Abstract. Differential algebra is the branch of algebra that studies differential equations from an algebraic standpoint,inspired by the way polynomial equation are studied using algebraic geometry.
A key topic in this field is the elimination of variables. More precisely, given a system with several variables, we aim to describe the differential relations satisfied by a selected subset of those variables. Elimination is used to analyze systems of differential equations, especially in the context of dynamical systems models where data is available only for a subset of the variables.
In this talk, we provide an overview of existing tools for differential elimination and discuss their potential applications. We present our main result: the characterization of the set of terms that can appear in the resulting polynomial after elimination. Based on this result, we introduce an algorithm and demonstrate that our implementation can handle problems that are beyond the capabilities of current state-of-the-art software for differential elimination. We also discuss a new approach to computing so-called mixed fiber polytopes in the context of polynomial elimination. We demonstrate the practical performance improvements of our algorithm compared to existing methods and highlight an application of this work to differential elimination.
Joint works with Gleb Pogudin and Rafael Mohr.

Abstract.

Abstract.

Abstract. There have been several works recently on getting sharper upper bounds for the tensor rank of matrix multiplication tensors for small and specific matrix formats. One of the techniques applied in this context is the flip graph method introduced by Jakob Moosbauer and the speaker at ISSAC'23. In the talk, we will explain this method, discuss its pros and cons, and summarize the results we obtained with it so far.

Abstract.

Abstract. Computing a Gröbner basis has been a fundamental task in computational algebra for more than 50 years. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the worst case. In this talk, I present a new paradigm for addressing such problems, i.e., a machine-learning approach using a Transformer model. This is a joint work with Hiroshi Kera, Yuki Ishihara, Yuta Kambe, Yota Maeda, et Kazuhiro Yokoyama. The learning approach does not require an explicit algorithm design and can return the solutions in (roughly) constant time. This talk covers our initial results on this approach and relevant computational algebraic and machine learning challenges. In particular, the learning approach needs a large-scale dataset of couples of non-Gröbner sets and Gröbner bases generating the same ideal. This gives rise to underexplored open problems: the random generation of Gröbner bases and their ideal-invariant transformation into non-Gröbner sets, the structure of GL_n (k[X_1,...,X_n]) and its relation to effective version of the Quillen-Suslin theorem.

Abstract. The central configurations of the Newtonian n-body problem are configurations of n point bodies, each with a positive mass, such that the Newtonian attraction may be exactly balanced by a centrifugal force. For example, a planar central configuration may rotate uniformly around its center of mass, thus defining a relative equilibrium of the n-body problem. Astronomical examples with n=3 are the Trojan asteroids and other bodies at some Lagrange point.
In 1995 I proved that there are only four types of central configurations of 4 bodies with equal masses, by using polynomial elimination. Here I wish to determine the central configurations of 5 bodies which have 3 bodies on a line. I will solve this problem in a symmetric case, where the existence result was published by Marian Gidea and Jaume Llibre in 2010, and independently by Kuo-Chang Chen and Jun-Shian Hsiao in 2011.

Abstract. We introduce a new family of rank metric codes. The construction of these codes relies on the "Chinese Remainder Theorem" for linearised polynomials. Linearised polynomials are polynomials in which the exponents of all the monomials are powers of q and the coefficients come from an extension field of the finite field of order q. The set of these polynomials forms a right Euclidean ring. We present the lifting of the isomorphism of the Chinese remainder theorem for linearised polynomials. We show how this lifting leads to a decoding algorithm for a special case of this family of codes: the case where the linearised polynomials have coefficients in the finite field of order q. (Joint work with Olivier Ruatta and Philippe Gaborit).
(The slides of this talk are available here.)

Abstract. We propose a new method for retrieving the algebraic structure of a generic alternant code given an arbitrary generator matrix, provided certain conditions are met. We then discuss how this challenges the security of the McEliece cryptosystem instantiated with this family of codes. The central object of our work is the quadratic hull related to a linear code, defined as the intersection of all quadrics passing through the columns of a given generator or parity-check matrix, where the columns are considered as points in the affine or projective space. The geometric properties of this object reveal important information about the internal algebraic structure of the code. This is particularly evident in the case of generalized Reed-Solomon codes, whose quadratic hull is deeply linked to a well-known algebraic variety called the rational normal curve. By utilizing the concept of Weil restriction of affine varieties, we demonstrate that the quadratic hull of a generic dual alternant code inherits many interesting features from the rational normal curve, on account of the fact that alternant codes are subfield-subcodes of generalized Reed-Solomon codes. If the rate of the generic alternant code is sufficiently high, this allows us to construct a polynomial-time algorithm for retrieving the underlying generalized Reed-Solomon code from which the alternant code is defined, which leads to an efficient key-recovery attack against the McEliece cryptosystem when instantiated with this class of codes. Finally, we discuss the generalization of this approach to Algebraic-Geometry codes and Goppa codes.
(The slides of this talk are available here.)

Abstract. Algbraic invariants plays a cental role in program analysis and verification, they provide a compact certificate for corrctness. A natural way to study them is through the algebraic closure of matrix monoids, which captures all the polynomial relations satisfied by the reachable configurations of linear or affine programs. Recent results show that given a finite set of rational matrices, the algebraic closure of the monoid they generate is effectively computable. This opens up new directions, ranging from the complexity of computing invariants to structural questions about orbit closures and algebraic groups. In this talk I will survey some of these problems and the techniques used to tackle them. I will begin with a polynomial-time algorithm for computing closures in the case of a single matrix, before moving on to the general case. I will then discuss the “reverse problem” for commutative matrices: given a set of polynomials, does their ideal define the orbit closure of an algebraic group? Finally, I will touch on connections with reachability and closure problems in one-counter systems (1-VASS). This talk is based on the following papers: arXiv:2407.09154, arXiv:2407.04626, arXiv:2507.09373.

Abstract. We introduce the class of P-finite automata. These are a generalisation of weighted automata, in which the weights of transitions can depend polynomially on the length of the input word. P-finite automata can also be viewed as simple tail-recursive programs in which the arguments of recursive calls can non-linearly refer to a variable that counts the number of recursive calls. The nomenclature is motivated by the fact that over a unary alphabet P-finite automata compute so-called P-finite sequences, that is, sequences that satisfy a linear recurrence with polynomial coefficients. Our main result shows that P-finite automata can be learned in polynomial time in Angluin's MAT exact learning model. This generalises the classical results that deterministic finite automata and weighted automata over a field are respectively polynomial-time learnable in the MAT model.

Abstract. In arithmetic geometry, Artin-Schreier-Witt theory gives a way to construct unramified covers of curves. However, nobody has ever made a full computation using these techniques. In this talk, we will explain how we implemented an algorithm which enables one to compute such covers. I will explain the motivation behind this problem, and the challenges that one faces using this theory. For our computations, it is necessary to have an algorithm which computes the fixed points of a Frobenius-semilinear map. I will also explain how to implement such an algorithm. This is a joint work with Christophe Levrat (Inria Saclay).

Abstract. Differential and difference algebra studies the algebraic aspect of differential and difference equations, which provides an algebraic foundation for symbolic computation of differential and difference equations. In this talk, we will first introduce a stability problem in differential and difference fields. Secondly, we present some stability criteria for several classes of special functions. At the end of this talk, we will prove that all D-finite functions or P-recursive sequences are eventually stable.
(The slides of this talk are available here.)

Abstract. This informal presentation provides an overview of the current state of small fast matrix multiplication algorithm development. We focus on finding compact rank tensor decompositions related to the bilinear map that computes the product of n×p and p×q matrices, where n,p and q are all less than 32. We review the main methods currently employed to construct the decompositions (with the smallest known tensor rank) and highlight the limitations of these approaches.

Archive [2019-2021] [2021-2024] [2024-2025]