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Identifier 000404561
Title An analogue of Hilbert's tenth problem for the ring of exponential sums
Alternative Title Ένα ανάλογο του 10ο προβλήματος του Χίλμπερτ για το δακτύλιο των εκθετικών αθροισμάτων
Author Χομπιτάκη, Δήμητρα
Thesis advisor Φειδάς, Αθανάσιος
Reviewer Kolountzakis, Mihalis
Vidaux, Xavier
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 L-structure. 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 (±e-z, ie-z). 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 (Z2-v - 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, Z-N]. 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
Language English, Greek
Subject Diophantine problem
Pell's equation
Positive existential theory
Διοφαντικό πρόβλημα
Εκθετικά αθροίσματα
Εξίσωση του Πελλ
Θετική υπαρξιακή θεωρία
Issue date 2016-11-18
Collection   School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Post-graduate theses
  Type of Work--Post-graduate theses
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