April 20, 2026

Decidability problems for global and local fields

Arno Fehm, TU Dresden, Germany RIMS Kyoto Room 110 + Zoom · JP: 15:30 · FR: 08:30

Can one algorithmically decide whether a system of polynomial equations has a solution in the field of rational numbers? While this question is open, several closely related questions are by now answered. In particular, the answer is YES for local fields (with some caveats in positive characteristic), and NO for global fields if we ask for something only slightly stronger.

In this talk I will give an introduction to this area and will discuss several variations of the question, for different fields (global and local), different kinds of equations (polynomial, linear, ...), and with arithmetic extra conditions (absolute values, heights).

  1. S. Anscombe, P. Dittmann and A. Fehm. Axiomatizing the existential theory of Fq ((t)). Algebra & Number Theory 17(11):2013–2032, 2023
  2. J. Koenigsmann. Decidability in local and global fields. Proc. Int. Cong. of Math. – 2018, Rio de Janeiro, Vol. 1 (45–60)
  3. B. Poonen. Undecidability in number theory. Notices of the Amer. Math. Soc. 55(3), 2008.

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