PRELIMINARIES

In this chapter we shall expand the results obtained in Chapter IV on the complete resolution of equations of genera 0 and 1 by determining explicit bounds on the heights of all their integral solutions, as expressed in Theorems 9 and 12. It is to be remarked that these bounds are linearly dependent on the height of the equation concerned, in contrast with the classical case when the bounds established by Baker and Coates [8] are of multiply exponential growth. Our method of proof consists of a detailed analysis of the construction of the algorithms derived in Chapter IV, coupled with an estimation of the various parameters involved at each stage thereof. Central to the constructions are Puiseux's theorem (see Chapter I) and the Puiseux expansions; in this section we shall establish the requisite bounds on the coefficients in any Puiseux expansion. First, however, we shall require a bound on the genus of any finite extension of k (z). Throughout this chapter we shall denote by L a sufficiently large finite extension of K, and, unless otherwise stated, for f in L H(f) will denote the sum − Σ min(0,v(f)) taken over all the valuations v on L. If K' is any field lying between K and L then we denote by GK, the integer [L:K'] (gK,−1), where gK, is the genus of K'/k and [L:K'] is the degree of L over K'; we also recall that the height in K' of any element f is given by HK'(f) = H(f)/[L:K'].