Abstract

Let F be a field of characteristic zero. We give the following answer to a generalization of a problem of Büchi over F[t]: A sequence of 92 or more cubes in F[t], not all constant, with constant third difference equal to 6, consists of cubes of successive elements x, x+1, …, for some x in F[t]. We use this, in conjunction to the negative answer to Hilbert’s tenth problem for F[t], to show that the solvability of systems of degree-one equations, where some of the variables are assumed to be cubes and (or) nonconstant, is an unsolvable problem over F[t].

Keywords

Buchi's problem, Hilbert's tenth problem, existential undecidability, cubic forms

Mathematical Subject Classification

Primary: 03C60, 12L05, 11U05, 11C08

Authors