Article

Henson and Rubel's Theorem for Zilber's Pseudoexponentiation

Journal of Symbolic Logic (Impact Factor: 0.47). 03/2009; DOI: 10.2178/jsl/1333566630
Source: arXiv

ABSTRACT This paper contains an alternate proof of the Schanuel Nullstellensatz for Zilber's Pseudoexponentiation. Furthermore, in an algebraically closed exponential field whose exponential map is surjective with standard kernel, this property follows from the field being exponentially algebraically closed.

1 Follower
 · 
74 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that finitely generated strong extensions are finitely presented, and these extensions are classified. An algebraic construction is given of Zilber's pseudo-exponential fields. As applications of the general results and methods of the paper, it is shown that Zilber's fields are not model-complete, answering a question of Macintyre, and a precise statement is given explaining how Schanuel's conjecture answers all transcendence questions about exponentials and logarithms. Connections with the Kontsevich-Zagier, Grothendieck, and Andr\'e transcendence conjectures on periods are discussed, and finally some open problems are suggested.
    Algebra and Number Theory 12/2009; DOI:10.2140/ant.2013.7.943 · 0.58 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp(g) for some exponential polynomial g, then p and q have only finitely many common zeros.
    Communications in Algebra 10/2009; 39(10). DOI:10.1080/00927872.2010.489918 · 0.39 Impact Factor

Preview

Download
3 Downloads
Available from