Hilbert’s Tenth Problem for function fields over valued fields in characteristic zero
نویسنده
چکیده
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 2-divisible; there exists a maximal subfield F in the valuation ring such that Gal(F̄ /F ) and Gal(k̄/k) have the same 2-cohomological 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 )).
منابع مشابه
Hilbert’s Tenth Problem for Function Fields of Characteristic Zero
In this article we outline the methods that are used to prove undecidability of Hilbert’s Tenth Problem for function fields of characteristic zero. Following Denef we show how rank one elliptic curves can be used to prove undecidability for rational function fields over formally real fields. We also sketch the undecidability proofs for function fields of varieties over the complex numbers of di...
متن کاملHILBERT’S TENTH PROBLEM FOR FUNCTION FIELDS OF VARIETIES OVER NUMBER FIELDS AND p-ADIC FIELDS
Let k be a subfield of a p-adic 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.
متن کاملHilbert’s Tenth Problem for Function Fields of Varieties over Algebraically Closed Fields of Positive Characteristic
Let K be the function field of a variety of dimension ≥ 2 over an algebraically closed field of odd characteristic. Then Hilbert’s Tenth Problem for K is undecidable. This generalizes the result by Kim and Roush from 1992 that Hilbert’s Tenth Problem for the purely transcendental function field Fp(t1, t2) is undecidable.
متن کاملHilbert’s Tenth Problem for Algebraic Function Fields of Characteristic 2
Let K be an algebraic function field of characteristic 2 with constant field CK . Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u, x of K with u transcendental over CK and x algebraic over C(u) and such that K = CK(u, x). Then Hilbert’s Tenth Problem over K is undecidable. Together with Shlapentokh’s result for ...
متن کاملDiophantine undecidability of function fields of characteristic greater than 2, finitely generated over fields algebraic over a finite field
Let F be a function field of characteristic p > 2, finitely generated over a field C algebraic over a finite field Cp and such that it has an extension of degree p. Then Hilbert’s Tenth Problem is not decidable over F .
متن کامل