The analogue of Büchi’s problem for cubes in rings of polynomials

Thanases Pheidas and Xavier Vidaux

Vol. 238 (2008), No. 2, 349–366

DOI: 10.2140/pjm.2008.238.349

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 ∈ 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

Büchi’s Problem, Hilbert’s Tenth Problem, existential undecidability, cubic Forms

Mathematical Subject Classification 2000

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

Milestones

Received: 10 January 2008

Accepted: 24 June 2008

Published: 1 December 2008