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On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents

Authors Pieter Collins , Bernard Hanzon , Eike Neumann

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Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Linear Dynamical Systems
  • Computability
  • Computable Numbers
  • Transcendental Numbers
  • Irrationality Measure
  • Irrationality Exponent

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Abstract

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.

Cite As Get BibTex

Pieter Collins, Bernard Hanzon, and Eike Neumann. On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.65

Author Details

Pieter Collins
  • Maastricht University, The Netherlands
Bernard Hanzon
  • University College Cork, Ireland
Eike Neumann
  • Swansea University, UK

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