Zero Estimates and Approximating Zeros; Themes from Transcendental Numbers

Abstract:

The object of this talk is to show how the use of elimination theory
arises in the theory of transcendental numbers.  We begin with simple
examples and classical resultants.

For higher dimensional situations, we introduce the notion of Chow
forms and the important notion of the resultant of a Chow form and an
ordinary homogeneous polynomial.  We outline Nesterenko's pioneering
work bounding the order of zeros of polynomials in a solution of a
linear system of differential equations.  We also indicate the lower
bounds on the maximum absolute values of polynomials in terms of their
degrees and the distance to common zeros.

In my third lecture, I plan to discuss bounds on the orders of zeros
on commutative algebraic groups.  Multihomogeneous polynomials and
multidegrees for ideals arise naturally, and a key role is played by
van der Waerden's version of Bezout's theorem.