arXiv subjects

Xavier Caruso

Publications and source records attributed to Xavier Caruso.

At least 19 recordsLinked to original sources

arXiv2026

Duality and decoding of linearized Algebraic Geometry codes

We design a polynomial time decoding algorithm for linearized Algebraic Geometry codes with unramified evaluation places, a family of sum-rank metric evaluation codes on division algebras over function fields. By establishing a Serre duality and a Riemann-Roch theorem for these algebras, we prove that the dual codes of such linearized Algebraic Geometry codes, that we term linearized Differential codes, coincide with the linearized Algebraic Geometry codes themselves over the adjoint algebra, and that our decoding algorithm is correct.

cs.IT↗

arXiv2026

Algebraic and Arithmetic Attributes of Hypergeometric Functions in SageMath

We report on implementations for algorithms treating algebraic and arithmetic properties of hypergeometric functions in the computer algebra system SageMath. We treat hypergeometric series over the rational numbers, over finite fields, and over the p-adics. Among other things, we provide implementations deciding algebraicity, computing valuations, and computing minimal polynomials in positive characteristic.

cs.SC↗

arXiv2026

Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions

We discuss algorithms for arithmetic properties of hypergeometric functions. Most notably, we are able to compute the p-adic valuation of a hypergeometric function on any disk of radius smaller than the p-adic radius of convergence. This we use, building on work of Christol, to determine the set of prime numbers modulo which it can be reduced. Moreover, we describe an algorithm to find an annihilating polynomial of the reduction of a hypergeometric function modulo p.

math.NT↗

arXiv2026

Diagonals and algebraicity modulo $p$: a sharper degree bound

In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in $\mathbb F_p$. Moreover, he conjectured that the algebraic degrees $d_p$ of these functions should grow at most polynomially in $p$. In this article, we provide a new and elementary proof of Deligne's theorem, which yields the first general polynomial bound on $d_p$ with an explicit and reasonable degree.

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arXiv2026

A computational approach to Drinfeld modules

This survey provides a practical and algorithmic perspective on Drinfeld modules over $\mathbb F_q[T]$. Starting with the construction of the Carlitz module, we present Drinfeld modules in any rank and some of their arithmetic properties. We emphasise the analogies with elliptic curves, and in the meantime, we also highlight key differences such as their rank structure and their associated Anderson motives. This document is designed for researchers in number theory, arithmetic geometry, algorithmic number theory, cryptography, or computer algebra, offering tools and insights to navigate the computational aspects of Drinfeld modules effectively. We include detailed SageMath implementations to illustrate explicit computations and facilitate experimentation. Applications to polynomial factorisation, isogeny computations, cryptographic constructions, and coding theory are also presented.

math.NT↗

arXiv2025

Galois Groups of Ap\'ery-like Series Modulo Primes

We compute the Galois groups of the reductions modulo the prime numbers $p$ of the generating series of Ap\'ery numbers, Domb numbers and Almkvist--Zudilin numbers. We observe in particular that their behavior is governed by congruence conditions on p.

math.NT↗

arXiv2025

Galois groups of reductions modulo p of D-finite series

The aim of this paper is to investigate the algebraicity behavior of reductions of $D$-finite power series modulo prime numbers. For many classes of D-finite functions, such as diagonals of multivariate algebraic series or hypergeometric functions, it is known that their reductions modulo prime numbers, when defined, are algebraic. We formulate a conjecture that uniformizes the Galois groups of these reductions across different prime numbers. We then focus on hypergeometric functions, which serves as a test case for our conjecture. Refining the construction of an annihilating polynomial for the reduction of a hypergeometric function modulo a prime number p, we extract information on the respective Galois groups and show that they behave nicely as p varies.

math.NT↗

arXiv2024

Wieferich primes for Drinfeld modules

The aim of this paper is to discuss the notion of Wieferich primes in the context of Drinfeld modules. Our main result is a surprising connection between the proprety of a monic irreducible polynomial $\mathfrak p$ to be Wieferich and the $\mathfrak p$-adic valuation of special $L$-values of Drinfeld modules. This generalizes a theorem of Thakur for the Carlitz module.We also study statistical distributions of Wieferich primes, proving in particular that a place of degree $d$ is Wieferich with the expected probability $q^{-d}$ when we average over large enough sets of Drinfeld modules.

math.NT↗

arXiv2024

Selfdual skew cyclic codes

Given a finite extension $K/F$ of degree $r$ of a finite field $F$, we enumerate all selfdual skew cyclic codes in the Ore quotient ring $K[X;\text{Frob}]/(X^{rk}-1)$ for any positive integer $k$ coprime to the characteristic $p$ (separable case). We also provide an enumeration algorithm when $k$ is a power of $p$ (purely inseparable case), at the cost of some redundancies. Our approach is based on an explicit bijection between skew cyclic codes, on the one hand, and certain families of $F$-linear subspaces of some extensions of $K$. Finally, we report on an implementation in SageMath.

cs.IT↗

arXiv2024

Algebraic Geometry codes in the sum-rank metric

We introduce the first geometric construction of codes in the sum-rank metric, which we called linearized Algebraic Geometry codes, using quotients of the ring of Ore polynomials with coefficients in the function field of an algebraic curve. We study the parameters of these codes and give lower bounds for their dimension and minimum distance. Our codes exhibit quite good parameters, respecting a similar bound to Goppa's bound for Algebraic Geometry codes in the Hamming metric. Furthermore, our construction yields codes asymptotically better than the sum-rank version of the Gilbert-Varshamov bound.

math.AG↗

arXiv2024

Reed-Muller codes in the sum-rank metric

We introduce the sum-rank metric analogue of Reed--Muller codes, which we called linearized Reed--Muller codes, using multivariate Ore polynomials. We study the parameters of these codes, compute their dimension and give a lower bound for their minimum distance. Our codes exhibit quite good parameters, respecting a similar bound to Reed--Muller codes in the Hamming metric. Finally, we also show that many of the newly introduced linearized Reed--Muller codes can be embedded in some linearized Algebraic Geometry codes, recently defined in arXiv:2303.08903, a property which could turn out to be useful in light of decoding.

cs.IT↗

arXiv2024

Computation of classical and $v$-adic $L$-series of $t$-motives

We design an algorithm for computing the $L$-series associated to an Anderson $t$-motives, exhibiting quasilinear complexity with respect to the target precision. Based on experiments, we conjecture that the order of vanishing at $T=1$ of the $v$-adic $L$-series of a given Anderson $t$-motive with good reduction does not depend on the finite place $v$.

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arXiv2023

Algorithms for computing norms and characteristic polynomials on general Drinfeld modules

We provide two families of algorithms to compute characteristic polynomials of endomorphisms and norms of isogenies of Drinfeld modules. Our algorithms work for Drinfeld modules of any rank, defined over any base curve. When the base curve is $\mathbb P^1_{\mathbb F_q}$, we do a thorough study of the complexity, demonstrating that our algorithms are, in many cases, the most asymptotically performant. The first family of algorithms relies on the correspondence between Drinfeld modules and Anderson motives, reducing the computation to linear algebra over a polynomial ring. The second family, available only for the Frobenius endomorphism, is based on a formula expressing the characteristic polynomial of the Frobenius as a reduced norm in a central simple algebra.

cs.SC↗

arXiv2023

A sharper multivariate Christol's theorem with applications to diagonals and Hadamard products

We provide a new proof of the multivariate version of Christol's theorem about algebraic power series with coefficients in finite fields, as well as of its extension to perfect ground fields of positive characteristic obtained independently by Denef and Lipshitz, Sharif and Woodcok, and Harase. Our proof is elementary, effective, and allows for much sharper estimates. We discuss various applications of such estimates, in particular to a problem raised by Deligne concerning the algebraicity degree of reductions modulo $p$ of diagonals of multivariate algebraic power series with integer coefficients.

math.NT↗

arXiv2023

Drinfeld modules in SageMath

We present the first implementation of Drinfeld modules fully integrated in the SageMath ecosystem. First features will be released with SageMath 10.0.

cs.SC↗

arXiv2023

Algebraic solutions of linear differential equations: an arithmetic approach

Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture.

math.NT↗

arXiv2022

Can we dream of a 1-adic Langlands correspondence?

After observing that some constructions and results in the $p$-adic Langlands programme are somehow independent from $p$, we formulate the hypothesis that this astonishing uniformity could be explained by a 1-adic Langlands correspondence.

math.NT↗

arXiv2022

On Polynomial Ideals And Overconvergence In Tate Algebras

In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gr{ö}bner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. We prove that ideals which are spanned by polynomials admit a Tate Gr{ö}bner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. As a result, the size of the output of this algorithm grows linearly with the precision. Following the same ideas, we propose an algorithm which computes an overconvergent basis for an ideal spanned by overconvergent series. Finally, we prove the existence of a universal analytic Gr{ö}bner basis for polynomial ideals in Tate algebras, compatible with all convergence radii.

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