We consider expansions of Presburger arithmetic with families of monadic polynomial predicates. (Examples of such predicates are the set of perfect squares, or the set of integers of the form 2n3-5n+3, etc.) Although the full attendant first-order theories are well known to be undecidable, very little is known when one restricts the number of variables. For single-variable theories, we obtain positive results for the following two families of predicates: (i) for perfect fixed powers, decidability of the corresponding theory follows from the solvability of hyperelliptic Diophantine equations; and (ii) for polynomials of degree at most three, we establish decidability by relying on the low genus of the resulting algebraic curves. Finally, we discuss limitations and hardness results (via encodings of longstanding open Diophantine problems) as soon as any of the above restrictions are lifted.
Submitted, 2026. 11 pages.
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© 2026 Piotr Bacik, Joris Nieuwveld, Joël Ouaknine, Mihir
Vahanwala, Madhavan Venkatesh, and Emil Rugaard Wieser.