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

Let F be a eld of zero characteristic. We give the following answer to a generalization of a problem of Buchi over F (t): A sequence of 92 or more cubes in F (t), not all constant, with third dierence constant and equal to 6, is of the form (x +n) 3 , n = 0;:::; 91, for some x2 F (t) (cubes of successive elements). We use this, in conjunction to the negative answer to the analogue of Hilbert's Tenth Problem for F (t) in order to show that the solvability of systems of degree-one equations, where some of the variables are assumed to be cubes and (or) non-constant, is an unsolvable problem over F (t). MSC: 03C60, 12L05, 11U05, 11C08