Transcendental Number Theory

Abstract

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Diophantine Properties of Linear Recursive Sequences I

“…Although they have also important applications in the theory of recurrences, see [ShT], we are dealing here only with the most classical case. For a collection of results proved by Baker's method and further references we refer to the monographs [Ba2,ShT,Sm].…”

Book Review: Transcendental numbers

“…This research is hardly mentioned in the book. Introductions to the part of transcendental number theory dealing with the GelfondSchneider method and its extensions have been given by Waldschmidt [13,14], Baker [1], Masser [8], and Feldman [4].…”

“…The most complete account on the subject in English up to now had been given by Mahler [7], Thus, the book under review does not deal with the general theory of transcendental numbers, covering topics as the Thue-SiegelRoth-Schmidt method, Gelfond-Baker method, Siegel-Shidlovskii method, and various other topics which belong to the field such as Mahler's classification, metrical theory, elliptic functions, and abelian varieties. For such books, see Siegel [12], Schneider [10], Gelfond [5], Lang [6], Baker [1], and Feldman [3]. 11], and others.…”

An Algorithm for Solving a Certain Class of Diophantine Equations. I

“…For more on Mordell's equation, see Baker [1], Hemer [2], Hilliker and Steiner [4], London and Finkelstein [7], and Mordell [10], [11]. (Note: R. Finkelstein is now known as R. Steiner.…”

“…An example of this would be 1/x + \/y = k. For k = 2, this equation has only the solution (1,1); but if k varies over the interval [1,2], in the infinite sequence k = 2, 3/2, 4/3, 5/4,..., there are infinitely many solutions (1,1), (1,2), (1,3), (1,4),... . Our objective will be to characterize the classes A and A, in terms of the existence of a certain quantity, q = q(xx, x2,...,x"), defined for all (x,, x2,...,x") in D. Then, based upon this characterization, we shall define another class A2, a subclass of A,, which will turn out to contain only Diophantine equations that possess a computable bound.…”