Unit Equations in Diophantine Number Theory

Abstract

Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.

On the sumsets of exceptional units in $$\mathbb {Z}_n$$ Z n

“…For other related results on units, one can refer to [3,4,7]. For any integers k, n, c with k, n ≥ 2, let…”

“…2014r029. We would like to thank Professor K. Gyory for pointing out the reference [3] and the anonymous referee for his/her useful comments.…”

$S$-parts of values of univariate polynomials,\\ binary forms and decomposable forms at integral points

“…We note that these conditions are satisfied by binary forms with at least three pairwise non-proportional linear factors, and also discriminant forms, index forms and a restricted class of norm forms in an arbitrary number of variables. As has been explained in [8,Chap. 9], conditions (2.11), (2.12) imply condition (i) of Theorem B.…”

Expansions of the p‐adic numbers that interpret the ring of integers

“…We say that G has the Mann property if for any equation like above there is finitely many nondegenerate solutions in G . Examples of groups with Mann property are the roots of unity in or any group of finite rank in a field of characteristic zero (cf., e.g., [, Theorem 6.3.1]). In particular, any subgroup of of finite rank has the Mann property, e.g., it is the case for .…”