Trace on imagep Original Research Article

Let p be a prime number, Qp the field of p-adic numbers, Qp a fixed algebraic closure of Qp, and imagep the completion of Qp. For elements Tset membership, variantimagep which satisfy a certain diophantine condition (*) we construct a power series F(T, Z) with coefficients in Qp and show that two elements T, U produce the same series F(T, Z)=F(U, Z) if and only if they are conjugate. We view the coefficient of Z in F(T, Z) as the trace of T. Further, we study F(T, Z) viewed as a rigid analytic function and prove that it is defined everywhere on imagep except on the set of conjugates of 1/T. The main result (Theorem 7.2) asserts that if {Tα}α is a family of elements of imagep which are non-conjugate, transcendental over Qp, and satisfy condition (*) then the functions {F(Tα, Z)}α are algebraically independent over imagep(Z). In particular, if T is an element of imagep which satisfies condition (*), then F(T, Z) is transcendental over imagep(Z) if and only if T is transcendental over Qp. In proving these results we develop some additional machinery, to be also used in a forthcoming paper which continues the study of orbits of elements in imagep.