Diophantine Correct Open Induction

TL;DR

This paper introduces an induction-free axiom system for diophantine correct open induction and links it to the value distribution of semialgebraic functions, identifying certain diophantine correct subrings of Puiseux series.

Contribution

It presents a novel axiom system for diophantine correct open induction and connects it to value distribution problems, expanding understanding of diophantine correctness in Puiseux series.

Findings

Established an induction-free axiom system for diophantine correct open induction.

Linked diophantine correctness of rings to value distribution of semialgebraic functions.

Identified classes of diophantine correct subrings of Puiseux series.

Abstract

We give an induction-free axiom system for diophantine correct open induction. We relate the problem of whether a finitely generated ring of Puiseux polynomials is diophantine correct to a problem about the value-distribution of a tuple of semialgebraic functions with integer arguments. We use this result, and a theorem of Bergelson and Leibman on generalized polynomials, to identify a class of diophantine correct subrings of the field of descending Puiseux series with real coefficients.