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DOI: 10.4230/LIPIcs.MFCS.2026.65

On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents

Authors: Pieter Collins, Bernard Hanzon, and Eike Neumann

Published in: LIPIcs, Volume 386, 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)

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.

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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)

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@InProceedings{collins_et_al:LIPIcs.MFCS.2026.65,
  author =	{Collins, Pieter and Hanzon, Bernard and Neumann, Eike},
  title =	{{On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents}},
  booktitle =	{51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
  pages =	{65:1--65:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-442-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{386},
  editor =	{Kouck\'{y}, Michal and Petrișan, Daniela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.65},
  URN =		{urn:nbn:de:0030-drops-274478},
  doi =		{10.4230/LIPIcs.MFCS.2026.65},
  annote =	{Keywords: Linear Dynamical Systems, Computability, Computable Numbers, Transcendental Numbers, Irrationality Measure, Irrationality Exponent}
}

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On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents

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