arXiv subjects

Tanguy Rivoal

Publications and source records attributed to Tanguy Rivoal.

At least 19 recordsLinked to original sources

arXiv2026

Transcendence measure for the values of the logarithmic derivative of the Bessel function $J_0$ at algebraic arguments

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any $\varepsilon>0$, there exists $c>0$ such that $\vert P(J_0'(α)/J_0(α))\vert>c/H^{4d^2δ+\varepsilon}$ where $J_0$ is the Bessel function. In this paper, we prove that the exponent $4d^2δ$ can be replaced by a smaller (explicit) quantity $μ(d,δ)\le 4d^2δ-2dδ-1$. A similar improvement holds more generally for the logarithmic derivative of any $E$-function $f$ of differential order 2 and such that $f$ and $f'$ are homogeneously algebraically independent. Our method rests upon the optimization of the size of a determinant that appears naturally in the classical Siegel-Shidlovskii method, following the steps of our previous improvement of the transcendence measure of the value $e^α$ for any $α\in \overline{\mathbb Q}^*$.

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arXiv2026

On the singularities of differential equations satisfied by $E$-functions

Let $ξ$ be a value, at an algebraic point, of a Siegel $E$-function. As a special case of a very general interpolation result, we prove that there exists an $E$-function $f$ such that $f(1)=ξ$, and such that 1 is not a singularity of the minimal differential equation satisfied by $f$. We prove that the same property does not hold at the point $0$, when $ξ$ is the value at a non-zero algebraic number of the Bessel function. This answers an analogue of a question asked by Yves Andr{é} for $G$-functions.

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arXiv2025

Representability of G-functions as rational functions in hypergeometric series

Fresán and Jossen have given a negative answer to a question of Siegel about the representability of every $E$-function as a polynomial with algebraic coefficients in $E$-functions of type ${}_pF_q[\underline{a};\underline{b};γx^{q-p+1}]$ with $q\geq p\geq 0$, $γ\in \overline{\mathbb Q}$ and rational parameters $\underline{a}, \underline{b}$. In this paper, we study, in a more general context, a similar question for $G$-functions asked by Fischler and the second author: can every $G$-function be represented as a polynomial with algebraic coefficients in $G$-functions of type $μ(x)\cdot {}_pF_{p-1}[\underline{a};\underline{b};λ(x)]$ with $p\ge 1$, rational parameters $\underline{a},\underline{b}$ and $μ,λ$ algebraic over $\mathbb Q(x)$ with $λ(0)=0$? They have shown the answer to be negative under a generalization of Grothendieck's Period Conjecture and a technical assumption on the~$λ$'s. Using differential Galois theory, we prove that, for every $N\in \mathbb N$, there exists a $G$-function which can not be represented as a rational function with coefficients in $\overline{\mathbb C(x)}$ of solutions of linear differential equations with coefficients in $\mathbb C(x)$ and at most $N$ singularities in $\mathbb{P}^1 (\mathbb C)$. As a corollary, we deduce that not all $G$-functions can be represented as a rational function in hypergeometric series of the above mentioned type, when the $λ$'s are rational functions with degrees of their numerators and denominators bounded by an arbitrarily large fixed constant. This provides an unconditional negative answer to the question asked by Fischler and the second author for such~$λ$'s.

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arXiv2025

Abel's problem, Gauss and Cartier congruences over number fields

Abel's problem consists in identifying the conditions under which the diferential equation $y'=ηy$, with $η$ an algebraic function in $\mathbb{C}(x)$, possesses a non-zero algebraic solution $y$. This problem has been algorithmically solved by Risch. In a previous paper, we have presented an alternative solution in the special arithmetic situation where $η$ has a Puiseux expansion with $\textit{rational}$ coefficients at the origin: there exists a non-trivial algebraic solution of $y'=ηy$ if and only if the coefficients of the Puiseux expansion of $xη(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. In this paper, we generalize this criterion to arbitrary $η$ algebraic over $\overline{\mathbb{Q}}(x)$, by means of a natural generalization to number fields of Gauss congruences and of the weaker Cartier congruences recently introduced in this context by Bostan. We then provide applications of this criterion in the hypergeometric setting and for Artin-Mazur zeta functions.

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arXiv2025

Rational approximations to values of $E$-functions

We solve a long standing problem in the theory of Siegel's $E$-functions, initiated by Lang for Bessel's function $J_0$ in the 60's and considered in full generality by G. Chudnovsky in the 80's: we prove that irrational values taken at rational points by $E$-functions with rational Taylor coefficients have irrationality exponent equal to 2. This result had been obtained before by Zudilin under strong assumptions on algebraic independence of $E$-functions, satisfied by $J_0$ but not by all hypergeometric $E$-functions for instance. We remove them using a new generalization of Shidlovskii's lemma, analogous to zero estimates on commutative algebraic groups in which obstructions come from algebraic subgroups.

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arXiv2025

Zeros of $E$-functions and of exponential polynomials defined over $\overline{\mathbb{Q}}$

Zeros of Bessel functions $J_α$ play an important role in physics. They are a motivation for studying zeros of exponential polynomials defined over $\overline{\mathbb{Q}}$, and more generally of $E$-functions. In this paper we partially characterize $E$-functions with zeros of the same multiplicity, and prove a special case of a conjecture of Jossen on entire quotients of $E$-functions, related to Ritt's theorem and Shapiro's conjecture on exponential polynomials. We also deduce from Schanuel's conjecture many results on zeros of exponential polynomials over $\overline{\mathbb{Q}}$, including $π$, logarithms of algebraic numbers, and zeros of $J_α$ when $2α$ is an odd integer. For the latter we define (if $α\neq\pm1/2$) an analogue of the minimal polynomial and Galois conjugates of algebraic numbers. At last, we study conjectural generalizations to factorization and zeros of $E$-functions.

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arXiv2025

A new transcendence measure for the values of the exponential function at algebraic arguments

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^α$: for any $\varepsilon\>0$, there exists $c\>0$ such that $\vert P(e^α)\vert\>c/H^{μ(d,δ)+\varepsilon}$ where the exponent $μ(d,δ)=(4d^2-2d)δ+2d-1$. Zheng obtained a better result in 1991 with $μ(d,δ)=(4d^2-2d)δ-1$. In this paper, we provide a new explicit exponent $μ(d,δ)$ which improves on Zheng's transcendence measure for all $δ\ge 2$ and all $d\ge 2$. When $δ=1$, we recover his bound for all $d\ge 2$, which had in fact already been obtained by Kappe in 1966. Our improvement rests upon the optimization of an accessory parameter in Siegel's classical determinant method applied to Hermite-Pad{é} approximants to powers of the exponential function.

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arXiv2025

Arithmetic properties of the Taylor coefficients of differentially algebraic power series

Let $f=\sum_{n=0}^\infty f_n x^n \in \overline{\mathbb Q}[[x]$ be a solution of an algebraic differential equation $Q(x,y(x), \ldots, y^{(k)}(x))=0$, where $Q$ is a multivariate polynomial with coefficients in $\overline{\mathbb Q}$. The sequence $(f_n)_{n\ge 0}$ satisfies a non-linear recurrence, whose expression involves a polynomial $M$ of degree $s$. When the equation is linear, $M$ is its indicial polynomial at the origin. We show that when $M$ is split over $\mathbb Q$, there exist two positive integers $δ$ and $ν$ such that the denominator of $f_n$ divides $δ^{n+1}(νn+ν)!^{2s}$ for all $n\ge 0\ $, generalizing a well-known property when the equation is linear. This proves in this case a strong form of a conjecture of Mahler that Pólya--Popken's upper bound $n^{\mathcal{O}(n\log(n))}$ for the denominator of $f_n$ is not optimal. This also enables us to make Sibuya and Sperber's bound $\vert f_n\vert_v\le e^{\mathcal{O}(n)}$, for all finite places $v$ of $\overline{\mathbb Q}$, explicit in this case. Our method is completely effective and rests upon a detailed $p$-adic analysis of the above mentioned non-linear recurrences. Finally, we present various examples of differentially algebraic functions for which the associated polynomial $M$ is split over $\mathbb Q$, among which are Weierstraß' elliptic $\wp$ function, solutions of Painlevé equations, and Lagrange's solution to Kepler's equation.

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arXiv2024

Transcendence of values of logarithms of $E$-functions

Let $f$ be an $E$-function (in Siegel's sense) not of the form $e^{βz}$, $β\in \overline{\mathbb{Q}}$, and let $\log$ denote any fixed determination of the complex logarithm. We first prove that there exists a finite set $S(f)$ such that for all $ξ\in \overline{\mathbb{Q}}\setminus S(f)$, $\log(f(ξ))$ is a transcendental number. We then quantify this result when $f$ is an $E$-function in the strict sense with rational coefficients, by proving an irrationality measure of $\ln(f(ξ))$ when $ξ\in \mathbb{Q}\setminus S(f)$ and $f(ξ)\gt0$. This measure implies that $\ln(f(ξ))$ is not an ultra-Liouville number, as defined by Marques and Moreira. The proof of our first result, which is in fact more general, uses in particular a recent theorem of Delaygue. The proof of the second result, which is independent of the first one, is a consequence of a new linear independence measure for values of linearly independent $E$-functions in the strict sense with rational coefficients, where emphasis is put on other parameters than on the height, contrary to the case in Shidlovskii's classical measure for instance.

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arXiv2023

Values of E-functions are not Liouville numbers

Shidlovskii has given a linear independence measure of values of $E$-functions with rational Taylor coefficients at a rational point, not a singularity of the underlying differential system satisfied by these $E$-functions. Recently, Beukers has proved a qualitative linear independence theorem for the values at an algebraic point of $E$-functions with arbitrary algebraic Taylor coefficients. In this paper, we obtain an analogue of Shidlovskii's measure for values of arbitrary $E$-functions at algebraic points. This enables us to solve a long standing problem by proving that the value of an $E$-function at an algebraic point is never a Liouville number. We also prove that values at rational points of $E$-functions with rational Taylor coefficients are linearly independent over $\overline{\mathbb{Q}}$ if and only if they are linearly independent over $\mathbb{Q}$. Our methods rest upon improvements of results obtained by André and Beukers in the theory of $E$-operators.

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arXiv2023

Minimization of differential equations and algebraic values of $E$-functions

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's $E$-functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments.

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arXiv2023

Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function

We investigate the relations between the rings ${\bf E}$, ${\bf G}$ and ${\bf D}$ of values taken at algebraic points by arithmetic Gevrey series of order either $-1$ ($E$-functions), $0$ (analytic continuations of $G$-functions) or $1$ (renormalization of divergent series solutions at $\infty$ of $E$-operators) respectively. We prove in particular that any element of ${\bf G}$ can be written as multivariate polynomial with algebraic coefficients in elements of ${\bf E}$ and ${\bf D}$, and is the limit at infinity of some $E$-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously $E$-functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for $E$-functions) and the conjecture ${\bf D}\cap{\bf E}=\overline{\mathbb Q}$ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of $E$-functions, the transcendance of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendance of Gompertz constant as well as the fact that Euler's constant is not in ${\bf E}$.

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arXiv2021

A note on G-operators of order 2

It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a $G$-function solution of an inhomogeneous equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, as well as that of a $G$-function $f$ of differential order 2 over $\overline{\mathbb{Q}}(z)$, and such that $f$ and $f'$ are algebraically dependent over $\mathbb{C}(z)$. Our results apply more generally to Nilsson-Gevrey arithmetic series of order 0 that encompass $G$-functions.

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arXiv2020

Explicit degree bounds for right factors of linear differential operators

If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominatorsof very high degree. When the base field is $\mathbb C$, we give a completely explicit bound for the degrees of the monic right factors in terms of the degree and the order of the original operator, as well as the largest modulus of the local exponents at all its singularities. As a consequence, if a differential operator $L$ has rational function coefficients over a number field, we get degree bounds for its monic right factors in terms of the degree, the order and the height of $L$, and of the degree of the number field.

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arXiv2018

Linear independence of values of G-functions, II. Outside the disk of convergence

Given any non-polynomial $G$-function $F(z)=\sum_{k=0}^\infty A_k z^k$ of radius of convergence $R$ and in the kernel a $G$-operator $L_F$, we consider the $G$-functions $F_n^{[s]}(z)=\sum_{k=0}^\infty \frac{A_k}{(k+n)^s}z^k$ for every integers $s\ge 0$ and $n\ge 1$. These functions can be analytically continued to a domain $\mathcal{D}_F$ star-shaped at $0$ and containing the disk $\{\vert z\vert 0$ and $v_F>0$. This appears to be the first Diophantine result for values of $G$-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [{\em Linear independence of values of G-functions}, 46 pages, J. Europ. Math. Soc., to appear], where $α\in \overline{\mathbb{Q}}^*$ was assumed to be such that $\vert α\vert <R$. Its proof relies on an explicit construction of a Padé approximation problem adapted to certain non-holomorphic functions associated to $F$, and it is quite different of that in the above mentioned paper. It makes use of results of André, Chudnovsky and Katz on $G$-operators, of a linear independence criterion à la Siegel over number fields, and of a far reaching generalization of Shidlovsky's lemma built upon the approach of Bertrand-Beukers and Bertrand.

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arXiv2018

A note on odd zeta values

Using a new construction of rational linear forms in odd zeta values and the saddle point method, we prove the existence of at least two irrational numbers amongst the 33 odd zeta values $ζ$(5), $ζ$(7),. .. , $ζ$(69).

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arXiv2017

Rational approximation to values of G-functions, and their expansions in integer bases

Building upon previous works of Andr{é} and Chudnovsky, we prove a general result concerning the approximations of values at rational points a/b of any G-function F with rational Taylor coefficients by fractions of the form n/(B $\times$b^m), where the integer B is fixed. As a corollary, we show that if F is not in Q(z), then for any $ε$ > 0, |F (a/b) -- n/b^m | $\ge$ 1/b^{m(1+$ε$)} provided b and m are large enough with respect to a, $ε$ and F. This enables us to obtain a new result on the repetition of patterns in the b-ary expansion of F (a/b) when b $\ge$ 2. In particular, defining N (n) as the number of consecutive equal digits in the b-ary expansion of F (a/b^s) starting from the n-th digit, we prove that lim sup N (n)/n $\le$ $ε$ provided the integer s $\ge$ 1 is such that b s is large enough with respect to a, $ε$ and F. This is a step towards the conjecture that this limit should be equal to 0 whenever F (a/b) is an irrational number. All our results are effective.

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arXiv2017

Exceptional values of E-functions at algebraic points

E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with rational functions coefficients. They were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function, and studied further by Shidlovskii in 1956. The celebrated Siegel-Shidlovskii Theorem deals with the algebraic (in)dependence of values at algebraic points of E-functions solutions of a differential system. However, somewhat paradoxically, this deep result may fail to decide whether a given E-fuction assumes an algebraic or a transcendental value at some given algebraic point. Building upon André's theory of E-operators, Beukers refined in 2006 the Siegel-Shidlovskii Theorem in an optimal way. In this paper, we use Beukers' work to prove the following result: there exists an algorithm which, given a transcendental E-function $f(z)$ as input, outputs the finite list of all exceptional algebraic points $α$ such that $f(α)$ is also algebraic, together with the corresponding list of values $f(α)$. This result solves the problem of deciding whether values of E-functions at algebraic points are transcendental.

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