Exponential sums, number of solutions of algebraic equations, and Poincaré series

Definitions of Gauss and Ramanujan sums over the algebras A and Lv are given, and their main properties are proved. Using these results an analogous of an old result of Libri on the number of solutions of algebraic equations with integral coecients modulo a prime power is obtained, and then used to compute the number of solutions of some equations with coecients in Lv. Finally, an analogous of a problem of Nageswara Rao on algebraic equations subject to partitions is solved for equations

with coecients in Lv.