On p-adic decomposable form inequalities

Abstract

The main concern of this thesis is the number of the solutions $N_F(m)$ of Decomposable form inequalities $F(x) \leq m$. In 2001, Thunder proved a conjecture of W.M. Schmidt, stating that, under suitable finiteness conditions, one has $N_F(m) \ll m^{n/d}$ where the implicit constant depends only on $n$ and $d$. The results in this thesis extend Thunder__s various results on Decomposable form inequalities to the p-adic setting (See Chapters 2, 4 and 5). In Chapter 3, we also show the effectivity of the condition under which the number of solutions of p-adic decomposable form inequalities is finite.