No abstract

We prove unsolvability of the analogue of Hilbert's Tenth Problem for rings of exponential polynomials. The technique of proof consists of an interaction between Arithmetic, Analysis, Logic, and Functional Transcendence.

We prove model completeness for the theory of addition and the Frobenius map for certain subrings of rational functions in positive characteristic. More precisely: Let p be a prime number, F p the prime field with p elements, F a field algebraic over F p and z a variable. We show that the structures of rings R, which are generated over F [z] by adjoining a finite set of inverses of irreducible polynomials of F [z] (e.g., R = F p [z, 1 z ]), with addition, the Frobenius map x → x p and the predicate '∈ F ' -together with function symbols and constants that allow building all elements of F p [z] -are model complete, i.e., each formula is equivalent to an existential formula. Further, we show that in these structures all questions, i.e., first order sentences, about the rings R may be, constructively, translated into questions about F .

Let pbe a prime number, Fp a finite field with pelements, Fan algebraic extension of Fp and z a variable. We consider the structure of addition and the Frobenius map (i.e., x →xp) in the polynomial rings F[z] and in fields F(z) of rational functions. We prove that any question about F[z] in the structure of addition and Frobenius map may be effectively reduced to questions about the similar structure of the field F. Furthermore, we provide an example which shows that a fact which is true for addition and the Frobenius map in the polynomial rings F[z] fails to be true in F(z). As a consequence, certain methods used to prove model completeness for polynomials do not suffice to prove model completeness for similar structures for fields of rational functions F(z), a problem that remains open even for F= Fp.