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DOI: 10.4230/LIPIcs.MFCS.2025.25

Polynomial-Time Tractable Problems over the p-Adic Numbers

Authors: Manuel Bodirsky and Arno Fehm

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

We study the computational complexity of fundamental problems over the p-adic numbers {ℚ}_p and the p-adic integers {ℤ}_p. Guépin, Haase, and Worrell [Florent Guépin et al., 2019] proved that checking satisfiability of systems of linear equations combined with valuation constraints of the form v_p(x) = c for p ≥ 5 is NP-complete (both over {ℤ}_p and over {ℚ}_p), and left the cases p = 2 and p = 3 open. We solve their problem by showing that the problem is NP-complete for {ℤ}₃ and for {ℚ}₃, but that it is in P for {ℤ}₂ and for {ℚ}₂. We also present different polynomial-time algorithms for solvability of systems of linear equations in {ℚ}_p with either constraints of the form v_p(x) ≤ c or of the form v_p(x) ≥ c for c ∈ {ℤ}. Finally, we show how our algorithms can be used to decide in polynomial time the satisfiability of systems of (strict and non-strict) linear inequalities over {ℚ} together with valuation constraints v_p(x) ≥ c for several different prime numbers p simultaneously.

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Manuel Bodirsky and Arno Fehm. Polynomial-Time Tractable Problems over the p-Adic Numbers. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bodirsky_et_al:LIPIcs.MFCS.2025.25,
  author =	{Bodirsky, Manuel and Fehm, Arno},
  title =	{{Polynomial-Time Tractable Problems over the p-Adic Numbers}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.25},
  URN =		{urn:nbn:de:0030-drops-241325},
  doi =		{10.4230/LIPIcs.MFCS.2025.25},
  annote =	{Keywords: p-adic numbers, existential theory, linear theory, constraint satisfaction, linear program feasibility, NP-hardness, polynomial-time algorithm}
}

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Polynomial-Time Tractable Problems over the p-Adic Numbers

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