© The London Mathematical Society
The Analogue of Büchi's Problem for Rational Functions
Department of Mathematics, University of Crete 71409 Heraklion, Greece pheidas{at}math.uoc.gr
Universidad de Concepción, Facultad de Ciencias Físicas y Matemáticas, Departamento de Matemática Casilla 160-C, Concepción, Chile xvidaux{at}udec.cl
Received 25 February 2005. Revision received 13 February 2006.
Büchi's problem asked whether there exists an integer M such that the surface defined by a system of equations of the form
 |
In this paper we prove the following:
(A) an analogue of Büchi's problem in rings of polynomials of characteristic either 0 or p 
17 and for fields of rational functions of characteristic 0; and
(B) an analogue of Büchi's problem in fields of rational functions of characteristic p 19, but only for sequences that satisfy a certain additional hypothesis.
As a consequence we prove the following result in logic.
Let F be a field of characteristic either 0 or at least 17 and let t be a variable. Let Lt be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property ‘x is a square’ and symbols for multiplication by each element of the image of 
[t] in F[t]. Let R be a subring of F(t), containing the natural image of [t] in F(t). Assume that one of the following is true:
(i) R 
F[t];
(ii) the characteristic of F is either 0 or p 19.
Then multiplication is positive-existentially definable over the ring R, in the language Lt. Hence the positive-existential theory of R in Lt is decidable if and only if the positive-existential ring-theory of R in the language of rings, augmented by a constant-symbol for t, is decidable.