The undecidability of fields of rational functions over fields of characteristic 2

Penzin, Yu. G.

doi:10.1007/bf02219294

Cited by 5 publications

(2 citation statements)

References 2 publications

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“…The ideas behind the machinery presented here originate with Ershov [21] and Penzin [22], J. Robinson [23,24] and Rumely [25]. They have been used extensively in this context by Shlapentokh (e.g.…”

Section: Integralitymentioning

confidence: 99%

A Note on Hilbert's ``Geometric'' Tenth Problem

Tyrrell

2022

Preprint

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This paper explores undecidability in theories of positive characteristic function fields in the “geometric” language of rings LF = {0, 1,+, ·, F}, with a unary predicate F for nonconstant elements. We indicate how to generalise existing machinery to prove the undecidability of the ∀1∃+-LF theory (without parameters) of any function field of a curve over an algebraic extension of Fp, not algebraically closed. We discuss the problem (and its geometric implications) further in this context too. MSC Classification: 03B25 , 12L05

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Section: Integralitymentioning

confidence: 99%

A Note on Hilbert's ``Geometric'' Tenth Problem

Tyrrell

2022

Preprint

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“…In [3], [8], [2] and [11] it was proved that the theory of F (z) as a field, with z being a constant-symbol of the language, is undecidable. In [10] and [18] it was proved that even the existential theory of any F p (z) is undecidable.…”

Section: Introductionmentioning

confidence: 99%

Notes on the decidability of addition and the Frobenius map for polynomials and rational functions

Chompitaki

Kamarianakis

Pheidas

2022

Reports on Mathematical Logic

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Let pbe a prime number, Fp a finite field with pelements, Fan algebraic extension of Fp and z a variable. We consider the structure of addition and the Frobenius map (i.e., x →xp) in the polynomial rings F[z] and in fields F(z) of rational functions. We prove that any question about F[z] in the structure of addition and Frobenius map may be effectively reduced to questions about the similar structure of the field F. Furthermore, we provide an example which shows that a fact which is true for addition and the Frobenius map in the polynomial rings F[z] fails to be true in F(z). As a consequence, certain methods used to prove model completeness for polynomials do not suffice to prove model completeness for similar structures for fields of rational functions F(z), a problem that remains open even for F= Fp.

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Hilbert's Tenth Problem for fields of rational functions over finite fields

Pheidas

1991

Invent Math

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The undecidability of fields of rational functions over fields of characteristic 2

Cited by 5 publications

References 2 publications

A Note on Hilbert's ``Geometric'' Tenth Problem

A Note on Hilbert's ``Geometric'' Tenth Problem

Notes on the decidability of addition and the Frobenius map for polynomials and rational functions

Hilbert's Tenth Problem for fields of rational functions over finite fields

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