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'].

INTRODUCTION

This chapter will be devoted to the construction of algorithms whereby may be determined the complete set of integral solutions of all equations of genera 0 and 1 over an arbitrary algebraic function field K. As usual [40] we shall assume that in the former case the curve associated with the equation possesses at least three infinite valuations. As for the Thue and hyperelliptic equations already solved, in each case we shall establish a simple criterion for the equation to possess an infinity of solutions in 0. In the next chapter a constructive examination of the algorithms will be used to establish explicit bounds on the heights of the solutions. The chief ingredient in those proofs is a direct recursive technique for determining the coefficients in a Puiseux series (see Lemma 5 and [10]). The bounds obtained are linear functions of the height of the original equation and thus are not exponential, as are the bounds established by Baker and Coates [8] in the case of algebraic numbers. This provides further evidence of the strength and power of our fundamental inequality which again plays the crucial role in the analysis. These results on the heights of solutions may be viewed in another way, as a complement to the celebrated theorem of Manin and Grauert, which had proved the analogue for function fields of Faltings' recent result. We recall that, as a consequence of this theorem, the heights of all solutions in K, not just those in 0, of any equation of genus 2 or more, are bounded.

Hitherto this book has effectively resolved the problem of determining the complete set of integral solutions to certain general families of equations. The fundamental inequality has formed the crux of the argument in each analysis, and it has led to the solution in Chapter II of the Thue equation, in Chapter III of the hyperelliptic equation, and in Chapter IV of equations of genera 0 and 1. In Chapter VII we succeeded in dealing with the Thue and hyperelliptic equations over fields of positive characteristic, and it was the appropriate extension of the fundamental inequality to such fields which again provided the crucial step. In the case of positive characteristic it is possible for the heights of the integral solutions to be unbounded, but this cannot occur over fields of characteristic 0, and for that case we determined explicit bounds for each of the various families solved. The fundamental inequality contributed the essence to each of those proofs also, and it is the purpose of this concluding chapter to illustrate a further range of applications for the inequality by studying briefly the superelliptic equation. Here the inequality is employed in a rather different fashion from previously, and this new approach will in fact lead to explicit bounds on the heights of all the solutions, not just those integral. Explicit bounds for non-integral solutions have only been obtained by Schmidt [36] in the case of certain Thue equations, and stronger bounds may be deduced from our methods as below.

Abstract. In this survey we discuss a common feature of some classical and recent results in number theory, graph theory, etc. We try to point out the fascinating relationship between the theory of uniformly distributed sequences and Ramsey theory by formulating the main results in both fields as statements about certain irregularities of partitions. Our approach leads to some new problems as well.

INTRODUCTION

In 1916 Hermann Weyl published his classical paper entitled “Ober die Gleichverteilung von Zahlen mod Eins”. This was intended to furnish a deeper understanding of the results in diophantine approximation and to generalize some basic results in this field. The theory of uniformly distributed sequences has originated with this paper. In the last decades this subject has developed into an elaborate theory related to number theory, geometry, probability theory, ergodic theory, etc.

Curiously enough, Issai Schur's paper entitled “Ober die Kongruenz xn+yn≡zn (mod p)” appeared in the very same year. He proved that if the positive integers are finitely colored, then there exist x, y, z having the same color so that x+y=z. Though Ramsey theory has various germs, Schur's theorem can be regarded as the first Ramsey-type theorem. Now literally the same applies to Ramsey theory as to the theory of uniform distribution:

In the last decades Ramsey theory became an elaborate theory related to number theory, geometry, probability theory, ergodic theory, etc.

This paper is dedicated to the memory of J. Howard Redfield.

INTRODUCTION

Several years ago in a lecture at a British Mathematical Colloquium splinter group (Lancaster meeting, 1978) Dr. E.K. Lloyd mentioned that he had reason to believe that J. Howard Redfield had written a second (but unpublished) paper. As is well-known Redfield's first paper [17], published in 1927, was badly neglected although at least published. Attention as far as one knows was first drawn to it by Littlewood [13] in 1950. The paper was first publicized by Harary in 1960 [10]. It seemed that Redfield had anticipated most of the later discoveries in the theory of unlabelled enumeration such as Pólya's Hauptsatz, Read's Superposition Theorem, the counting of nonisomorphic graphs and the counting of self-complementary subsets of a set with respect to a group. At the end of 1981 I received an exciting letter from Lloyd stating that this second paper [18] had indeed been found. Apparently the paper had been submitted for publication in the American Journal of Mathematics on October 19th, 1940 and was rejected by the editors in a brief letter of January 7th, 1941. Harary and Robinson [12] have written a brief account of the circumstances leading up to the discovery of Redfield's second paper and happily a special edition of the Journal of Graph Theory is to be published entirely dedicated to Redfield.