Éric Delaygue
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View article: Abel’s Problem, Gauss and Cartier Congruences Over Number Fields
Abel’s Problem, Gauss and Cartier Congruences Over Number Fields Open
Abel’s problem consists in identifying the conditions under which the differential equation $y^{\prime}=\eta y$, with $\eta $ an algebraic function in $\mathbb{C}(x)$, possesses a non-zero algebraic solution $y$. This problem has been algo…
View article: On Ruzsa's conjecture on congruence preserving functions
On Ruzsa's conjecture on congruence preserving functions Open
Ruzsa's conjecture asserts that any sequence $(a_n)_{n \geq 0}$ of integers that preserves congruences, $\textit{i.e.}$, satisfies $ a_{n+k} \equiv a_n \mod k $, and has the growth condition $\limsup_{n \to +\infty} |a_n|^{1/n} < e$, must …
View article: A Lindemann–Weierstrass theorem for 𝐸-functions
A Lindemann–Weierstrass theorem for 𝐸-functions Open
𝐸-functions were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function. After developments of Siegel’s methods by Shidlovskii, Nesterenko and André, Beukers proved in 2006 an optimal result on the al…
View article: Cyclotomic valuation of $q$-Pochhammer symbols and $q$-integrality of basic hypergeometric series
Cyclotomic valuation of $q$-Pochhammer symbols and $q$-integrality of basic hypergeometric series Open
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View article: On Abel’s Problem and Gauss Congruences
On Abel’s Problem and Gauss Congruences Open
A classical problem due to Abel is to determine if a differential equation $y^{\prime}=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta $ is a given algebraic function over $\mathbb C(x)$. Risch designed a…
View article: A Lindemann-Weierstrass theorem for $E$-functions
A Lindemann-Weierstrass theorem for $E$-functions Open
$E$-functions were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function. After developments of Siegel's methods by Shidlovskii, Nesterenko and Andr\'e, Beukers proved in 2006 an optimal result on th…
View article: On primary pseudo-polynomials (around Ruzsa’s conjecture)
On primary pseudo-polynomials (around Ruzsa’s conjecture) Open
Every polynomial [Formula: see text] satisfies the congruences [Formula: see text] for all integers [Formula: see text]. An integer valued sequence [Formula: see text] is called a pseudo-polynomial when it satisfies these congruences. Hall…
View article: On primary pseudo-polynomials (Around Ruzsa's Conjecture)
On primary pseudo-polynomials (Around Ruzsa's Conjecture) Open
Every polynomial $P(X)\in \mathbb Z[X]$ satisfies the congruences $P(n+m)\equiv P(n) \mod m$ for all integers $n, m\ge 0$. An integer valued sequence $(a_n)_{n\ge 0}$ is called a pseudo-polynomial when it satisfies these congruences. Hall …
View article: Some supercongruences of arbitrary length
Some supercongruences of arbitrary length Open
We prove supercongruences modulo $p^2$ for values of truncated hypergeometric\nseries at some special points. The parameters of the hypergeometric series are\n$d$ copies of $1/2$ and $d$ copies of $1$ for any integer $d\\ge2$.\n
View article: Some supercongruences of arbitrary length
Some supercongruences of arbitrary length Open
We prove supercongruences modulo $p^2$ for values of truncated hypergeometric series at some special points. The parameters of the hypergeometric series are $d$ copies of $1/2$ and $d$ copies of $1$ for any integer $d\ge2$.
View article: Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series
Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series Open
We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their $q$-anal…
View article: On Dwork’s 𝑝-adic formal congruences theorem and hypergeometric mirror maps
On Dwork’s 𝑝-adic formal congruences theorem and hypergeometric mirror maps Open
Using Dwork's theory, we prove a broad generalisation of his famous p-adic\nformal congruences theorem. This enables us to prove certain p-adic congruences\nfor the generalized hypergeometric series with rational parameters; in\nparticular…
View article: Algebraic independence of $G$-functions and congruences "à la Lucas"
Algebraic independence of $G$-functions and congruences "à la Lucas" Open
We develop a new method for proving algebraic independence of $G$-functions. Our approach rests on the following observation: $G$-functions do not always come with a single linear differential equation, but also sometimes with an infinite …