Nonstandard polynomials: algebraic properties and elementary equivalence

TL;DR

This paper classifies rings elementarily equivalent to polynomial and Laurent polynomial rings over infinite fields or integers, revealing their algebraic structure via a novel interpretability method and linking their logic to weak second-order logic.

Contribution

It introduces a new method of regular bi-interpretations to classify and analyze the algebraic and logical properties of polynomial rings and their non-standard models.

Findings

Rings elementarily equivalent to polynomial rings are characterized as non-standard models.

Polynomial rings are bi-interpretable with superstructures of the base field or integers.

The logical complexity of these rings matches weak second-order logic over the base set.

Abstract

We solve the first-order classification problem for rings $R$ of polynomials $F[x_1, \ldots,x_n]$ and Laurent polynomials $F[x_1,x_1^{-1}, \ldots,x_n,x_n^{-1}]$ 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 $F[x_1, \ldots,x_n]$ and $F[x_1,x_1^{-1}, \ldots,x_n,x_n^{-1}]$ 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 $R = F[x_1, \ldots,x_n]$ or $R = F[x_1,x_1^{-1}, \ldots,x_n,x_n^{-1}]$ 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 $S\equiv R$. 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 $R$.