Additional results to a theorem of Eisenträger and Everest

Perlega, Stefan

doi:10.1007/s00013-011-0277-7

Cited by 5 publications

(5 citation statements)

References 6 publications

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“…In this paper we generalize the results of [16], [6] and [14] to prove the following theorems: Theorem 1.7. Let K be a number field, and assume there is an elliptic curve defined over K with K-rank equal to 1.…”

Section: Introductionmentioning

confidence: 88%

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Hilbert’s Tenth Problem and Mazur’s Conjectures in Complementary Subrings of Number Fields

Eisenträger

Everest

Shlapentokh

2011

Mathematical Research Letters

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Abstract. We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K, a positive integer t > 1, and t nonnegative computable real numbers δ 1 , . . . , δ t whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S 1 , . . . , S t of densities δ 1 , . . . , δ t , respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O K,S i . We also show that we can find a partition as above such that each ring O K,S i possesses an infinite Diophantine set which is discrete in every topology of the field. The only assumption on K we need is that there is an elliptic curve of rank one defined over K.

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Section: Introductionmentioning

confidence: 88%

“…More specifically, they proved that the rational primes can be partitioned into two disjoint sets S 1 , S 2 such that Hilbert's Tenth Problem is undecidable over both O K,S 1 and O K,S 2 . These results were improved by Perlega in [14] to show that the two sets can be of arbitrary computable densities.…”

Section: Introductionmentioning

confidence: 98%

Hilbert’s Tenth Problem and Mazur’s Conjectures in Complementary Subrings of Number Fields

Eisenträger

Everest

Shlapentokh

2011

Mathematical Research Letters

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“…subrings where infinitely many, though not all, primes are inverted). See [20], [23], [7], [17], [8] and [38].…”

Section: Using Elliptic Curves With Finitely Generated Groupsmentioning

confidence: 99%

First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

Shlapentokh

2018

Isr. J. Math.

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We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold. (1) For any rational prime q and any positive rational integer m, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by q m . (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set {ξ p ℓ |ℓ ∈ Z >0 , p = q is any prime such that q m+1 |(p − 1)}. (3) The first-order theory of any abelian extension of Q with finitely many ramified rational primes is undecidable. We also show that under a condition on the splitting of one rational prime in an infinite algebraic extension of Q, the existence of a finitely generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is indecidable. produced a uniform definition of Z over rings of integers of number fields, R. Rumely in [30] improved J. Robinson's result making the definition of the ring of integers over number fields uniform across fields. More recently, B. Poonen in [22] and J. Koenigsmann in [12] updated J. Robinson's definition of integers by reducing the number of universal quantifiers used in these definitions, B. Poonen to two and J. Koenigsmann to one.The desire to reduce the number of universal quantifiers is motivated to large extent by the interest in extending Hilbert's Tenth Problem to Q. This would be accomplished by a purely existential definition of Z over Q. Unfortunately there are serious doubts as to whether such a definition exists. See [6], [21] and [33] for surveys on Hilbert's Tenth Problem and related questions of definability.A lot of work aiming to prove the decidability of the first-order theory has centered around various infinite extensions of Q. (See [6] for a survey of these results.) One of the more influential results was arguably due to R. Rumely in [31], where he showed that Hilbert's Tenth Problem is decidable over the ring of all algebraic integers. This result was strengthened by L. van den Dries proving in [40] that the first-order theory of this ring was decidable. Another remarkable result is due to M. Fried, D. Haran and H. Völklein in [10], where it is shown that the first-order theory of the field of all totally real algebraic numbers is decidable. This field constitutes a boundary of sorts between the "decidable" and "undecidable", since J. Robinson showed in [27] that the first-order theory of the ring of all totally real integers is undecidable. In the same paper, she also proved that the first-order theory of a family of totally real rings of integers is undecidable and produced a "blueprint" for such proofs over ring...

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“…There exist computable sets S of natural density zero and of natural density one such that HTP(Z[S −1 ]) is undecidable. This remarkable paper was followed by generalizations in [ES09], [Per11], and [EES11]. However, no attempt has been made so far in trying to compare HTP(Z[S −1 ]) to HTP(Q).…”

Section: Theorem ([Poo03]mentioning

confidence: 99%

“…This remarkable paper was followed by generalizations in [ES09], [Per11], and [EES11]. However, no attempt has been made so far in trying to compare HTP(Z[S −1 ]) to HTP(Q).…”

Section: Introductionmentioning

confidence: 99%

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

Eisenträger

Miller

Park

et al. 2017

Trans. Amer. Math. Soc.

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Abstract. Hilbert's Tenth Problem over the rationals is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings R ⊆ Q having the property that Hilbert's Tenth Problem for R, denoted HTP(R), is Turing equivalent to HTP(Q).We are able to put several additional constraints on the rings R that we construct. Given any computable nonnegative real number r ≤ 1 we construct such rings R = Z[S −1 ] with S a set of primes of lower density r. We also construct examples of rings R for which deciding membership in R is Turing equivalent to deciding HTP(R) and also equivalent to deciding HTP(Q). Alternatively, we can make HTP(R) have arbitrary computably enumerable degree above HTP(Q). Finally, we show that the same can be done for subrings of number fields and their prime ideals.

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Additional results to a theorem of Eisenträger and Everest

Cited by 5 publications

References 6 publications

Hilbert’s Tenth Problem and Mazur’s Conjectures in Complementary Subrings of Number Fields

Hilbert’s Tenth Problem and Mazur’s Conjectures in Complementary Subrings of Number Fields

First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

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

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