On Variable-Bounded Non-Linear Expansions of Presburger Arithmetic

2026-05-16

AI Summary

This study investigates the decidability of the first-order theory of Presburger arithmetic extended with unary nonlinear predicates, such as fixed powers and polynomials of degree at most three. By integrating tools from algebraic geometry—specifically the theory of low-genus curves—with number-theoretic methods for solving Diophantine equations, the authors establish, for the first time, that this restricted nonlinear extension remains decidable. The work delineates a sharp boundary of decidability: even slight relaxations of the constraints lead to undecidability. Moreover, it reveals deep connections between more general extensions and classical open problems in Diophantine analysis, thereby underscoring both the tightness and theoretical significance of the obtained decidability result.

Abstract

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 $2n^3-5n+3$, 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. In the case of single-variable theories, we obtain positive results for the following two families of predicates: (i) for perfect fixed powers, decidability ofthe corresponding theory follows from the solvability of hyperellipticDiophantine 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.

Problem

• Presburger arithmetic
• decidability
• monadic polynomial predicates
• Diophantine equations
• first-order theories

Innovation

• Presburger arithmetic
• monadic polynomial predicates
• decidability
• Diophantine equations
• algebraic curves