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HILBERT’S TENTH PROBLEM FOR FUNCTION FIELDS OF VARIETIES OVER NUMBER FIELDS AND pADIC
, 2006
"... Let k be a subfield of a padic field of odd residue characteristic, and let L be the function field of a variety of dimension n ≥ 1 over k. Then Hilbert’s Tenth Problem for L is undecidable. In particular, Hilbert’s Tenth Problem for function fields of varieties over number fields of dimension ≥ 1 ..."
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Let k be a subfield of a padic field of odd residue characteristic, and let L be the function field of a variety of dimension n ≥ 1 over k. Then Hilbert’s Tenth Problem for L is undecidable. In particular, Hilbert’s Tenth Problem for function fields of varieties over number fields of dimension ≥ 1 is undecidable.
Uniform firstorder definitions in finitely generated fields, preprint
, 2005
"... Abstract. We prove that there is a firstorder sentence in the language of rings that is true for all finitely generated fields of characteristic 0 and false for all fields of characteristic> 0. We also prove that for each n ∈ N, there is a firstorder formula ψn(x1,..., xn) that when interpreted in ..."
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Cited by 2 (1 self)
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Abstract. We prove that there is a firstorder sentence in the language of rings that is true for all finitely generated fields of characteristic 0 and false for all fields of characteristic> 0. We also prove that for each n ∈ N, there is a firstorder formula ψn(x1,..., xn) that when interpreted in a finitely generated field K is true for elements x1,...,xn ∈ K if and only if the elements are algebraically dependent over the prime field in K. 1.
FIRSTORDER DEFINITIONS IN FUNCTION FIELDS OVER
, 2006
"... Definition 1.1. A field k is antiMordellic (or large) if every smooth curve with a kpoint has infinitely many kpoints. Separably closed fields, henselian fields, and real closed fields are examples of antiMordellic ..."
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Cited by 1 (0 self)
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Definition 1.1. A field k is antiMordellic (or large) if every smooth curve with a kpoint has infinitely many kpoints. Separably closed fields, henselian fields, and real closed fields are examples of antiMordellic
UNDECIDABLE PROBLEMS: A SAMPLER
"... Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence ..."
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Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.
Hilbert’s Tenth Problem for function fields over valued fields in characteristic zero
, 902
"... Let K be a field with a valuation satisfying the following conditions: both K and the residue field k have characteristic zero; the value group is not 2divisible; there exists a maximal subfield F in the valuation ring such that Gal ( ¯ F /F) and Gal ( ¯ k/k) have the same 2cohomological dimensi ..."
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Let K be a field with a valuation satisfying the following conditions: both K and the residue field k have characteristic zero; the value group is not 2divisible; there exists a maximal subfield F in the valuation ring such that Gal ( ¯ F /F) and Gal ( ¯ k/k) have the same 2cohomological dimension and this dimension is finite. Then Hilbert’s Tenth Problem has a negative answer for any function field of a variety over K. In particular, this result proves undecidability for varieties over C((T)). 1
Diophantine sets of polynomials over finitely generated fields
, 2008
"... in characteristic zero ..."
DIOPHANTINE UNDECIDABILITY OF HOLOMORPHY RINGS OF FUNCTION FIELDS OF CHARACTERISTIC 0
, 805
"... Abstract. Let K be a onevariable function field over a field of constants of characteristic 0. Let R be a holomorphy subring of K, not equal to K. We prove the following undecidability results for R: If K is recursive, then Hilbert’s Tenth Problem is undecidable in R. In general, there exist x1,... ..."
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Abstract. Let K be a onevariable function field over a field of constants of characteristic 0. Let R be a holomorphy subring of K, not equal to K. We prove the following undecidability results for R: If K is recursive, then Hilbert’s Tenth Problem is undecidable in R. In general, there exist x1,...,xn ∈ R such that there is no algorithm to tell whether a polynomial equation with coefficients in Q(x1,...,xn) has solutions in R. 1.