Rubén Muñoz--Bertrand

Maître de conférences in mathematics

Curriculum vitae

I'm working both in arithmetic geometry and computer algebra. More precisely, I study Witt vectors, overconvergent de Rham–Witt cohomology and its coefficients, and their applications to number theory. I've also become interested in algorithmic applications of these theories.

I have defended my PhD thesis in 2020.

Articles

1. Isocrystals and de Rham–Witt connections NEW (30/04/2026) accepted at Documenta Mathematica (2025), 28 pages, arXiv:2502.01902

   We introduce the notion of integrable connections for a sheaf of differential graded algebras on a topological space. We then describe them in the finite locally projective setting, when the sheaf is either the de Rham complex of a formal or a weakly formal scheme, or for the convergent or the overconvergent de Rham-Witt complex on a smooth scheme over a perfect field of positive characteristic. This enables us to give a new description of convergent and overconvergent isocrystals with a Frobenius structure.
2. Characterising properties of commutative rings using Witt vectors, online first at Rendiconti del Seminario Matematico della Università di Padova (2026), 14 pages, doi:10.4171/rsmup/192

   We give equivalences between given properties of a commutative ring, and other properties on its ring of Witt vectors. Amongst them, we characterise all commutative rings whose rings of Witt vectors are Noetherian. We define a new category of commutative rings called preduced rings, and explain how it is the category of rings whose ring of Witt vectors has no p-torsion. We then extend this characterisation to the torsion of the de Rham–Witt complex.
3. Pseudovaluations on the de Rham–Witt complex, Bulletin de la Société Mathématique de France 150 (2022), no. 1, pp. 53–75, doi:10.24033/bsmf.2844

   For a polynomial ring over a commutative ring of positive characteristic, we define on the associated de Rham–Witt complex a set of functions, and show that they are pseudovaluations in the sense of Davis, Langer and Zink. To achieve this, we explicitly compute products of basic elements on the complex. We also prove that the overconvergent de Rham–Witt complex can be recovered using these pseudovaluations.

Preprints

1. Effective Artin–Schreier–Witt theory for curves (2025, joint with Christophe Levrat), 31 pages, arXiv:2509.10633

   We present an algorithm which, given a connected smooth projective curve 𝑋 over an algebraically closed field of characteristic 𝑝>0 and its Hasse–Witt matrix, as well as a positive integer 𝑛, computes all étale Galois covers of 𝑋 with group ℤ/𝑝^𝑛ℤ. We compute the complexity of this algorithm when 𝑋 is defined over a finite field, and we provide a complete implementation of this algorithm in SageMath, as well as some explicit examples. We then apply this algorithm to the computation of the cohomology complex of a locally constant sheaf of ℤ/𝑝^𝑛ℤ-modules on such a curve.
2. Faster computation of Witt vectors over a polynomial ring (2025), 11 pages, arXiv:2504.01834

   We describe an algorithm which computes the ring laws for Witt vectors of finite length over a polynomial ring with coefficients in a finite field. This algorithm uses an isomorphism of Illusie in order to compute in an adequate polynomial ring. We also give an implementation of the algorithm in SageMath, which turns out to be faster that Finotti's algorithm, which was until now the most efficient one for these operations.
3. Local structure of the overconvergent de Rham–Witt complex (2023), 54 pages, arXiv:2311.15449

   We give a general description of the structure of the relative de Rham–Witt complex on a polynomial ring, seen as an algebra over its integral part. After giving a control of the overconvergence of Lazard's morphism, we similarly give the structure of the overconvergent complex for a finite étale algebra over a perfect Noetherian ring. We then deduce a generalization of the usual decomposition as a direct sum, which is compatible with overconvergence on the projections.

Software

I have also implemented scripts related to my mathematical work. I am actively involved in SageMath, so do not hesitate to write me about packages features related to the work below. Some more code can be found in my curriculum vitae.

1. Witt vectors (2025), SageMath package (since version 10.7)

   Based on a PR by Jacob Dennerlein, and with the help of Xavier Caruso and Frédéric Chapoton, I implemented Witt vectors in SageMath. My implementation includes the new phantom algorithm, which is much faster when the coefficient ring is a polynomial ring over a finite field.
2. Effective Artin–Schreier–Witt theory for curves (2025, joint with Christophe Levrat), SageMath code of our article

   This is the implementation of our article. Given a connected smooth projective curve 𝑋 over an algebraically closed field of characteristic 𝑝>0, as well as a positive integer 𝑛, it computes all étale Galois covers of 𝑋 with group ℤ/𝑝^𝑛ℤ.
3. Faster computation of Witt vectors over a polynomial ring (2025), SageMath code of my article

   This is the implementation of my article. It is a new algorithm computing the ring laws of Witt vectors for polynomial rings over a finite field in positive characteristic.

Teaching

You can find below some lecture notes of my past teachings.

1. p-adic cohomology theories and point counting, draft updated on June 26, 2023, comments welcome!
2. Introduction à la théorie des groupes, online version updated on September 06, 2024.

How to spell my name

I have noticed that my name is often misspelt. This is because of its two diacritical signs, but mostly because of the two dashes in my family name. It is indeed a double dash, and not a simple one, nor a long one, and certainly not a typo!

If you use LaTeX2e with a version released after 2018, you simply have to use that code:

Rubén Muñoz-\relax-Bertrand

In Plain TeX, or for obsolete versions of LaTeX2e, you can use the following code:

Rub\'en Mu\~noz-\relax-Bertrand

For BibTeX, you can use:

Mu{\~n}oz-{}-Bertrand, Rub{\'e}n

For Markdown, you can try:

Rubén Muñoz-\-Bertrand

I would be very grateful if you use the proper spelling of my name.