The Absolute Subspace Theorem and linear equations with unknowns from a multiplicative group

Abstract

They both gave essentially the same proof, based on the Subspace Theorem (more precisely, Schlickewei’s generalisation to p-adic absolute values and number fields [30] of the Subspace Theorem proved by Schmidt in 1972 [41]). In 1984, Evertse [7] showed that if G is the group of S-units in K and a, b ∈ K∗, then the equation ax+ by = 1 has at most 3×7 solutions in x, y ∈ G, where s is the cardinality of S. The significant feature of this bound is its uniformity. It does not depend upon the coefficients a and b and it involves only the cardinality of the set S but not the particular primes belonging to S. Schmidt’s pioneering work from 1989 [42] in which he obtained a quantitative version of his Subspace Theorem from 1972 giving an explicit upper bound for the number of subspaces involved, opened the possibility to determine explicit upper bounds for the number of solutions of Diophantine equations from several classes, including eq. (0.1) in n ≥ 3 unknowns. In fact, many of the generalisations and improvements of Schmidt’s result obtained later were motivated by the desire to derive good explicit uniform upper bounds for