Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.39810

An introduction to the strategy of transcendence proofs

Michel Waldschmidt
View original source ↗Read source document ↗Export metadata ↓

Abstract

The proof by Ch. Hermite of the transcendence of the number e in 1873 is the model on which most transcendence proofs for constants of analysis are based. The Schneider-Lang Theorem gives an opportunity to understand the strategy behind the proof. We only outline the main ideas, without providing technical details, for which we refer to the bibliography.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michel Waldschmidt. 2026-10-29. An introduction to the strategy of transcendence proofs. https://arxiv.org/abs/2610.39810

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Modularity over $\mathbb{C}$ implies modularity over $\mathbb{Q}$

We give an account of Mazur's proof that, for an elliptic curve over $\mathbb{Q}$, if it admits a nonconstant mapping from $X(N)$ defined over the complex numbers $\mathbb{C}$, for some $N$, then it also admits a nonconstant mapping from $X_0(M)$ for some (possibly different) $M$ defined over the rational numbers $\mathbb{Q}$. We also briefly discuss two open questions of Khare concerning uniformisations of elliptic curves by noncongruence modular curves. This expository note is based on the author's expository talk in March 2022 during the 2nd Trimester Program on "Modularity and the Generalised Fermat Equation" held online and organised by Bhaskaracharya Pratishthana in Pune, India.

math.NT↗
arXiv2026

On pairs of $r$-primitive and $k$-normal elements with prescribed traces over finite fields

Given $\mathbb{F}_{q^{n}}$, a field with $q^n$ elements, where $q $ is a prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2 \in \mathbb{N}$, $k_1,k_2 \in \mathbb{N}\cup \{0\}$, a rational function $F = \frac{F_1}{F_2}$ in $\mathbb{F}_{q}[x]$ with deg($F_i$) $\leq m_i$; $i=1,2,$ satisfying some conditions, and $a,b \in \mathbb{F}_{q}$, we construct a sufficient condition on $(q,n)$ which guarantees the existence of an $r_1$-primitive, $k_1$-normal element $ε\in \mathbb{F}_{q^n}$ such that $F(ε)$ is $r_2$-primitive, $k_2$-normal with $\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(ε) = a$ and $\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(ε^{-1}) = b$. For $m_1=10, \; m_2=11,\; r_1 = 3, \; r_2 = 2, \; k_1=2,\;k_2 = 1$, we establish bounds on $q$, for various $n$, to determine the existence of such elements in $\mathbb{F}_{q^{n}}$. Furthermore, we identify all such pairs $(q,n)$ excluding 10 possible values of $(q,n)$, in fields of characteristics 13.

math.NT↗
arXiv2026

Uniform existential definitions of valuations in function fields in one variable

We study function fields of curves over a base field $K$ which is either a global field or a large field having a separable field extension of degree divisible by $4$. We show that, for any such function field, Hilbert's 10th Problem has a negative answer, the valuation rings containing $K$ are uniformly existentially definable, and the finitary holomorphy rings are definable by a universal-existential formula. In order to obtain these results, we develop further the usage of local-global principles for quadratic forms in function fields to definability of certain subrings. We include a first systematic presentation of this general method, without restriction on the characteristic.

math.NT↗