Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T21:21:12.606Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPARING $\mathbb{C}$ AND ZILBER’S EXPONENTIAL FIELDS: ZERO SETS OF EXPONENTIAL POLYNOMIALS

Part of: Model theory Number theory: Connections with logic

Published online by Cambridge University Press:  04 August 2014

P. D’Aquino ,
A. Macintyre  and
G. Terzo
P. D’Aquino
Affiliation:
Departments of Mathematics and Physics, Seconda Università di Napoli, viale Lincoln 5, 81100 Caserta, Italy (paola.daquino@unina2.it; giuseppina.terzo@unina2.it)
A. Macintyre
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK (A.Macintyre@qmul.ac.uk)
G. Terzo
Affiliation:
Departments of Mathematics and Physics, Seconda Università di Napoli, viale Lincoln 5, 81100 Caserta, Italy (paola.daquino@unina2.it; giuseppina.terzo@unina2.it)
Get access Rights & Permissions [Opens in a new window]

Abstract

We continue the research programme of comparing the complex exponential with Zilberś exponential. For the latter, we prove, using diophantine geometry, various properties about zero sets of exponential functions, proved for $\mathbb{C}$ using analytic function theory, for example, the Identity Theorem.

Keywords

exponential fieldsZilber fieldsSchanuel’s ConjectureIdentity Theorem

MSC classification

Primary: 03C60: Model-theoretic algebra 11U09: Model theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 1 , January 2016 , pp. 71 - 84
Copyright
© Cambridge University Press 2014 

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

Bombieri, E. and Gubler, W., Heights in Diophantine Geometry (Cambridge University Press, 2006).Google Scholar
Bombieri, E., Masser, D. and Zannier, U., Anomalous subvarieties — structure theorems and applications, Int. Math. Res. Not. 2007 (2007), 1–33.CrossRefGoogle Scholar
Bourbaki, N., Algebra II, Chapters 4–7, (Springer, 1990).Google Scholar
Cassels, J. W. S. and Fröhlich, A., Algebraic Number Theory, second edition (London Mathematical Society, 2010).Google Scholar
Conway, J. B., Functions Of One Complex Variable I, second edition (Springer, 1978).Google Scholar
D’Aquino, P., Macintyre, A. and Terzo, G., Schanuel Nullstellensatz for Zilber fields, Fund. Math. 207 (2010), 123–143.CrossRefGoogle Scholar
D’Aquino, P., Macintyre, A. and Terzo, G., From Schanuel Conjecture to Shapiro Conjecture, Comment. Math. Helv. to appear.Google Scholar
D’Aquino, P. and Terzo, G., A theorem of finite reducibility for exponential polynomials, submitted (2012).Google Scholar
van der Poorten, A. J. and Tijdeman, R., On common zeros of exponential polynomials, L’Enseign. Math. 21 (1975), 57–67.Google Scholar
Evertse, J. H., Schilickewei, H. P. and Schmidt, W. M., Linear equations in variables which lie in a multiplicative group, Ann. of Math. 155 (2002), 807–836.CrossRefGoogle Scholar
Gunaydin, A., Rational solutions of polynomial–exponential equations, Internat. J. Number Theory 8(6) 1391–1399.Google Scholar
Györy, K. and Schinzel, A., On a Conjecture of Posner and Rumsey, J. Number Theory 47 (1994), 63–78.Google Scholar
Henson, C. W. and Rubel, L. A., Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions, Trans. Amer. Math. Soc. 282(1) (1984), 1–32.Google Scholar
Kirby, J., Macintyre, A. and Onshuus, A., The algebraic numbers definable in various exponential fields, J. Inst. Math. Jussieu 11(4) (2012), 825–834.Google Scholar
Lang, S., Elliptic Functions, second edition (Springer, 1987).Google Scholar
Merel, L., Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437–449.Google Scholar
Pólya, G., Geometiisches ber die Verteilung der Nullstellen gewisser ganzer transzendenter Funktionen, Munch. Sitzungsber. 50 (1920), 285–290.Google Scholar
Ritt, J. F., A factorization theorem of functions ∑i=1na ie i𝛼z, Trans. Amer. Math. Soc. 29 (1927), 584–596.Google Scholar
Zilber, B., Pseudo-exponentiation on algebraically closed fields of characteristic zero, Ann. Pure Appl. Logic 132(1) (2005), 67–95.CrossRefGoogle Scholar