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In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field K. For example, we show that under some conditions on rational functions $f_1, \ldots, f_n\in K(X)$, there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets.
This book provides the first thorough treatment of effective results and methods for Diophantine equations over finitely generated domains. Compiling diverse results and techniques from papers written in recent decades, the text includes an in-depth analysis of classical equations including unit equations, Thue equations, hyper- and superelliptic equations, the Catalan equation, discriminant equations and decomposable form equations. The majority of results are proved in a quantitative form, giving effective bounds on the sizes of the solutions. The necessary techniques from Diophantine approximation and commutative algebra are all explained in detail without requiring any specialized knowledge on the topic, enabling readers from beginning graduate students to experts to prove effective finiteness results for various further classes of Diophantine equations.
We give an overview of the history of the classical Diophantine equations treated in our book and describe the ineffective finiteness results for those. We consider unit equations, Thue equations, hyper- and superelliptic equations, the Schinzel–Tijdeman equation, Catalan’s equation, decomposable form equations, and discriminant equations.
We complete the proofs of our results for decomposable form equations and discriminant equations, by making a reduction to the effective finiteness results for unit equations proved in Chapter 9. In the course of the proof, we apply the estimates for degree-height estimates, proved in Chapter 8.
We give an overview of the techniques from commutative algebra that we need. In particular, we recall explicit results of Aschenbrenner concerning the solutions of linear equations over polynomial rings over the integers.
We complete the proofs of our effective finiteness results for unit equations, Thue equations, hyper- and superelliptic equations, the Schinzel–Tijdeman equation, and Catalan’s equation. We apply the specialization method from Chapter 7 and make a reduction to the results from Chapters 4 and 5.
we introduce degree-height estimates for elements of finitely generated domains and of algebraic extensions thereof, which may be seen as an analogue of the naive height of an algebraic number, and prove some properties of those.
We describe in full detail our effective specialization method, which allows us to reduce effective finiteness results for Diophantine equations over finitely generated domains to effective results over number fields and function fields, discussed in Chapters 4 and 5.
We recall the basics for function fields of characteristic 0, state and prove the Mason–Stothers abc-theorem for function fields, and deduce effective finiteness results for various classes of Diophantine equations.