2026-08-21T18:34:53Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27447 2026-08-21T14:42:40Z ddc:004 open_access

On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents Collins, Pieter Hanzon, Bernard Neumann, Eike Linear Dynamical Systems Computability Computable Numbers Transcendental Numbers Irrationality Measure Irrationality Exponent 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Pieter Collins and Bernard Hanzon and Eike Neumann 2026 Is Part Of LIPIcs, Volume 386, 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2026.65 urn:nbn:de:0030-drops-274478 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.65 eng https://creativecommons.org/licenses/by/4.0/legalcode