1st Brazilian Workshop in Analytic Number Theory, Rio de Janeiro;
  • Italy - France Analytic Number Theory Workshop, Genova;
  • Séminaire de Théorie des Nombres de l'Institut Fourier, Grenoble;
  • One World Numeration Seminar (online)

11.1.5 Leadership within the scientific community

Alban Quadrat co-organized with Eva Zerz an invited session "Algebraic and Geometric Approaches to Systems Structure and Control", at the 26th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Cambridge (U.K.), 19-23/09/24.

11.1.6 Scientific expertise

• Yves Guiraud is an elected member of the Comité National de la Recherche Scientifique (the evaluation body of the CNRS), Section 41 (Mathematics), and an appointed member of the section board (2021-2025).
• Yves Guiraud was a member of the HCERES evaluation committee of the Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA, UMR CNRS 7352, Amiens), visited in October 2024.

11.1.7 Research administration

• Elisha Falbel was director of the "École Doctorale Sciences Mathématiques de Paris Centre - ED 386" until Sept. 2024
• Yves Guiraud is an elected member of the laboratory council of IMJ-PRG, since 2021.
• Yves Guiraud is an elected member of the “comité de centre” of INRIA Paris, since 2019.
• Fabrice Rouillier is a member of the scientific commitee of the Indo French Centre for Applied Mathematics.
• Elias Tsigaridas is an elected member of the Commission d'évaluation d'Inria (CE) since 2019.

11.2.1 Teaching

• Antonin Guilloux, Alban Quadrat, Elias Tsigaridas, Master 1, Effective Linear Algebra and Polynomials. (24h course + 36h exercises).
• Antonin Guilloux, Fabrice Rouillier, Master 1, Introduction to Algebraic geometry (24h course + 36h exercises).
• Jean-Claude Bajard, Antonin Guilloux, Pierre-Vincent Koseleff and Fabrice Rouillier take part to the "agrégation de mathématiques - option C" at Sorbonne Université.
• Pierre-Vincent Koseleff : Master 1 Maths - Sorbonne Université : Algebraic Cryptography (36H) at Sorbonne Université.
• Pierre-Vincent Koseleff : Master 2 EducFellow in Maths - Computer Algebra (120H) at Sorbonne Université.
• Pierre-Vincent Koseleff : Master 1 Maths - Sorbonne Université : Algebraic Algorithmic (36H) at Sorbonne Université.
• Pascal Molin manages the Master math-info spécialités crypto et big-data at Paris Université.
• Pascal Molin : teaches codes et crypto and théorie de l'information in Master 1 at Paris Université.
• Elias Tsigaridas : Algorithms and Competitive Programming, Ingénieur 2A, modal. 20h lectures and 25h TD. Department of Informatics (LIX), École Polytechnique, France.
• Elias Tsigaridas : Algorithms for data analysis in C++, Ingénieur 2A. 40h TD. Department of Informatics (LIX), École Polytechnique, France.

11.2.2 Supervision

• Antonin Guilloux supervises (in collaboration with Pierre Charollois) the PhD of Pierre Morrain
• Yves Guiraud supervises, with Vladimir Dotsenko (Univ. Strasbourg) and Najib Idrissi (Univ. Paris-Cité, IMJ-PRG), the PhD thesis of José São João, since October 2024.
• Yves Guiraud supervises, with Najib Idrissi (Univ. Paris-Cité, IMJ-PRG), the two-year postdoc of Victor Roca i Lucio, since October 2024 (funded by the ANR project SHoCoS).
• Alban Quadrat supervises the PhD of Camille Pinto since 10/2022.
• Fabrice Rouillier supervises the PhD of Alexandre Lê (Safran CIFRE Grant) (co-dupervision with Damien Chablat - LS2N Nantes) since 01/2021.
• Fabrice Rouillier supervises the PhD of Joao Ruiz since 09/2023.
• Fabrice Rouillier supervises the PhD of Florent Corniquel since 10/2024.
• Elias Tsigaridas, Alban Quadrat and Fabrice Rouillier supervise the PhD of Chaoping Zhu since 09/2023.
• Elias Tsigaridas and Fabrice Rouillier supervise the PhD of Jules Tsukahara since 09/2024
• Elias Tsigaridas and Antonin Guilloux supervise the PhD of Ennio Grammatica since 11/2024.

11.2.3 Juries

• Yves Guiraud was a referee for the PhD thesis of Owen Garnier, Garside groupoids and complex braid groups, Univ. Picardie Jules-Verne, Amiens, November 2024.
• Fabrice Rouillier was a referee for the PhD thesis of Natacha Daoud Assessment of Chemical Risks and Circular Economy Implications of Recycled PET in Food Packaging with Functional Barriers., Université Paris-Saclay, September 2024
• Fabrice Rouillier was the president of the jury of the PhD thesis of Antoine Béreau « Systemes polynomiaux tropicaux et theorie des jeux », École Polytechnique.

11.3.1 Specific official responsibilities in science outreach structures

• Antonin Guilloux and Cathy Swaenepoel took part to the organization of the action Math-C-pour-L
• Fabrice Rouillier :
  • chairman of the association Animath,
  • member of the Comité de Culture Mathématique (CCM) of the Institut Henri Poincaré.
  • member of the conseil d'administration of Maths.en.Jeans
  • member of the conseil scientifique des Irems
  • member of the Commission Française pour l'Enseignement des Mathématiques (CFEM)
  • the scientific referent for the mediation activities of the Inria center in Paris
  • member of the national Jury of national olympiads of Mathematics

11.3.2 Productions (articles, videos, podcasts, serious games, ...)

• Cathy Swaenepoel did contribute to two articles :
  • C. Laurens, Le mystère des nombres, la grande quête de la relation entre + et ×. Epsiloon, hors série n°10 (2024).
  • C. Mauger, Les nombres premiers révèlent certains de leurs mystères. Mediapart, Éclats de Sciences, 7 décembre 2024.

11.3.3 Participation in Live events

• Antonin Guilloux, Pierre-Vincent Koseleff, Pascal Molin, Fabrice Rouillier, and Elias Tsigaridas organized a half-day presentation of OURAGAN research themes for Master 1 mathematics students from the University of Saclay.
• Fabrice Rouillier organized and animated the visit at Inria of a group of young students (High School level) participating to a week organized by the association Science Ouverte.

11.3.4 Others science outreach relevant activities

• All the team did participate actively to wellcome high school (3 girls) and college (1 girl) students for their stage d'observation (an observation period to learn more about working in a research institute).

International journals

• 18 articleJ.-C.Jean-Claude Bajard, J.Jérémy Marrez, T.Thomas Plantard and P.Pascal Véron. On Polynomial Modular Number Systems over $/p$.Advances in Mathematics of Communications183June 2024, 674-695HALDOIback to text
• 19 articleF.François Brunault, A.Antonin Guilloux, M.Mahya Mehrabdollahei and R.Riccardo Pengo. Limits of Mahler measures in multiple variables.Annales de l'Institut Fourier744July 2024, 1407-1450HALDOIback to text
• 20 articleA.Apostolos Chalkis, V.Vissarion Fisikopoulos, E.Elias Tsigaridas and H.Haris Zafeiropoulos. dingo : a Python package for metabolic flux sampling.Bioinformatics Advances41March 2024HALDOIback to text
• 21 articleC.Cécile Dartyge, B.Bruno Martin, J.Joël Rivat, I. E.Igor E. Shparlinski and C.Cathy Swaenepoel. Reversible primes.Journal of the London Mathematical Society1093March 2024, e12883HALback to text
• 22 articleM.Martin Deraux. Mirror stabilizers for lattice complex hyperbolic triangle groups.Geometriae Dedicata2183March 2024, 58HALDOIback to text
• 23 articleM. L.M. Levent Dogan, A.Alperen Ergür and E.Elias Tsigaridas. On the Complexity of Chow and Hurwitz Forms.ACM Communications in Computer Algebra574March 2024, 167-199HALDOIback to text
• 24 articleE.Elisha Falbel, A.Antonin Guilloux and P.Pierre Will. Slim curves, limit sets and spherical CR uniformisations.Groups, Geometry, and Dynamics184May 2024, 1507-1557HALDOIback to text
• 25 articleM.Martin Helmer and E.Elias Tsigaridas. Segre-driven radicality testing.Journal of Symbolic Computation122May 2024, 102262HALDOIback to text
• 26 articleC.Constantin Ickstadt, T.Thorsten Theobald and E.Elias Tsigaridas. Semidefinite games.International Journal of Game Theory533June 2024, 827-857HALDOIback to text
• 27 articleK.Khazhgali Kozhasov and J.Josué Tonelli-Cueto. Probabilistic bounds on best rank-one approximation ratio.Linear and Multilinear AlgebraMarch 2024, 1-29HALDOIback to text
• 28 articleA.Alexandre Lê, F.Fabrice Rouillier, G.Guillaume Rance and D.Damien Chablat. Certified Kinematic Tools for the Design and Control of Parallel Robots.Mechanism and Machine Theory205March 2025, 105865HALDOIback to text
• 29 articleA.Alban Quadrat, F.Fabrice Rouillier and G.Grace Younes. $L^{}$-norm computation for linear time-invariant systems depending on parameters.Maple Transactions41June 2024, 18HALDOIback to text

International peer-reviewed conferences

• 30 inproceedingsT.Thomas Cluzeau, G.Guillaume Moroz and A.Alban Quadrat. Towards the Computation of Stabilizing Controllers of Multidimensional Systems.IFAC-Papers onlineMTNS 2024 - 6th International Symposium on Mathematical Theory of Networks and Systems5817Cambdridge, United KingdomElsevierOctober 2024HALDOIback to text
• 31 inproceedingsT.Thomas Cluzeau, C.Camille Pinto and A.Alban Quadrat. Effective characterization of evaluation ideals of the ring of integro-differential operators.ISSAC 2024 - International Symposium on Symbolic and Algebraic ComputationRaleigh, NC, United StatesJuly 2024, 117 - 125HALDOIback to text
• 32 inproceedingsA.Alexandre Lê, G.Guillaume Rance, F.Fabrice Rouillier and D.Damien Chablat. Inertial line-of-sight stabilization using a 3-dof spherical parallel manipulator with coaxial input shafts.11th International Symposium on Optronics in Defense and SecurityOPTRO2024 - 11th International Symposium on Optronics in defence & securityBordeaux, FranceJanuary 2024HALback to text
• 33 inproceedingsA.Alexandre Lê, F.Fabrice Rouillier, G.Guillaume Rance and D.Damien Chablat. Certified Kinematic Tools for the Design and Control of Parallel Robots (Extended abstract).Mechanism and Machine Theory SymposiumGuimarães, Portugal2024HAL

Conferences without proceedings

• 34 inproceedingsJ.Josué Tonelli-Cueto and E.Elias Tsigaridas. Condition-based Low-Degree Approximation of Real Polynomial Systems I: The Zero-Dimensional Case.Effective Methods in Algebraic Geometry (MEGA)Leipzig, Germany2024HALback to text

Scientific book chapters

• 35 inbookA.Alexandre Lê, G.Guillaume Rance, F.Fabrice Rouillier, A.Arnaud Quadrat and D.Damien Chablat. On a Software Joint Velocity Limitation of a Spherical Parallel Manipulator with Coaxial Input Shafts.31Advances in Robot KinematicsSpringer Proceedings in Advanced RoboticsSpringer2024, 43 - 52HALDOIback to text

Reports & preprints

• 36 miscM. R.Matías R Bender, L.Laurent Busé, C.Carles Checa and E.Elias Tsigaridas. Multigraded Castelnuovo-Mumford regularity and Groebner bases.July 2024HAL
• 37 miscM. R.Matías R Bender, L.Laurent Busé, C.Carles Checa and E.Elias Tsigaridas. Solving bihomogeneous polynomial systems with a zero-dimensional projection.January 2025HAL
• 38 miscM. R.Matías R Bender, E.Elias Tsigaridas and C.Chaoping Zhu. Rational SOS certificates of any polynomial over its zero-dimensional gradient ideal.January 2025HAL
• 39 miscG.Gilles Courtois and A.Antonin Guilloux. Hausdorff dimension, diverging Schottky representations and the infinite dimensional hyperbolic space.October 2024HAL
• 40 miscE.Elisha Falbel, A.Antonin Guilloux and P.Pierre Will. A Hilbert metric for bounded symmetric domains.March 2024HAL
• 41 miscE.Elisha Falbel, M.Martin Mion-Mouton and J. M.Jose M. Veloso. Reductions of path structures and classification of homogeneous structures in dimension three.June 2024HAL
• 42 miscA.Antonin Guilloux and T.Theodore Weisman. Limits of limit sets in rank-one symmetric spaces.July 2024HAL
• 43 miscB. E.Boulos El Hilany, M.Martin Helmer and E.Elias Tsigaridas. Stratification of Projection Maps From Toric Varieties.2024HALDOI
• 44 miscB. E.Boulos El Hilany and E.Elias Tsigaridas. Bounds on the infimum of polynomials over a generic semi-algebraic set using asymptotic critical values.July 2024HALDOI
• 45 miscP.Pierre Morain. Elliptic Units Above Fields With Exactly One Complex Place.June 2024HAL
• 46 reportA.Alban Quadrat. On a general robust stability test based on the homological perturbation lemma.RR-9553Inria Paris; Sorbonne UniversiteSeptember 2024, 33HALback to text
• 47 miscG.Grégoire Sergeant-Perthuis, N.Nils Ruet, D.David Rudrauf, D.Dimitri Ognibene and Y.Yvain Tisserand. Influence of the Geometry of the world model on Curiosity Based Exploration.2024HAL