On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents

We establish Diophantine hardness results for the decidability of the Positivity Problem for exponential-trigonometric polynomials over computable discrete subfields of the real numbers, and for related questions. We show that any algorithm for deciding either non-negativity, eventual non-negativity, the existence of a zero, or the existence of infinitely many zeros of exponential-trigonometric polynomials over a computable discrete subfield K of the reals containing the number π can be translated into an algorithm for computing the irrationality exponents of all elements of K. As a consequence, we exhibit a computable discrete subfield K of the reals such that all of the aforementioned questions about exponential-trigonometric polynomials over K are undecidable. In particular, we provide the first example of a natural generalisation of the Continuous Skolem Problem that is provably undecidable.