A Powerful Determinant

Abstract

We study the construct ion of auxiliary functions likely to aid in obtaining improved irrationality measures for cubic irrationalities and thence for arbitrary algebraic numbers. Specifically, we note that the construction of curves with singularities appropri ately prescribed for our purpose leads to a simultaneous Pade approximation problem. The first step towards an explicit construction appears to be the evaluation of certain determinants. Our main task here is the computation of an example determinant, which turns out indeed to be a product of a small number of factors each to high multiplicity—whence the adjective ‘powerful’. Our evaluation confirms a computational conjecture of Bombieri, Hunt and van der Poorten.

Keywords:

• Diophantine approximation
• Roth's Theorem
• Simultaneous Pad'eapproximation