Forcing the truth of a weak form of Schanuel’s conjecture

Matteo Viale

doi:10.5802/cml.33

Matteo Viale ¹

¹ Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy

Confluentes Mathematici, Tome 8 (2016) no. 2, pp. 59-83.

Résumé

Schanuel’s conjecture states that the transcendence degree over $ℚ$ of the $2 n$ -tuple $(λ_{1}, \dots, λ_{n}, e^{λ_{1}}, \dots, e^{λ_{n}})$ is at least $n$ for all $λ_{1}, \dots, λ_{n} \in ℂ$ which are linearly independent over $ℚ$ ; if true it would settle a great number of elementary open problems in number theory, among which the transcendence of $e$ over $π$ .

Wilkie [11], and Kirby [4, Theorem 1.2] have proved that there exists a smallest countable algebraically and exponentially closed subfield $K$ of $ℂ$ such that Schanuel’s conjecture holds relative to $K$ (i.e. modulo the trivial counterexamples, $ℚ$ can be replaced by $K$ in the statement of Schanuel’s conjecture). We prove a slightly weaker result (i.e. that there exists such a countable field $K$ without specifying that there is a smallest such) using the forcing method and Shoenfield’s absoluteness theorem.

This result suggests that forcing can be a useful tool to prove theorems (rather than independence results) and to tackle problems in domains which are apparently quite far apart from set theory.

Reçu le : 2015-03-11
Révisé le : 2016-07-15
Accepté le : 2016-07-15
Publié le : 2017-03-19

DOI : 10.5802/cml.33

Classification : 03E57, 03C60, 11U99
Mots clés : Schanuel’s conjecture, forcing and generic absoluteness

Affiliations des auteurs :

Matteo Viale ¹

¹ Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy

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Matteo Viale. Forcing the truth of a weak form of Schanuel’s conjecture. Confluentes Mathematici, Tome 8 (2016) no. 2, pp. 59-83. doi : 10.5802/cml.33. https://cml.centre-mersenne.org/articles/10.5802/cml.33/

Bibliographie
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