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Integration is indispensable, not only in mathematics, but also in a wide range of other fields. A deep learning method has recently been developed and shown to be capable of integrating mathematical functions that could not previously be…

Transformer-based models excel in various tasks but their generalization capabilities, especially in arithmetic reasoning, remain incompletely understood. Arithmetic tasks provide a controlled framework to explore these capabilities, yet…

The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…

The proof assistant Lean has support for abstract polynomials, but this is not necessarily the same as support for computations with polynomials. Lean is also a functional programming language, so it should be possible to implement…

We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of first-order functional programs. We explain their semantics and prove that they form a…

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…

High-throughput computational screening of polymers offers a powerful way to address the imbalance between the vast number of polymers synthesised for diverse applications and the relatively small subset that can be studied using atomistic…

We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in…

We show a new algorithm and its implementation for multiplying bit-polynomials of large degrees. The algorithm is based on evaluating polynomials at a specific set comprising a natural set for evaluation with additive FFT and a high order…

A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…

Motivated by existing blend-to-zero techniques, a formal framework is developed for defining and constructing blend-to-zero operators on closed intervals for the generation of sufficiently smooth transitions between functions. Such…

Transmission system operators need to adapt their decision-making tools to the technological evolutions of the twenty first century. A computation inherent to most tools seeks to find alternating-current power flows that minimize power loss…

When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…

In this paper, I introduce the retrieval problem, a simple yet common reasoning task that can be solved only by transformers with a minimum number of layers, which grows logarithmically with the input size. I empirically show that large…

This paper describes a general-purpose programming technique, called the Simulation of Simplicity, which can be used to cope with degenerate input data for geometric algorithms. It relieves the programmer from the task to provide a…

Modular exponentiation is crucial to number theory and cryptography, yet remains largely unexplored from a mechanistic interpretability standpoint. We train a 4-layer encoder-decoder Transformer model to perform this operation and…

Constructing accurate and automatic solvers of math word problems has proven to be quite challenging. Prior attempts using machine learning have been trained on corpora specific to math word problems to produce arithmetic expressions in…

We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete…

Developing robust representations of chemical structures that enable models to learn topological inductive biases is challenging. In this manuscript, we present a representation of atomistic systems. We begin by proving that our…

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement…