MODNET

Research Training Network in Model Theory

Publications > Preprint server > Preprint Number 1694
Preprint Number 1694
1694. Alessandro Berarducci and Marcello Mamino Asymptotic analysis of Skolem's exponential functions E-mail: Submission date: 18 November 2019 Abstract: Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f+g,fg and f^g are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2^{2^x}. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type ω. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below 2^{n^x}. We deduce an epsilon-zero upper bound for the fragment below 2^{x^x}, improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations. Mathematics Subject Classification: 03C64, 03E10, 16W60, 26A12, 41A58 Keywords and phrases: |

Last updated: December 24 2019 16:16 | Please send your corrections to: |