2024/09/22 by Alexei Myasnikov, Myasnikov, Alexei, Andrey Nikolaev +1
Mathematics · #03B10 #03B16 #03H05 (Primary) #12L15 (Secondary) #FOS: Mathematics #History and Theory of Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2409.14467
openalex publication_date 2024/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We solve the first-order classification problem for rings R of polynomials F[x1, …,xn] and Laurent polynomials F[x1,x1-1, …,xn,xn-1] with coefficients in an infinite field F or the ring of integers \mathbb Z, that is, we describe the algebraic structure of all rings S that are first-order equivalent to R. Our approach is based on a new and very powerful method of regular bi-interpretations, or more precisely, regular invertible interpretations. Namely, we prove that F[x1, …,xn] and F[x1,x1-1, …,xn,xn-1] are regularly bi-interpretable with the list superstructure \mathbb S(F,\mathbb N) of F, which is equivalent to regular bi-interpretation with the superstructure HF(F) of hereditary finite sets over F. The expressive power of \mathbb S(F,\mathbb N) is the same as that of the weak second-order logic over F. Hence, the first-order logic in R = F[x1, …,xn] or R = F[x1,x1-1, …,xn,xn-1] is equivalent to the weak second-order logic in F (following the terminology of Kharlampovich, Myasnikov, and Sohrabi [16], such structures are necessarily rich), which allows one to describe the algebraic structure of all rings S with S≡ R. In fact, these rings S are precisely the ``non-standard'' models of R, like in non-standard arithmetic or non-standard analysis. This is particularly straightforward when F is regularly bi-interpretable with \mathbb N, in this case the ring R is also bi-interpretable with \mathbb N. Using our approach, we describe various, sometimes rather surprising, algebraic and model-theoretic properties of the non-standard models of R.