Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T19:50:09.225Z Has data issue: false hasContentIssue false
  • English
  • Français

A PSEUDOEXPONENTIAL-LIKE STRUCTURE ON THE ALGEBRAIC NUMBERS

Published online by Cambridge University Press:  22 December 2015

VINCENZO MANTOVA
VINCENZO MANTOVA*
Affiliation:
DIVISION OF MATHEMATICS, SCHOOL OF SCIENCE AND TECHNOLOGY UNIVERSITY OF CAMERINO VIA MADONNA DELLE CARCERI 9, 62032, CAMERINO (MC)ITALYE-mail: vincenzo.mantova@unicam.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Pseudoexponential fields are exponential fields similar to complex exponentiation which satisfy the Schanuel Property, i.e., the abstract statement of Schanuel’s Conjecture, and an adapted form of existential closure.

Here we show that if we remove the Schanuel Property and just care about existential closure, it is possible to create several existentially closed exponential functions on the algebraic numbers that still have similarities with complex exponentiation. The main difficulties are related to the arithmetic of algebraic numbers, and they can be overcome with known results about specialisations of multiplicatively independent functions on algebraic varieties.

Keywords

exponential fieldsexponential-algebraic closuremultiplicative dependencies
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1339 - 1347
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Hyttinen, Tapani, Random logarithm and homogeneity, Logic and Its Applications (Blass, Andreas and Zhang, Yi, editors), Contemporary Mathematics, pp. 137–166, American Mathematical Society, Providence, RI, 2005.CrossRefGoogle Scholar
Kirby, Jonathan, A Note on the Axioms for Zilber’s Pseudo-Exponential Fields. Notre Dame Journal of Formal Logic, vol. 54, no. 3–4, pp. 509–520, 2013.CrossRefGoogle Scholar
Kirby, Jonathan, Finitely presented exponential fields. Algebra & Number Theory, vol. 7, no. 4, pp. 943–980, 2013.CrossRefGoogle Scholar
Mantova, Vincenzo, Involutions on Zilber fields, September 2011. http://arxiv.org/abs/1109.6155 arXiv:1109.6155.Google Scholar
Marker, David, A remark on Zilber’s pseudoexponentiation. Journal of Symbolic Logic, vol. 71, no. 3, pp. 791–798, 2006.CrossRefGoogle Scholar
Masser, David W., Specialisations of finitely generated subgroups of abelian varieties. Transactions of the American Mathematical Society, vol. 311, no. 1, pp. 413–424, 1989.CrossRefGoogle Scholar
Zilber, Boris, Pseudo-exponentiation on algebraically closed fields of characteristic zero. Annals of Pure and Applied Logic, vol. 132, no. 1, pp. 67–95, 2005.CrossRefGoogle Scholar