Abstract 
At a glance: We prove that the positive existential theory of the ring of exponential sums is undecidable.
Define the set of exponential sums, EXP(C), to be the the set of expres¬sions
a = αο + α1βμιζ + + aw βμΝ z
where α^,μ.,• G C. We ask whether the positive existential first order theory of EXP(C), as a structure of the language
L = {+, •, 0,1,ez}
is decidable or undecidable. Our result may be considered as an analogue of Hilbert's Tenth Problem for this structure and as a step to answering the similar problem for the ring of exponential polynomials, which is still open. We prove:
Theorem 1 The ring of gaussian integers Z[i] is positive existentially defin¬able over EXP(C), as an Lstructure. Hence the positive existential theory of this structure is undecidable.
In order to prove Theorem 1 we adapt techniques of [8] and we show Theorem 2:
(0.0.16)
We consider the equation
(e2z  1)y2 = x2  1
where x,y Ε EXP(C).
Let (a,b) and (a2,b2) be solutions of (0.0.16). We define the law Θ by
bi) Θ (a2, b2) = (aia2 + (e2z  1)bib2, aib2 + a2bi)
The pair (a,b) = (ai,bi) Θ (a2,b2) is also a solution of (0.0.16).
We denote by κ Θ (a, b) = (a, b) Θ • • • Θ (a, b). ((a, b) added to itself by Θ κ
times.)
Theorem 2 The solutions of the equation (0.0.16) are given by (x, y) = κ Θ (±ez, 1) Θ λ Θ (±ez, iez). The proof uses techniques of [38], [28] and [23]. Important points of the proof
We would like to characterise all the solutions of Equation (0.0.16) over EXP(C). Observe that, by the definition of EXP(C), x and y lay in some ring of the form R = C[ewz, β~μιΖ,... , eMfcZ,e_MfcZ], where k is a natural number and each μ Ε C.
In [28] it is shown that one can choose the μ in such a way that μ1 = , for some natural number N, and the set {1,μ2, • • • , μ^;} is linearly indepen¬dent over the field Q. By results of [38] it follows that the set {βμιΖ,..., eMfcz} is algebraically independent over C. So the question about solutions of (0.0.16) becomes
Given a natural number N, find the solutions of
(Z2v  1)y2 = x2  1 (0.0.17)
over the ring
C[Z,Z 1 ,t2,t21, ■ ■ ■ ,ίί,ίρ1],
where Z = eJ1 and the elements t2, ■ ■ ■ te are variables over may be consid¬ered as variables over C[Z, ZAt a first stage we show that any solu¬tion of (0.0.17) does not depend on the varables tj, i.e. is over C[Z, Z Then, extending techniques of [23] we show that any solution is over the ring C[ZN, ZN]. Finally we give the characterization of solutions as in Theorem 2. Subsequently the set of integers is positive existentially definable, by tech¬niques of [8] and [7].
The results of Theorem 2 may be stated as
The set of solutions of
(T2  1)y2 = x2  1 over the tower of rings
uN C[T Μ ,T  Μ ]
stabilizes at the level of C[T, T
