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.

翻译：我们考虑Presburger算术在单变量多项式谓词族上的展开。（此类谓词的例子包括完全平方数集，或形如$2n^3-5n+3$的整数集等。）尽管相应完备的一阶理论已知是不可判定的，但当限制变量数量时，所知甚少。在单变量理论的情况下，我们针对以下两类谓词族获得了肯定性结果：(i) 对于完美固定幂次，相关理论的可判定性由超椭圆丢番图方程的可解性得出；(ii) 对于次数至多为三的多项式，我们通过依赖所得到代数曲线的低亏格确立了可判定性。最后，我们讨论了一旦上述任一限制被解除时存在的局限性及困难结果（通过编码长期未解决的开放丢番图问题）。