P-adic logarithmic forms and group varieties III

Let alpha(1), . . . , alpha(n) be non-zero algebraic numbers and K be a number field containing alpha(1), (.) (.) (.) , alpha(n). Denote by p a prime ideal of the ring of integers in K. We present completely explicit upper bounds for ord(p)(Xi - 1), where Xi = alpha(b1)(1) . . . alpha(bn)(n) with b(1), . . . ,b(n), being rational integers and Xi not equal 1. The cost n!/2(n-1) of the classical Kummer descent has been removed from the previous best upper bounds.