P-adic logarithmic forms and group varieties III

Let α1, ... , α n be non-zero algebraic numbers and K be a number field containing α1, ... , α n . Denote by p a prime ideal of the ring of integers in K. We present completely explicit upper bounds for , where with b 1, ... , b n being rational integers and Ξ ≠ 1. The cost n!/2 n−1 of the classical Kummer descent has been removed from the previous best upper bounds.