Decidability of the theory of addition and the Frobenius map in rings of rational functions (original) (raw)

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, Fp the prime field with p elements, F a field algebraic over Fp 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 = Fp[z, 1 z ]), with addition, the Frobenius map x 7→ x and the predicate ‘∈ F ’ together with function symbols and constants that allow building all elements of Fp[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 .

Sign up for access to the world's latest research.

checkGet notified about relevant papers

checkSave papers to use in your research

checkJoin the discussion with peers

checkTrack your impact