The Decision Problem for Exponential Diophantine Equations

Zur Theorie Der Spektralen Darstellung Von Prädikaten Durch Ausdrücke Der Prädikatenlogik 1. Stufe

In this paper we investigate the algebraic extensions $K$ of $\mathbb{Q}$ in which we cannot existentially or universally define the ring of integers $\mathcal{O}_K$. A complete answer to this question would have important consequences. For example, the existence of an existential definition of $\mathbb{Z}$ in $\mathbb{Q}$ would imply that Hilbert's Tenth Problem for $\mathbb{Q}$ is undecidable, resolving one of the biggest open problems in the area. However, a conjecture of Mazur implies that the integers are not existentially definable in the rationals. Although proving that an existential definition of $\mathbb{Z}$ in $\mathbb{Q}$ does not exist appears to be out of reach right now, we show that when we consider all algebraic extensions of $\mathbb{Q}$, this is the generally expected outcome. Namely, we prove that in most algebraic extensions of the rationals, the ring of integers is not existentially definable. To make this precise, we view the set of algebraic extensions of $\mathbb{Q}$ as a topological space homeomorphic to Cantor space. In this light, the set of fields which have an existentially definable ring of integers is a meager set, i.e. is very small. On the other hand, by work of Koenigsmann and Park, it is possible to give a universal definition of the ring of integers in finite extensions of the rationals, i.e. in number fields. Still, we show that their results do not extend to most algebraic infinite extensions: the set of algebraic extensions of $\mathbb{Q}$ in which the ring of integers is universally definable is also a meager set.

Additional excerpts

A topological approach to undefinability in algebraic extensions of $\mathbb{Q}$

For a ring R, Hilbert’s Tenth Problem HTP(R) is the set of polynomial equations over R, in several variables, with solutions in R. We consider computability of this set for subrings R of the rationals. Applying Baire category theory to these subrings, which naturally form a topological space, relates their sets HTP(R) to the set HTP$(\mathbb {Q})$, whose decidability remains an open question. The main result is that, for an arbitrary set C, HTP$(\mathbb {Q})$ computes C if and only if the subrings R for which HTP(R) computes C form a nonmeager class. Similar results hold for 1-reducibility, for admitting a Diophantine model of $\mathbb {Z}$, and for existential definability of $\mathbb {Z}$.

background

Baire category theory and Hilbert's Tenth Problem inside $\mathbb{Q}$

The computability, definability, and proof theory of Artinian rings

The first edition of MOSCA, the Meeting on String Constraints and Applications was held in May 2019 in the otherworldly location of Bertinoro [D'Antoni et al. 2019]. For one week, 43 participants presented surveys and discussed recent research into the analysis of string constraints.

The Decision Problem for Exponential Diophantine Equations

Citations

Zur Theorie Der Spektralen Darstellung Von Prädikaten Durch Ausdrücke Der Prädikatenlogik 1. Stufe

A topological approach to undefinability in algebraic extensions of $\mathbb{Q}$

Baire category theory and Hilbert's Tenth Problem inside $\mathbb{Q}$

The computability, definability, and proof theory of Artinian rings

Strings at MOSCA

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