The Number of Solutions of Bounded Height to a System of Linear Equations

AbstractWe Count the number of solutions with height less than or equal to B to a system of linear equations over a number field. We give explicit asymptotic estimates for the number of such solutions as B goes to infinity, where the constants involved depend on the classical invariants of the number field (degree, discriminant, class number. etc.). The problem is reformulated as an estimate for the number of lattice points in a certain bounded domain