p-adic exponential ring, p-adic Schanuel's conjecture and decidability

TL;DR

This paper explores the implications of a p-adic version of Schanuel's conjecture on the decidability of the exponential ring structure over p-adic integers, linking number theory conjectures with logical decidability.

Contribution

It establishes that assuming a p-adic Schanuel's conjecture, the theory of the p-adic exponential ring becomes decidable, connecting deep number theory conjectures with logical properties.

Findings

Conditional proof of decidability based on p-adic Schanuel's conjecture

Defines exponential functions on p-adic integers

Links number theory conjectures to logical decidability

Abstract

Let exp(x) be the function determined by the classical power series of the exponentiation. Then E_p(x):=exp(px) is well-defined on Zp, the ring of p-adic integer (for p not equal to 2, we set E_2(x)=exp(4x)). Furthermore, E_p determines a structure of exponential ring on Zp. In this paper, we prove that if a p-adic version of Schanuel's conjecture is true then the theory of (Zp, +, ., 0, 1, E_p) is decidable.