RECENT PROGRESS IN TRANSCENDENCE THEORY

摘要

1. Introduction In the last few years, methods from commutative algebra, algebraic geometry, complex analysis of several variables, and even cohomology theory have been used to solve problems in transcendence theory which had long been regarded as inaccessible. One of the central objects is the so-called zero-estimates, or more generally multiplicity-estimates, on certain algebraic objects. Using these estimates and the techniques developed to obtain them, many open problems in transcendence theory have been solved. On the other hand, many new problems have arisen, and it seems that transcendence theory has finally become a theory. In this article, we would like to describe this development, and for this we have to begin with a short description of the theory of multiplicity-estimates. Let us begin with two very elementary examples to describe what we mean by multiplicity-estimates.