2023Activity reportProject-TeamOURAGAN

2.1.1 Basic computable objects and algorithms

The basic computable objects and algorithms we study, use, optimize or develop are among the most classical ones in computer algebra and are studied by many people around the world: they mainly focus on basic computer arithmetic, linear algebra, lattices, and both polynomial system and differential system solving.

In the context of OURAGAN, it is important to avoid reinventing the wheel and to re-use wherever possible existing objects and algorithms, not necessarily developed in our team so that the main effort is focused on finding good formulations/modelisations for an efficient use. Also, our approach for the development of basic computable objects and algorithms is application-driven and follows a simple strategy: use the existing tools in priority, develop missing tools when required and then optimize the critical operations. First, for some selected problems, we do propose and develop general key algorithms (isolation of real roots of univariate polynomials, parametrisations of solutions of zero-dimensional polynomial systems, solutions of parametric equations, equidimensional decompositions, etc.) in order to complement the existing set computable objects developed and studied around the world (Gröbner bases, resultants 70, subresultants 92, critical-point methods 47, etc.) which are also deeply used in our developments. Second, for a selection of well-known problems, we propose different computational strategies (for example the use of approximate arithmetic to speed up LLL algorithm or root isolators, still certifying the final result). Last, we propose specialized variants of known algorithms optimized for a given problem (for example, dedicated solvers for degenerated bivariate polynomials to be used in the computation of the topology of plane curves).

In the activity of OURAGAN, many key objects or algorithms around the resolution of algebraic systems are developed or optimized within the team, such as the resolution of polynomials in one variable with real coefficients 11117, rational parameterizations of solutions of zero-dimensional systems with rational coefficients 5516 or discriminant varieties for solving systems depending on parameters 14, but we are also power users of existing software (mainly Sage 1, Maple 2, Pari-GP 3,Snappea 4) and libraries (mainly gmp 5, mpfr 6, flint 7, arb 8, etc.) to which we contribute when it makes sense.

For our studies in number theory and applications to the security of cryptographic systems, our team works on three categories of basic algorithms: discrete logarithm computations 106 (for example to make progress on the computation of class groups in number fields 93), network reductions by means of LLL variants 81 and, obviously, various computations in linear algebra, for example dedicated to almost sparse matrices 107.

For the algorithmic approach to algebraic analysis of functional equations 51109110, we developed the effective study of both module theory and homological algebra 141 over certain noncommutative polynomial rings of functional operators 4, of Stafford's famous theorems on the Weyl algebras 131, of the equidimensional decomposition of functional systems 128, etc.

Finally, we study effective methods in algebraic topology, with a view towards the computation of normal forms or bases, and the construction of small resolutions of various algebraic structures: monoids and groups, algebras and operads, categories and higher structures, etc. The construction methods can come from combinatorial group theory (rewriting, Garside structures), combinatorial algebra (Gröbner bases), or homological algebra (Koszul duality, Morse theory). We explore potential deep foundational connexions between these different points of view, to unify, generalise and improve them.

2.1.2 Computational number theory

Many frontiers between computable objects, algorithms (above section), computational number theory and applications, especially in cryptography are porous. However, one can classify our work in computational number theory into two classes of studies: computational algebraic number theory and (rigorous) numerical computations in number theory.

Our work on rigorous numerical computations is somehow a transverse activity in Ouragan: floating point arithmetic is used in many basic algorithms we develop (root isolation, LLL) and is thus present in almost all our research directions. However there are specific developments that could be labelized Number Theory, in particular contributions to numerical evaluations of $L$-functions which are deeply used in many problems in number theory (for example the Riemann Zeta function). We participate, for example to the L-functions and Modular Forms Database9 a world wide collaborative project.

Our work in computational algebraic number theory is driven by the algorithmic improvement to solve presumably hard problems relevant to cryptography. The use of number-theoretic hard problems in cryptography dates back to the invention of public-key cryptography by Diffie and Hellman 77, where they proposed a first instantiation of their paradigm based on the discrete logarithm problem in prime fields. The invention of RSA 139, based on the hardness of factoring came as a second example. The introduction of discrete logarithms on elliptic curves 11245 only confirmed this trend.

These crypto-systems attracted a lot of interest on the problems of factoring and discrete log. Their study led to the invention of fascinating new algorithms that can solve the problems much faster than initially expected:

• the elliptic curve method (ECM) 147,
• the quadratic field for factoring 125 and its variant for discrete log called the Gaussian integers method 120,
• the number field sieve (NFS) 148.

Since the invention of NFS in the 90’s, many optimizations of this algorithm have been performed. However, an algorithm with better complexity hasn’t been found for factoring and discrete logarithms in large characteristic.

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.

Members of OURAGAN started working on the topic of discrete logarithms around 1998, with several computation records that were announced on the NMBRTHRY mailing list. In large characteristic, especially for the case of prime fields, the best current method is the number field sieve (NFS) algorithm. In particular, they published the first NFS based record computation13. Despite huge practical improvements, the prime field case algorithm hasn't really changed since that first record. Around the same time, we also presented small characteristic computation record based on simplifications of the Function Field Sieve (FFS) algorithm  105.

In 2006, important changes occurred concerning the FFS and NFS algorithms, indeed, while the algorithms only covered the extreme case of constant characteristic and constant extension degree, two papers extended their ranges of applicability to all finite fields. At the same time, this permitted a big simplification of the FFS, removing the need for function fields.

Starting from 2012, new results appeared in small characteristic. Initially based on a simplification of the 2006 result, they quickly blossomed into the Frobenial representation methods, with quasi-polynomial time complexity 106, 94.

An interesting side-effect of this research was the need to revisit the key sizes of pairing-based cryptography. This type of cryptography is also a topic of interest for OURAGAN. In particular, it was introduced in 2000 12.

The computations of class groups in number fields have strong links with the computations of discrete logarithms or factorizations using the NFS (number field sieve) strategy which as the name suggests is based on the use of number fields. Roughly speaking, the NFS algorithm uses two number fields and the strategy consists in choosing number fields with small sized coefficients in their definition polynomials. On the contrary, in class group computations, there is a single number field, which is clearly a simplification, but this field is given as input by some fixed definition polynomial. Obviously, the degree of this polynomial as well as the size of its coefficients are both influencing the complexity of the computations so that finding other polynomials representing the same class group but with a better characterization (degree or coefficient's sizes) is a mathematical problem with direct practical consequences. We proposed a method to address the problem 93, but many issues remain open.

Computing generators of principal ideals of cyclotomic fields is also strongly related to the computation of class groups in number fields. Ideals in cyclotomic fields are used in a number of recent public-key cryptosystems. Among the difficult problems that ensure the safety of these systems, there is one that consists in finding a small generator, if it exists, of an ideal. The case of cyclotomic fields is considered 50.

2.1.3 Topology in small dimension

2.1.4 Algebraic analysis of functional systems

Systems of functional equations or simply functional systems are systems whose unknowns are functions, such as systems of ordinary or partial differential equations, of differential time-delay equations, of difference equations, of integro-differential equations, etc.

Numerical aspects of functional systems, especially differential systems, have been widely studied in applied mathematics due to the importance of numerical simulation issues.

Complementary approaches, based on algebraic methods, are usually upstream or help the numerical simulation of systems of functional systems. These methods also tackle a different range of questions and problems such as algebraic preconditioning, elimination and simplification, completion to formal integrability or involution, computation of integrability conditions and compatibility conditions, index reduction, reduction of variables, choice of adapted coordinate systems based on symmetries, computation of first integrals of motion, conservation laws and Lax pairs, Liouville integrability, study of the (asymptotic) behavior of solutions at a singularity, etc. Although not yet very popular in applied mathematics, these theories have lengthy been studied in fundamental mathematics and were developed by Lie, Cartan, Janet, Ritt, Kolchin, Spencer, etc. 101109110113138126.

Over the past years, certain of these algebraic approaches to functional systems have been investigated within an algorithmic viewpoint, mostly driven by applications to engineering sciences such as mathematical systems theory and control theory. We have played a role towards these effective developments, especially in the direction of an algorithmic approach to the so-called algebraic analysis109, 110, 51, a mathematical theory developed by the Japanese school of Sato, which studies linear differential systems by means of both algebraic and analytic methods. To develop an effective approach to algebraic analysis, we first have to make algorithmic standard results on rings of functional operators, module theory, homological algebra, algebraic geometry, sheaf theory, category theory, etc., and to implement them in computer algebra systems. Based on elimination theory (Gröbner or Janet bases 101, 68, 140, differential algebra 5382, Spencer's theory 126, etc.), in 4, 5, we have initiated such a computational algebraic analysis approach for general classes of functional systems (and not only for holonomic systems as done in the literature of computer algebra 68). Based on the effective aspects to algebraic analysis approach, the parametrizability problem 4, the reduction and (Serre) decomposition problems 5, the equidimensional decomposition 128, Stafford's famous theorems for the Weyl algebras 131, etc., have been studied and solutions have been implemented in Maple, Mathematica, and GAP675. But these results are only the first steps towards computational algebraic analysis, its implementation in computer algebra systems, and its applications to mathematical systems, control theory, signal processing, mathematical physics, etc.

Rigorous numerical computations

Some studies in this area will be driven by some other directions, for example, the rigorous evaluation of non algebraic functions on algebraic varieties might become central for some of our work on topology in small dimension (volumes of varieties, drawing of amoeba) or control theory (approximations of discriminant varieties) are our two main current sources of interesting problems. In the same spirit, the work on $L$-functions computations (extending the computation range, algorithmic tools for computing algebraic data from the $L$ function) will naturally follow.

On the other hand, another objective is to extend existing results on periods of algebraic curves to general curves and higher dimensional varieties is a general promising direction. This project aims at providing tools for integration on higher homology groups of algebraic curves, ie computing Gauss-Manin connections. It requires good understanding of their topology, and more algorithmic tools on differential equations.

Character varieties

The brute force approach to computable objects from topology of small dimension will not allow any significant progress. As explained above, the systems that arise from these problems are simply outside the range of doable computations. We still continue the work in this direction by a four-fold approach, with all three directions deeply inter-related. First, we focus on a couple of especially meaningful (for the applications) cases, in particular the 3-dimensional manifold called Whitehead link complement. At this point, we are able to make steps in the computation and describe part of the solutions 86, 98; we hope to be able to complete the computation using every piece of information to simplify the system. Second, we continue the theoretical work to understand more properties of these systems 83. These properties may prove how useful for the mathematical understanding is the resolution of such systems - or at least the extraction of meaningful information. This approach is for example carried on by Falbel and his work on configuration of flags 87, 89. Third, we position ourselves as experts in the know-how of this kind of computations and natural interlocutors for colleagues coming up with a question on such a computable object (see 95 and 98). This also allows us to push forward the kind of computation we actually do and make progress in the direction of the second point. We are credible interlocutors because our team has the blend of theoretical knowledge and computational capabilities that grants effective resolutions of the problems we are presented. And last, we use the knowledge already acquired to pursue our theoretical study of the CR-spherical geometry 75, 88, 84.

Another direction of work is the help to the community in experimental mathematics on new objects. It involves downsizing the system we are looking at (for example by going back to systems coming from hyperbolic geometry and not CR-spherical geometry) and get the most out of what we can compute, by studying new objects. An example of this research direction is the work of Guilloux around the volume function on deformation varieties. This is a real-analytic function defined on the varieties we specialized in computing. Being able to do effective computations with this function led first to a conjecture 97. Then, theoretical discussions around this conjecture led to a paper on a new approach to the Mahler measure of some 2-variables polynomials 96. In turn, this last paper gave a formula for the Mahler measure in terms of a function akin to the volume function applied at points in an algebraic variety whose moduli of coordinates are 1. The OURAGAN team has the expertise to compute all the objects appearing in this formula, opening the way to another area of application. This area is deeply linked with number theory as well as topology of small dimension. It requires all the tools at disposition within OURAGAN.

Knot theory

We will carry on the exhaustive search for the lexicographic degrees for the rational knots. They correspond to trigonal space curves: computations in the braid group $B_3$, explicit parametrization of trigonal curves corresponding to "dessins d'enfants", etc. The problem seems much more harder when looking for more general knots.

On the other hand, a natural direction would be: given an explicit polynomial space curve, determine the under/over nature of the crossings when projecting, draw it and determine the known knot 16 it is isotopic to.

Visualization and computational geometry

As mentioned above, the drawing of algebraic curves and surfaces is a critical action in OURAGAN since it is a key ingredient in numerous developments. In some cases, one will need a fully certified study of the variety for deciding existence of solutions (for example a region in a robot's parameter's space with solutions to the DKP above or deciding if some variety crosses the unit polydisk for some stability problems in control-theory), in some other cases just a partial but certified approximation of a surface (path planning in robotics, evaluation of non algebraic functions over an algebraic variety for volumes of knot complements in the study of character varieties).

On the one hand, we will contribute to general tools like ISOTOP 17 under the supervision of the GAMBLE project-team and, on the other hand, we will propose ad-hoc solutions by gluing some of our basic tools (problems of high degrees in robust control theory). The priority is to provide a first software that implements methods that fit as most as possible the very last complexity results we got on several (theoretical) algorithms for the computation of the topology of plane curves.

A particular effort will be devoted to the resolution of overconstraint bivariate systems which are useful for the studies of singular points and to polynomials systems in 3 variables in the same spirit : avoid the use of Gröbner basis and propose a new algorithm with a state-of-the-art complexity and with a good practical behavior.

In parallel, one will have to carefully study the drawing of graphs of non algebraic functions over algebraic complex surfaces for providing several tools which are useful for mathematicians working on topology in small dimension (a well-known example is the drawing of amoebia, a way of representing a complex curve on a sheet of paper).

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
• Authors:
  Fabrice Rouillier, Alexander Kobel, Michael Sagraloff
• 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/
• Authors:
  Damien Chablat, Fabrice Rouillier, Guillaume Moroz, Philippe Wenger
• 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

7.2.1 Visualisation of limit sets

Character varieties are studied in the team as an interesting algebraic object. This study is completed by an effort to understand the geometrical meaning of each points in some carefully chosen character varieties. One approach to this problem is the construction of geometric structures.

Another approach is the study of limit sets associated to such points and their deformations when moving in the character variety. Those are fractal objects in the 3-sphere ${𝕊}^3$, which shares numerous properties with the usual fractal from complex dynamics. In our work, we study this limit sets first and foremost in an experimental way, leading to a ( Landscape of limit sets). The computation of such visualisations leverages the theoretical properties of these limit sets, certified numerical approximations. An improved version of the visualisation is in progress, and will use the theory of automatic groups as well as new and more efficient parametrizations of the objects of study. The experimental approach in turns inform the theoretical one. One important new step, bridging the two approaches, is done in 85.

Fast verification and public key storage optimization for unstructured lattice-based signatures

A recent work of Sipasseuth, Plantard and Susilo proposed to accelerate lattice-based signature verifications and compress public key storage at the cost of a precomputation on a public key. This first approach, which focused on a restricted type of key, did not include most NIST candidates or most lattice representations in general. In 18, we first present a way to improve even further both their verification speed and their public key compression capability by using a generator of numbers that better suit the method needs. We then also generalize their framework to apply to q-ary lattice schemes as well as classical lattices using Hermite Normal Form, improving their security and applicable scope, thus exhibiting potential trade-offs to accelerate lattice-based signature verification in general and compression of the public key on the verifier side for unstructured lattices.

Shared permutation for syndrome decoding: new zero-knowledge protocol and code-based signature

Zero-knowledge proofs are an important tool for many cryptographic protocols and applications. The threat of a coming quantum computer motivates the research for new zero-knowledge proof techniques for (or based on) post-quantum cryptographic problems. One of the few directions is code-based cryptography for which the strongest problem is the syndrome decoding (SD) of random linear codes. This problem is known to be NP-hard and the cryptanalysis state of affairs has been stable for many years. A zero-knowledge protocol for this problem was pioneered by Stern in 1993. As a simple public-coin three-round protocol, it can be converted to a post-quantum signature scheme through the famous Fiat-Shamir transform. The main drawback of this protocol is its high soundness error of $2/3$, meaning that it should be repeated 1.7*$\lambda$ times to reach a $\lambda$-bit security. In 28, we improve this three-decade-old state of affairs by introducing a new zero-knowledge proof for the syndrome decoding problem on random linear codes. Our protocol achieves a soundness error of $1/n$ for an arbitrary $n$ in complexity $O\left(n\right)$. Our construction requires the verifier to trust some of the variables sent by the prover which can be ensured through a cut-and-choose approach. We provide an optimized version of our zero-knowledge protocol which achieves arbitrary soundness through parallel repetitions and merged cut-and-choose phase. While turning this protocol into a signature scheme, we achieve a signature size of 17 KB for a 128-bit security. This represents a significant improvement over previous constructions based on the syndrome decoding problem for random linear codes.

On the Hardness of the Finite Field Isomorphism Problem

The finite field isomorphism (FFI) problem was introduced in PKC'18, as an alternative to average-case lattice problems (like LWE, SIS, or NTRU). As an application, the same paper used the FFI problem to construct a fully homomorphic encryption scheme. In 35, we prove that the decision variant of the FFI problem can be solved in polynomial time for any field characteristics $q=\Omega \left(\beta n^2\right)$, where $q,\beta ,n$ parametrize the FFI problem. Then we use our result from the FFI distinguisher to propose polynomial-time attacks on the semantic security of the fully homomorphic encryption scheme. Furthermore, for completeness, we also study the search variant of the FFI problem and show how to state it as a q-ary lattice problem, which was previously unknown. As a result, we can solve the search problem for some previously intractable parameters using a simple lattice reduction approach.

On subgroups of finite index in complex hyperbolic lattice triangle groups

In24, we study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.

Torsion in 1-cusped Picard modular groups

In 25 present a systematic effective method to construct coarse fundamental domains for the action of the Picard modular groups $PU(2,1,{𝒪}_d)$ where ${𝒪}_d$ has class number one, i.e. $d=1,2,3,7,11,19,43,67,163$. The computations can be performed quickly up to the value $d=19$. As an application of this method, we classify conjugacy classes of torsion elements, deduce short presentations for the groups, and construct neat subgroups of small index.

Computing the non-properness set of real polynomial maps in the plane

In 26, we introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call the Jelonek set, is a subset of ${𝕂}^2$ where a dominant polynomial map $f:{𝕂}^2\mapsto {𝕂}^2$ is not proper; $𝕂$ could be either $ℂ$ or $ℝ$. Unlike all the previously known approaches we make no assumptions on $f$ whenever $𝕂=ℝ$; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in Maple.

Trifocal Relative Pose From Lines at Points

In 27, we present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of ( i ) three points and one line and the novel case of ( ii ) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Gröbner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver framework MINUS, which dramatically speeds up previous HC solving by specializing hc methods to generic cases of our problems. We characterize their number of solutions and show with simulated experiments that our solvers are numerically robust and stable under image noise, a key contribution given the borderline intractable degree of nonlinearity of trinocular constraints. We show in real experiments that ( i ) sift feature location and orientation provide good enough point-and-line correspondences for three-view reconstruction and ( ii ) that we can solve difficult cases with too few or too noisy tentative matches, where the state of the art structure from motion initialization fails.

Algorithmic study of the algebraic parameter estimation problem for a class of perturbations

In 21, we consider the algebraic parameter estimation problem for a class of standard perturbations. We assume that the measurement $z\left(t\right)$ of a solution $x\left(t\right)$ of a linear ordinary differential equation, whose coefficients depend on a set $\theta =\{{\theta }_1,...,{\theta }_r\}$ of unknown constant parameters,is affected by a perturbation $\gamma \left(t\right)$ whose structure is supposed to be known (e.g., an unknown bias, an unknown ramp), i.e., $z\left(t\right)=x(t,\theta )+\gamma \left(t\right)$. We investigate the problem of obtaining closed-form expressions for the parameters $\theta$ ’s in terms of repeated $i$ indefinite integrals or convolutions of $z$. We illustrate the different results with explicit examples computed using the NonA package, developed in Maple, in which we have implemented our main contributions.

Further results on the computation of the annihilators of integro-differential operators

In 34, we expose some effective aspects of the algebra of linear ordinary integro-differential operators with polynomial coefficients. More precisely, we prove that the annihilator of an evaluation operator is a finitely generated ideal which can be explicitly characterized and computed. This is an advance towards the development of an effective elimination theory for ordinary integro-differential operators and an effective study of linear systems of integro-differential equations with polynomial coefficients.

Coherent presentations of monoids with a right-noetherian Garside family

In 22, the authors show how to construct coherent presentations (presentations by generators, relations and relations among relations) of monoids admitting a right-noetherian Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of construction of coherent presentations for spherical Artin-Tits monoids: to general Artin-Tits monoids, and to Garside monoids. The result is applied to some monoids which are neither Artin-Tits nor Garside.

Geometric algorithms for sampling the flux space of metabolic networks

Metabolic networks and their reconstruction set a new era in the analysis of metabolic and growth functions in the various organisms. By modeling the reactions occurring inside an organism, metabolic networks provide the means to understand the underlying mechanisms that govern biological systems. Constraint-based approaches have been widely used for the analysis of such models and led to intriguing geometry-oriented challenges. In this setting, sampling uniformly points from polytopes derived from metabolic models (flux sampling) provides a representation of the solution space of the model under various conditions. However, the polytopes that result from such models are of high dimension (in the order of thousands) and usually considerably skinny. Therefore, to sample uniformly at random from such polytopes shouts for a novel algorithmic and computational framework specially tailored for the properties of metabolic models. In 19, we present a complete software framework to handle sampling in metabolic networks. Its backbone is a Multiphase Monte Carlo Sampling (MMCS) algorithm that unifies rounding and sampling in one pass, yielding both upon termination. It exploits an optimized variant of the Billiard Walk that enjoys faster arithmetic complexity per step than the original. We demonstrate the efficiency of our approach by performing extensive experiments on various metabolic networks. Notably, sampling on the most complicated human metabolic network accessible today, Recon3D, corresponding to a polytope of dimension 5335, took less than 30 hours. To the best of our knowledge, that is out of reach for existing software.

Truncated Log-concave Sampling for Convex Bodies with Reflective Hamiltonian Monte Carlo

In 20, we introduce Reflective Hamiltonian Monte Carlo (ReHMC), an HMC-based algorithm to sample from a log-concave distribution restricted to a convex body. The random walk is based on incorporating reflections to the Hamiltonian dynamics such that the support of the target density is the convex body. We develop an efficient open source implementation of ReHMC and perform an experimental study on various high-dimensional datasets. The experiments suggest that ReHMC outperforms Hit-and-Run and Coordinate-Hit-and-Run regarding the time it needs to produce an independent sample, introducing practical truncated sampling in thousands of dimensions.

Segre-driven radicality testing

In 29, wee 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.

Randomized geometric tools for anomaly detection in stock markets

In 33, we propose novel randomized geometric tools to detect low-volatility anomalies in stock markets; a principal problem in financial economics. Our modeling of the (detection) problem results in sampling and estimating the (relative) volume of geodesically non-convex and non-connected spherical patches that arise by intersecting a non-standard simplex with a sphere. To sample, we introduce two novel Markov Chain Monte Carlo (MCMC) algorithms that exploit the geometry of the problem and employ state-of-the-art continuous geometric random walks (such as Billiard walk and Hit-and-Run) adapted on spherical patches. To our knowledge, this is the first geometric formulation and MCMC-based analysis of the volatility puzzle in stock markets. We have implemented our algorithms in C++ (along with an R interface) and we illustrate the power of our approach by performing extensive experiments on real data. Our analyses provide accurate detection and new insights into the distribution of portfolios’ performance characteristics. Moreover, we use our tools to show that classical methods for low-volatility anomaly detection in finance form bad proxies that could lead to misleading or inaccurate results.

On Isolating Roots in a Multiple Field Extension

In 36, we address univariate root isolation when the polynomial's coefficients are in a multiple field extension. We consider a polynomial $F\in L\left[Y\right]$, where $L$ is a multiple algebraic extension of $\mathbb{Q}$. We provide aggregate bounds for $F$ and algorithmic and bit-complexity results for the problem of isolating its roots. For the latter problem we follow a common approach based on univariate root isolation algorithms. For the particular case where $F$ does not have multiple roots, we achieve a bit-complexity in ${\tilde{O}}_B\left(n{d}^{2n+2}(d+n\tau )\right)$, where $d$ is the total degree and $\tau$ is the bitsize of the involved polynomials. In the general case we need to enhance our algorithm with a preprocessing step that determines the number of distinct roots of $F$. We follow a numerical, yet certified, approach that has bit-complexity ${\tilde{O}}_B(n^2{d}^{3n+3}\tau +n^3{d}^{2n+4})$.

On the Certification of the Kinematics of 3-DOF Spherical Parallel Manipulators

In 31 we aims to study a specific kind of parallel robot: Spherical Parallel Manipulators (SPM) that are capable of unlimited rolling. A focus is made on the kinematics of such mechanisms, especially taking into account uncertainties (e.g. on conception & fabrication parameters, measures) and their propagations. Such considerations are crucial if we want to control our robot correctly without any undesirable behavior in its workspace (e.g. effects of singularities). In this paper, we will consider two different approaches to study the kinematics and the singularities of the robot of interest: symbolic and semi-numerical. By doing so, we can compute a singularity-free zone in the work- and joint spaces, considering given uncertainties on the parameters. In this zone, we can use any control law to inertially stabilize the upper platform of the robot.

10.1.1 ANR

• ANR JCJC GALOP (Games through the lens of ALgebra and OPptimization)

  Coordinator: Elias Tsigaridas

  Duration: 2018 – 2023

  GALOP is a Young Researchers (JCJC) project with the purpose of extending the limits of the state- of-the-art algebraic tools in computer science, especially in stochastic games. It brings original and innovative algebraic tools, based on symbolic-numeric computing, that exploit the geometry and the structure and complement the state-of-the-art. We support our theoretical tools with a highly efficient open-source software for solving polynomials. Using our algebraic tools we study the geometry of the central curve of (semi-definite) optimization problems. The algebraic tools and our results from the geometry of optimization pave the way to introduce algorithms and precise bounds for stochastic games.

• 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.

10.1.2 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.

10.1.3 Technological Development Actions

• ACE (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

11.1.2 Scientific events: selection

• Elias Tsigaridas co-supervised the edition of the Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation 39.

11.1.3 Journal

11.1.4 Research administration

• Elisha Falbel is director of the "École Doctorale Sciences Mathématiques de Paris Centre - ED 386".
• 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 an elected member of the scientific council of the department of mathematics of Université Paris 7 / Université de Paris / Université Paris Cité (2019-2022).
• Yves Guiraud is an elected member of the laboratory council of IMJ-PRG (since 2021).
• 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, 24h “Algebraic techniques in optimation”, Master 2 Mathématiques fondamentales, Sorbonne 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

• Jean-Claude Bajard supervised the PhD of Thibauld Feneuil until 09/2023 (defense date).
• Elias Tsigaridas supervised the PhD of Carles Checa, since (co-supervision with Ioannis Emiris).
• Pierre-Vincent Koseleff supervises the PhD of Andrea Negro , since 10/2021, (co-supervision with Julien Marché IMJ-PRG).
• Elias Tsigaridas and Fabrice Rouillier supervised the PhD of Christina Katsamaki until 07/2023 (defense date).
• Yves Guiraud supervised, with Pierre-Louis Curien (CNRS, Univ. Paris Cité, IRIF) the PhD thesis of Alen Đurić until 04/2023 (Defense).
• 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.
• Elias Tsigaridas, Alban Quadrat and Fabrice Rouillier supervised the PhD of Chaoping Zhu since 09/2023.

11.2.3 Juries

• Yves Guiraud was a referee for the PhD thesis of Sophie d'Espalungue, Operads in categories and models of structure interchange, Univ. Lille, December 2023.
• Alban Quadrat was a referee for the Habilitation thesis of Michaël Di Loreto Propriétés structurelles des systèmes dynamiques pour le contrôle, Institut National des Sciences Appliquées de Lyon, October, 2023.

International journals

• 18 articleJ.-C.Jean-Claude Bajard, K.Kazuhide Fukushima, T.Thomas Plantard and A.Arnaud Sipasseuth. Fast verification and public key storage optimization for unstructured lattice-based signatures.Journal of Cryptographic EngineeringJanuary 2023HALDOIback to text
• 19 articleA.Apostolos Chalkis, I. Z.Ioannis Z. Emiris, V.Vissarion Fisikopoulos, E.Elias Tsigaridas and H.Haris Zafeiropoulos. Geometric algorithms for sampling the flux space of metabolic networks.Journal of Computational Geometry1412023HALDOIback to text
• 20 articleA.Apostolos Chalkis, V.Vissarion Fisikopoulos, M.Marios Papachristou and E.Elias Tsigaridas. Truncated Log-concave Sampling for Convex Bodies with Reflective Hamiltonian Monte Carlo.ACM Transactions on Mathematical Software492June 2023, 1-25HALDOIback to text
• 21 articleM.Maya Chartouny, T.Thomas Cluzeau and A.Alban Quadrat. Algorithmic study of the algebraic parameter estimation problem for a class of perturbations.Maple Transactions3February 2023HALDOIback to text
• 22 articleP.-L.Pierre-Louis Curien, A.Alen Ðurić and Y.Yves Guiraud. Coherent presentations of monoids with a right-noetherian Garside family.Journal of Homotopy and Related Structures182023, 115-152HALDOIback to text
• 23 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 Society2024HAL
• 24 articleM.Martin Deraux. On Subgroups Finite Index in Complex Hyperbolic Lattice Triangle Groups.Experimental Mathematics2023, 1-26HALDOIback to text
• 25 articleM.Martin Deraux and M.Mengmeng Xu. Torsion in 1-Cusped Picard Modular Groups.Transformation GroupsJanuary 2023HALDOIback to text
• 26 articleB.Boulos El Hilany and E.Elias Tsigaridas. Computing the non-properness set of real polynomial maps in the plane.Vietnam Journal of MathematicsJune 2023HALDOIback to text
• 27 articleR.Ricardo Fabbri, T.Timothy Duff, H.Hongyi Fan, M.Margaret Regan, D.David da Costa de Pinho, E.Elias Tsigaridas, C.Charles Wampler, J.Jonathan Hauenstein, P.Peter Giblin, B.Benjamin Kimia, A.Anton Leykin and T.Tomas Pajdla. Trifocal Relative Pose From Lines at Points.IEEE Transactions on Pattern Analysis and Machine Intelligence456June 2023, 7870-7884HALDOIback to text
• 28 articleT.Thibauld Feneuil, A.Antoine Joux and M.Matthieu Rivain. Shared permutation for syndrome decoding: new zero-knowledge protocol and code-based signature.Designs, Codes and Cryptography912February 2023, 563-608HALDOIback to text
• 29 articleM.Martin Helmer and E.Elias Tsigaridas. Segre-driven radicality testing.Journal of Symbolic Computation122May 2024, 102262HALDOIback to text
• 30 articleK.Khazhgali Kozhasov and J.Josué Tonelli-Cueto. Probabilistic bounds on best rank-one approximation ratio.Linear and Multilinear AlgebraMarch 2024, 1-29HALDOI
• 31 articleA.Alexandre Lê, D.Damien Chablat, G.Guillaume Rance and F.Fabrice Rouillier. On the Certification of the Kinematics of 3-DOF Spherical Parallel Manipulators.Maple Transactions32August 2023HALDOIback to text
• 32 articleM.Manuel Radons and J.Josué Tonelli-Cueto. Generalized Perron Roots and Solvability of the Absolute Value Equation.SIAM Journal on Matrix Analysis and Applications444October 2023, 1645-1666HALDOI

International peer-reviewed conferences

• 33 inproceedingsC.Cyril Bachelard, A.Apostolos Chalkis, V.Vissarion Fisikopoulos and E.Elias Tsigaridas. Randomized geometric tools for anomaly detection in stock markets.Proceedings of Machine Learning Research26th International Conference on Artificial Intelligence and Statistics (AISTATS)Valencia, SpainApril 2023HALback to text
• 34 inproceedingsT.Thomas Cluzeau, C.Camille Pinto and A.Alban Quadrat. Further results on the computation of the annihilators of integro-differential operators.2023 International Symposium on Symbolic and Algebraic ComputationTromso, NorwayJuly 2023, 9HALDOIback to text
• 35 inproceedingsD.Dipayan Das and A.Antoine Joux. On the Hardness of the Finite Field Isomorphism Problem.LNCS - Lecture Notes in Computer ScienceEurocrypt 2023 - Annual International Conference on the Theory and Applications of Cryptographic Techniques14008Lecture Notes in Computer ScienceLyon, FranceSpringer Nature SwitzerlandApril 2023, 343-359HALDOIback to text
• 36 inproceedingsC.Christina Katsamaki and F.Fabrice Rouillier. On Isolating Roots in a Multiple Field Extension.ISSAC '23: 2023 International Symposium on Symbolic and Algebraic ComputationTromso, NorwayACM; ACMAugust 2023, 363-371HALDOIback to text
• 37 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 2024HAL

Scientific books

• 38 bookD.Dimitri Ara, A.Albert Burroni, Y.Yves Guiraud, P.Philippe Malbos, F.François Métayer and S.Samuel Mimram. Polygraphs: From Rewriting to Higher Categories.London Mathematical Society Lecture Note Series2023HALback to text

Edition (books, proceedings, special issue of a journal)

• 39 proceedingsA.Alicia DickensteinG.Gabriela JeronimoE.Elias TsigaridasISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation.ACM; ACMJuly 2023HALDOIback to text

Doctoral dissertations and habilitation theses

• 40 thesisT.Thibauld Feneuil. Post-Quantum Signatures from Secure Multiparty Computation.Sorbonne UniversitéOctober 2023HALback to text
• 41 thesisC.Christina Katsamaki. Exact Algebraic and Geometric Computations for Parametric Curves.Sorbonne UniversitéJune 2023HALback to text

Reports & preprints

• 42 miscE.Elisha Falbel and J. M.Jose Miguel Veloso. A global invariant for path structures and second order differential equations.June 2023HAL
• 43 miscC.Christina Katsamaki, F.Fabrice Rouillier and E.Elias Tsigaridas. Exact Convex Hull Computation for Plane and Space Parametric Curves.December 2023HAL
• 44 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.February 2024HAL