A Diophantine problem for Laurent polynomial rings

Abstract

Let R R be an integral domain of characteristic zero. We prove that the diophantine problem for the Laurent polynomial ring R [ T , T − 1 ] R[T,{T^{ - 1}}] with coefficients in Z [ T ] {\mathbf {Z}}[T] is unsolvable. Under suitable conditions on R R we then show that either Z {\mathbf {Z}} or Z [ i ] {\mathbf {Z}}[i] is diophantine over R [ T , T − 1 ] R[T,{T^{ - 1}}] .