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HILBERT’S TENTH PROBLEM FOR FUNCTION FIELDS OF VARIETIES OVER NUMBER FIELDS AND pADIC Fields
, 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
, 2005
"... 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 fi ..."
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Cited by 2 (1 self)
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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.
FIRSTORDER UNDECIDABILITY IN FUNCTION FIELDS OF POSITIVE CHARACTERISTIC
, 709
"... Abstract. We prove that the firstorder theory of any function field K of characteristic p> 2 is undecidable. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. 1. ..."
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Abstract. We prove that the firstorder theory of any function field K of characteristic p> 2 is undecidable. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. 1.
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.
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... 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. ..."
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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.