Hilbert's Tenth Problem for function fields over valued fields in characteristic zero

TL;DR

This paper proves the undecidability of Hilbert's Tenth Problem for certain function fields over valued fields in characteristic zero, extending known results to a broad class of fields including complex Laurent series.

Contribution

It establishes conditions under which Hilbert's Tenth Problem is undecidable for function fields over valued fields in characteristic zero, generalizing previous undecidability results.

Findings

Hilbert's Tenth Problem is undecidable for function fields over certain valued fields

The result applies to fields like C((T)) and similar valued fields

Undecidability holds under specific Galois cohomological conditions

Abstract

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(\bar{F}/F) and Gal(\bar{k}/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)).