As easy as $\mathbb {Q}$: Hilbert’s Tenth Problem for subrings of the rationals and number fields

Eisenträger, Kirsten; Miller, Russell; Park, Jennifer; Shlapentokh, Alexandra

doi:10.1090/tran/7075

Cited by 7 publications

(26 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result, stated formally below, essentially follows from work of Julia Robinson in [9]. For a proof by Eisenträger, Park, Shlapentokh, and the author, see [2].…”

Section: Subrings Of the Rationalsmentioning

confidence: 88%

See 1 more Smart Citation

Measure Theory and Hilbert’s Tenth Problem Inside ℚ

Miller¹

2017

Sets and Computations

Self Cite

View full text Add to dashboard Cite

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. When R = Z, it is known that the jump Z is Turing-reducible to HTP(Z). We consider computability of HTP(R) for subrings R of the rationals. Applying measure theory to these subrings, which naturally form a measure space, relates their sets HTP(R) to the set HTP(Q), whose decidability remains an open question. We raise the question of the measure of the topological boundary of the solution set of a polynomial within this space, and show that if these boundaries all have measure 0, then for each individual oracle Turing machine Φ, the reduction R = Φ HTP(R) fails on a set of subrings R of positive measure. That is, no Turing reduction of the jump R of a subring R to HTP(R) holds uniformly on a set of measure 1.

show abstract

“…This result, stated formally below, essentially follows from work of Julia Robinson in [9]. For a proof by Eisenträger, Park, Shlapentokh, and the author, see [2].…”

Section: Subrings Of the Rationalsmentioning

confidence: 88%

“…A stronger version of this question appears in [2]. There Eisenträger, Park, Shlapentokh, and the author ask (in essence) whether a polynomial f could have the properties that C(f ) = ∅, yet that there also exists ε > 0 such that for every W ∈ A(f ), there is an n for which |W ∩{0,...,n}| n+1 > ε.…”

Section: Questionsmentioning

confidence: 99%

Measure Theory and Hilbert’s Tenth Problem Inside ℚ

Miller¹

2017

Sets and Computations

Self Cite

View full text Add to dashboard Cite

show abstract

“…This result, which follows from Corollary 2.2 below, began with work of Julia Robinson in [11]. A proof by Eisenträger, Park, Shlapentokh, and the author appears in [4], based in turn on work by Koenigsmann in [6]. Proposition 2.1 (see Proposition 5.4 in [4]) For every prime p, there is a polynomial g p (Z, X 1 , X 2 , X 3 ) such that for all rationals q, we have q ∈ R (P−{p}) ⇐⇒ g p (q, X) ∈ HT P (Q).…”

Section: Subrings Of the Rationalsmentioning

confidence: 89%

The Hilbert’s-Tenth-Problem Operator

Kramer

Miller

2019

Isr. J. Math.

Self Cite

View full text Add to dashboard Cite

For a ring R, Hilbert's Tenth Problem HT P (R) is the set of polynomial equations over R, in several variables, with solutions in R. We view HT P as an operator, mapping each set W of prime numbers to HT P (Z[W −1 ]), which is naturally viewed as a set of polynomials in Z[X 1 , X 2 , . . .]. For W = ∅, it is a famous result of Matiyasevich, Davis, Putnam, and Robinson that the jump ∅ ′ is Turing-equivalent to HT P (Z). More generally, HT P (Z[W −1 ]) is always Turing-reducible to W ′ , but not necessarily equivalent. We show here that the situation with W = ∅ is anomalous: for almost all W , the jump W ′ is not diophantine in HT P (Z[W −1 ]). We also show that the HT P operator does not preserve Turing equivalence: even for complementary sets U and U , HT P (Z[U −1 ]) and HT P (Z[U −1 ]) can differ by a full jump. Strikingly, reversals are also possible, with V < T W but HT P (Z[W −1 ]) < T HT P (Z[V −1 ]).

show abstract

“…(If such an existential definition exists, then HT P (Z) itself 1-reduces to HT P (Q), and therefore so does the Halting Problem.) Before that, Sections 2 and 3 provide lemmas established earlier in [2] and [8], that will be of use in the subsequent sections. For basic information about computability theory, [18] remains an excellent source, and [16] is also helpful.…”

Section: Introductionmentioning

confidence: 99%

HTP-complete rings of rational numbers

Miller¹

2019

Preprint

Self Cite

View full text Add to dashboard Cite

For a ring R, Hilbert's Tenth Problem HT P (R) is the set of polynomial equations over R, in several variables, with solutions in R. We view HT P as an enumeration operator, mapping each set W of prime numbers to HT P (Z[W −1 ]), which is naturally viewed as a set of polynomials in Z[X 1 , X 2 , . . .]. It is known that for almost all W , the jump W ′ does not 1-reduce to HT P (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 HT P (Z[W −1 ]) can succeed uniformly on a set of measure 1, and regarding the consequences for the boundary sets of the HT P operator in case Z has an existential definition in Q.

show abstract

As easy as $\mathbb {Q}$: Hilbert’s Tenth Problem for subrings of the rationals and number fields

Cited by 7 publications

References 34 publications

Measure Theory and Hilbert’s Tenth Problem Inside ℚ

Measure Theory and Hilbert’s Tenth Problem Inside ℚ

The Hilbert’s-Tenth-Problem Operator

HTP-complete rings of rational numbers

Contact Info

Product

Resources

About