A powerful determinant

We study the construction of auxiliary functions likely to aid in obtaining improved irrationality measures for cubic irrationalities and thence for a rbitrary algebraic numbers. Specifically, we note that the construction of curves with singularities appropriately prescribed for our purpose leads to a simultaneous Fade approximation problem. The first step towards an expli cit construction appears to be the evaluation of certain determinants. Our main task here is the computation of an example determinant, which turns ou t indeed to be a product of a small number of factors each to high multipli city - whence the adjective 'powerful'. Our evaluation confirms a computati onal conjecture of Bombieri, Hunt and van der Poorten.