Hilbert’s Tenth Problem for algebraic function fields over infinite fields of constants of positive characteristic

Alexandra Shlapentokh

Vol. 193 (2000), No. 2, 463–500

DOI: 10.2140/pjm.2000.193.463

Abstract

Let K be an algebraic function field of characteristic p > 2. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree p. Assume also that K contains a subfield K₁, possibly equal to C, and elements u,x such that u is transcendental over K₁, x is algebraic over C(u) and K = K₁(u,x). Then the Diophantine problem of K is undecidable.

Let G be an algebraic function field in one variable whose constant field is algebraic over a finite field and is not algebraically closed. Then for any prime p of G, the set of elements of G integral at p is Diophantine over G.

Milestones

Received: 23 September 1998

Revised: 15 January 1999

Published: 1 April 2000