Results and open problems related to Schmidt’s Subspace Theorem

Abstract

Assumptions being as above, there is an effectively computable, proper linear subspace T exc of Q such that (1) has only finitely many solutions outside T . Moreover, T exc can be chosen from a finite collection independent of c1, . . . , cn. Here, the method of proof does not allow to compute the solutions outside T . But one can prove the following ‘semi-effective’ result. Assume that the coefficients of L1, . . . , Ln have heights at most H, and that they generate a number field K of degree D. Further, let c1 + · · · + cn ≤ −δ with 0 < δ < 1, and max(c1, . . . , cn) = 1. Then one can show that for the solutions x ∈ Z of (1) outside T exc one has ‖x‖ ≤ max ( B(n,K, δ), H eff(n,D,δ) ) ,