Suppose we are given a “Thue equation” f(x, y) = 1, where f is a binary form with coefficients in a function field K of characteristic zero. A typical result is that if f is of degree at least 5 and has no multiple factors, then every solution x = (x, y) of the equation with components in K has H(x)≤90H(f) + 250g. Here g is the genus of K and H(x), H(f) are suitably defined heights. No assumption is made that x be “integral” in some sense. As an application, bounds are derived for “integral” solutions of hyperelliptic equations over K.

There is an old conjecture that every integer can be decomposed into four (positive or negative) perfect cubes. More specifically one would like to know the asymptotic number of solutions of when a large bound N is placed on the parts mi. Using the circle method it is shown that the number of such representations of n when N → ∞ is asymptotically equal to C(n). N for a certain positive constant C(n), provided that the contribution of the minor arcs can be neglected.

This is the text of an invited address given at the Summer Research Institute of the Australian National University in January 1978. It provides a survey of the subject matter of the title.

In the present paper, we derive transcendence measures for the numbers log α, eβ, αβ, (log α1)/(log α2) from a previous lower bound of ours on linear forms in the logarithms of algebraic numbers.

We give several results concerning the simultaneous approximation of certain complex numbers. For instance, we give lower bounds for |a–ξo |+ | ea – ξ1 |, where a is any non-zero complex number, and ξ are two algebraic numbers. We also improve the estimate of the so-called Franklin Schneider theorem concerning | b – ξ | + | a – ξ | + | ab – ξ. We deduce these results from an estimate for linear forms in logarithms.

In this note a number of related problems about divisors are studied, and partial solutions obtained by elementary means. The problems are rather unconventional and seem to suggest interesting developments.

The Markoff spectrum of an algebraic number field is defined and it is proved that the spectrum of Q(✓5) is not discrete.

Let x0, x1, x2, x3 be polynomials in a variable t and with coefficients in a field k of character of characteristic 0. If and , then x0 = x1 = x2 = x3 = 0. This partially answers a question of Pjatetskii-Š;apiro and Šafarevič about the K3-surface . The proof uses a technique of M. R. Christie.

The greatest lower bound (in fact, ½) is found of constants k such that for an infinity of rationals p/q.

A link between norms from quadratic fields and —det (AB−BA) for 2 × 2 matrices is reformulated via central polynomials and thereby generalized.

The asymptotic behaviour of second-order recurring seequences is studied.