Categoricity properties for computable algebraic fields

Hirschfeldt, Denis R.; Kramer, Kenneth; Miller, Russell; Shlapentokh, Alexandra

doi:10.1090/s0002-9947-2014-06094-7

Cited by 10 publications

(13 citation statements)

References 35 publications

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“…On the other hand, the current work [11], by Hirschfeldt, Kramer, and the present authors, shows that relative computable categoricity does have a fairly reasonable structural characterization for computable algebraic fields. Their criterion does involve computability, just as does ours in this paper, but it can be expressed in a generally understandable way.…”

Section: Introductionsupporting

confidence: 44%

“…It considers relative computable categoricity for computable algebraic fields, and also examines the possible computable dimensions of such fields. Its most relevant results for us, however, are negative ones: it is shown in [11,Theorem 4.5] that there exists a computable algebraic field with computable orbit relation which is not computably categorical, and it is shown in [11,Theorem 5.1] that there exists a computably categorical algebraic field F such that B F is not even Σ 0 2 , let alone computable. One might have hoped for Proposition 7.2 to generalize to all computable algebraic fields; alternatively, one might have rephrased Theorem 7.1 to say that computable categoricity is equivalent to computable enumerability of B F (which is exactly the content of the proof, B F being Π 0 1 for any computable field with a splitting algorithm).…”

Section: Further Notesmentioning

confidence: 99%

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Computable categoricity for algebraic fields with splitting algorithms

Miller

Shlapentokh

2014

Trans. Amer. Math. Soc.

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A computably presented algebraic field F has a splitting algorithm if it is decidable which polynomials in F [X] are irreducible there. We prove that such a field is computably categorical iff it is decidable which pairs of elements of F belong to the same orbit under automorphisms. We also show that this criterion is equivalent to the relative computable categoricity of F . 3955 Licensed to Univ of Mississippi. Prepared on Tue Jul 14 02:15:54 EDT 2015 for download from IP 130.74.92.202.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3956 RUSSELL MILLER AND ALEXANDRA SHLAPENTOKH versions essentially study how much information about the structures is needed to compute isomorphisms.Among these, computable categoricity remains the most widely studied concept. It is often equivalent to relative computable categoricity, but exceptions are known to exist; see [15] for an exception, and [9] for conditions implying equivalence. Traditionally, the main question has been to determine, for a particular class of structures, some structural criterion which is equivalent to computable categoricity. In early examples, from around 1980, Dzgoev, Goncharov, and Remmel showed (independently; see [10, 24]) that a linear order is computably categorical iff it has only finitely many pairs of adjacent elements, and Remmel also showed in [25] that a Boolean algebra is computably categorical iff it has only finitely many atoms. In both cases, the structural criterion identifies the obstacle to computing isomorphisms, in the class of structures under consideration. On the other hand, the criteria equivalent to computable categoricity for trees (viewed either as partial orders, or under the meet relation, and also with distinguished subtrees), established in [16] and [18] by Lempp, McCoy, Miller, and Solomon and in [13] by Kogabaev, Kudinov, and Miller, are not easy to describe in any known way, even though they are "structural", in any reasonable sense of the word. In terms of computational complexity, they are Σ 0 3 , as defined in Section 2, just like the conditions for linear orders and Boolean algebras. Indeed, for all of these structures, computable categoricity is equivalent to relative computable categoricity, and it is known from work by Ash, Knight, Manasse, and Slaman in [1], and independently by Chisholm in [2], that the computational complexity of relative computable categoricity is always Σ 0 3 . Our intention in this paper is to give a criterion for computable categoricity for algebraic fields with splitting algorithms. This should be viewed as a first step towards a criterion for computable fields in general, for which the question of computable categoricity has long been studied and has proven highly intractable. The basic definitions regarding computable fields appear in Section 3. Apart from algebraically closed fields, we believe that ours is the first result to characterize computable categoricity for any natural class of fields. (An algebraically closed fie...

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Section: Introductionsupporting

confidence: 44%

Section: Further Notesmentioning

confidence: 99%

Computable categoricity for algebraic fields with splitting algorithms

Miller

Shlapentokh

2014

Trans. Amer. Math. Soc.

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“…Each of these polynomials has the same positive probability of having exactly one root in a given field, and linear disjointness ensures that these probabilities are all independent: the number of roots of one such polynomial in F is independent of the number of roots of any of the others. We will not go further into the details here; the reader may refer to [12,Prop. 2…”

Section: Measure and Category For Field Propertiesmentioning

confidence: 99%

Isomorphism and classification for countable structures

Miller

2019

COM

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We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we use a computable homeomorphism to define a measure on the space of isomorphism types of algebraic fields, and examine the prevalence of relative computable categoricity under this measure. * This project grew out of the extended abstract [20], although the results presented there and here are disjoint.

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“…THEOREM 4.1 [15]. There is an algebraic field (i.e., a subfield of the algebraic closure of Q) that is computably categorical but not relatively computably categorical.…”

Section: Fieldsmentioning

confidence: 99%

Preserving Categoricity and Complexity of Relations

et al. 2015

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Keywords: Δ 0 α categorical structure, structure that is not relatively Δ 0 α categorical, field.In [3,6,7], it was proved that for each computable ordinal α, there is a structure that is Δ 0 α categorical but not relatively Δ 0 α categorical. The original examples were not familiar algebraic kinds of structures. In [11], it was shown that for α = 1, there are further examples in several familiar classes of structures, including rings and 2-step nilpotent groups. Similar example for all computable successor ordinals were constructed in [12]. In the present paper, this result is extended to computable limit ordinals. We know of an example of an algebraic field that is computably categorical but not relatively computably categorical. Here we show that for each computable limit ordinal α > ω, there is a field which is Δ 0 α categorical but not relatively Δ 0 α categorical. Examples on dimension and complexity of relations are given.

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Categoricity properties for computable algebraic fields

Cited by 10 publications

References 35 publications

Computable categoricity for algebraic fields with splitting algorithms

Computable categoricity for algebraic fields with splitting algorithms

Isomorphism and classification for countable structures

Preserving Categoricity and Complexity of Relations

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