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, Volume 8 (2016) no. 2, pp. 59-83.

Abstract

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.

Received: 2015-03-11
Revised: 2016-07-15
Accepted: 2016-07-15
Published online: 2017-03-19

DOI: 10.5802/cml.33

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

Author's affiliations:

Matteo Viale ¹

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

@article{CML_2016__8_2_59_0,
     author = {Matteo Viale},
     title = {Forcing the truth of a weak form of {Schanuel{\textquoteright}s} conjecture},
     journal = {Confluentes Mathematici},
     pages = {59--83},
     publisher = {Institut Camille Jordan},
     volume = {8},
     number = {2},
     year = {2016},
     doi = {10.5802/cml.33},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.33/}
}

TY  - JOUR
AU  - Matteo Viale
TI  - Forcing the truth of a weak form of Schanuel’s conjecture
JO  - Confluentes Mathematici
PY  - 2016
SP  - 59
EP  - 83
VL  - 8
IS  - 2
PB  - Institut Camille Jordan
UR  - https://cml.centre-mersenne.org/articles/10.5802/cml.33/
DO  - 10.5802/cml.33
LA  - en
ID  - CML_2016__8_2_59_0
ER  -

%0 Journal Article
%A Matteo Viale
%T Forcing the truth of a weak form of Schanuel’s conjecture
%J Confluentes Mathematici
%D 2016
%P 59-83
%V 8
%N 2
%I Institut Camille Jordan
%U https://cml.centre-mersenne.org/articles/10.5802/cml.33/
%R 10.5802/cml.33
%G en
%F CML_2016__8_2_59_0

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

References
Cited by

[1] James Ax On Schanuel’s conjectures, Ann. of Math. (2), Volume 93 (1971), pp. 252-268

[2] Thomas Jech Set theory, Spring Monographs in Mathematics, Springer, 2003 (3rd edition)

[3] Thomas J. Jech Abstract theory of abelian operator algebras: an application of forcing, Trans. Amer. Math. Soc., Volume 289 (1985) no. 1, pp. 133-162 | DOI

[4] Jonathan Kirby Exponential algebraicity in exponential fields, Bull. Lond. Math. Soc., Volume 42 (2010) no. 5, pp. 879-890 | DOI

[5] Jonathan Kirby; Boris Zilber Exponentially closed fields and the conjecture on intersections with tori, Ann. Pure Appl. Logic, Volume 165 (2014) no. 11, pp. 1680-1706 | DOI

[6] Kenneth Kunen Set theory, Studies in Logic and the Foundations of Mathematics, 102, North-Holland Publishing Co., Amsterdam-New York, 1980, xvi+313 pages (An introduction to independence proofs)

[7] David Mumford Algebraic geometry. I, Classics in Mathematics, Springer-Verlag, Berlin, 1995, x+186 pages (Complex projective varieties, Reprint of the 1976 edition)

[8] Masanao Ozawa A classification of type $I$ $A W^{*}$ -algebras and Boolean valued analysis, J. Math. Soc. Japan, Volume 36 (1984) no. 4, pp. 589-608 | DOI

[9] Andrea Vaccaro $C^{*}$ -algebras and $B$ -names for complex numbers, University of Pisa, September (2015) (Thesis for the master degree in mathematics)

[10] Andrea Vaccaro; Matteo Viale Generic absoluteness and boolean names for elements of a Polish space (2016) (To appear in Bollettino Unione Matematica Italiana)

[11] A. J. Wilkie Some local definability theory for holomorphic functions, Model theory with applications to algebra and analysis. Vol. 1 (London Math. Soc. Lecture Note Ser.), Volume 349, Cambridge Univ. Press, Cambridge, 2008, pp. 197-213 | DOI

[12] B. Zilber Pseudo-exponentiation on algebraically closed fields of characteristic zero, Ann. Pure Appl. Logic, Volume 132 (2005) no. 1, pp. 67-95 | DOI

Cited by Sources: