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This chapter reviews the origin and development of the fundamental principles of the contemporary analysis of diophantine equations, from the perspective of the theory of diophantine approximation.
This chapter covers some general lemmas on algebraic numbers, algebraic number fields and analytic functions in p-adic fields. More specialised lemmas will be adduced as needed in subsequent chapters.
This chapter is of an auxiliary nature, being mainly concerned with the relationship between bounds for linear forms in the logarithms of algebraic numbers in different (archimedean and non-archimedean) metrics. This material will later be used in the analysis of Thue and Thue-Mahler equations. Elliptic and hyperelliptic equations, and equations of hyperelliptic type, will be analysed using direct...
At last, we pass to the analysis of diophantine equations, starting with the central problem of the representation of numbers by binary forms. We return as in Chapter I, to the connection between the magnitude of solutions of Thue's equation and rational approximation of algebraic numbers: but now our approach is the opposite of Thue's: we obtain bounds for the approximation as a corollary to bounds...
We develop and deepen the arguments of the previous chapter mainly by further applications of p-adic analysis. This analysis allows one to observe qualitatively new facts, for example, that the speed of growth of the maximal prime divisor of a binary form can be bounded from below. And we can deepen the bounds for rational approximations of algebraic numbers by including the p-adic metrics. We begin...
The equations considered in this chapter are in essence different both from the Thue equation and from its direct generalisations. It is possible to prove the existence of an effective bound for solutions of these equations by purely arithmetic methods, by reducing them to the Thue equation or to its generalisations over relative fields. However the bounds obtained in this way are not quite satisfactory...
We apply the methods and considerations described in the previous chapter to more general situations so as to obtain new facts generalising our former results. Then we proceed to a new type of equations in which at least one of the unknowns is a power of an unknown integer. Our aim is to bound the unknown exponent so as to reduce these new equations to those of the kind considered before. In this...
It was ascertained in previous chapters that upper bounds for the solutions of diophantine equations under our consideration depend essentially on the regulators of certain algebraic number fields related to the equation. Now we concentrate our attention on this phenomenon and relate it to the general problem of the magnitude of ideal class numbers. We show that algebraic number fields with ‘small’...
In this final chapter, we deal with a ‘meta’-topic, one that lies ‘over’ the theory of diophantine equations, as it were; namely arithmetic specialisation of polynomials. The main result asserts that under such specialisations the multiplicative structure of the numbers obtained goes some considerable way towards determining the multiplicative structure of the original polynomials. This allows one...
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