The complexity of the theory of p-adic numbers

This paper addresses the question of the complexity of the decision problem for the theory Th(Q_p) of p-adic numbers. The best known lower bound for the theory is double exponential alternating time with a linear number of alternations. I have designed an algorithm that determines the truth value of sentences of the theory requiring double exponential space. My algorithm is based on techniques used by G.E. Collins (1975) for the theory Th(R) of the reals, and on J. Denef´s work (1986) on semi-algebraic sets and cell decomposition for p-adic fields. No elementary upper bound had been previously established