# CONFLUENTES MATHEMATICI

Forcing the truth of a weak form of Schanuel’s conjecture
Confluentes Mathematici, Tome 8 (2016) no. 2, pp. 59-83.

Schanuel’s conjecture states that the transcendence degree over $ℚ$ of the $2n$-tuple $\left({\lambda }_{1},\cdots ,{\lambda }_{n},{e}^{{\lambda }_{1}},\cdots ,{e}^{{\lambda }_{n}}\right)$ is at least $n$ for all ${\lambda }_{1},\cdots ,{\lambda }_{n}\in ℂ$ 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 $\pi$.

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 : 2015-03-12
Révisé le : 2016-07-16
Accepté le : 2016-07-16
Publié le : 2017-03-20
DOI : https://doi.org/10.5802/cml.33
Classification : 03E57,  03C60,  11U99
Mots clés: Schanuel’s conjecture, forcing and generic absoluteness
@article{CML_2016__8_2_59_0,
author = {Matteo Viale},
title = {Forcing the truth of a weak form of Schanuel's conjecture},
journal = {Confluentes Mathematici},
publisher = {Institut Camille Jordan},
volume = {8},
number = {2},
year = {2016},
pages = {59-83},
doi = {10.5802/cml.33},
language = {en},
url = {cml.centre-mersenne.org/item/CML_2016__8_2_59_0/}
}
Viale, Matteo. 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/item/CML_2016__8_2_59_0/

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