HTP-COMPLETE RINGS OF RATIONAL NUMBERS

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 view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP(\mathbb {Z}[W^{-1}])$ , which is naturally viewed as a set of polynomials in $\ma...

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Bibliographic Details
Published inThe Journal of symbolic logic Vol. 87; no. 1; pp. 252 - 272
Main Author MILLER, RUSSELL
Format Journal Article
LanguageEnglish
Published New York, USA Cambridge University Press 01.03.2022
Subjects
03C40
Hilbert’s Tenth Problem
12L05
boundary sets
subrings of the rational numbers
03D45
computability theory
HTP-completeness
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ISSN0022-4812
1943-5886
DOI10.1017/jsl.2021.96

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Summary: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 view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP(\mathbb {Z}[W^{-1}])$ , which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$ . It is known that for almost all W, the jump $W'$ does not $1$ -reduce to $HTP(R_W)$ . In contrast, we show that every Turing degree contains a set W for which such a $1$ -reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP(\mathbb {Z}[W^{-1}])$ can succeed uniformly on a set of measure $1$ , and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb {Z}$ has an existential definition in $\mathbb {Q}$ .
ISSN:0022-4812
1943-5886
DOI:10.1017/jsl.2021.96