Results 1 
2 of
2
doi:10.1112/S0024610706023283 THE ANALOGUE OF BÜCHI’S PROBLEM FOR RATIONAL FUNCTIONS
"... Büchi’s problem asked whether there exists an integer M such that the surface defined by a system of equations of the form x 2 n + x2n−2 =2x2n−1 +2, n =2,...,M − 1, has no integer points other than those that satisfy ±xn = ±x0 + n (the ± signs are independent). If answered positively, it would imply ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Büchi’s problem asked whether there exists an integer M such that the surface defined by a system of equations of the form x 2 n + x2n−2 =2x2n−1 +2, n =2,...,M − 1, has no integer points other than those that satisfy ±xn = ±x0 + n (the ± signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system Q =(q1,...,qr) of integral quadratic forms and an arbitrary rtuple B =(b1,...,br) of integers, whether Q represents B (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171–194). Thus it would imply the following strengthening of the negative answer to Hilbert’s tenth problem: the positiveexistential theory of the rational integers in the language of addition and a predicate for the property ‘x is a square ’ would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Büchi’s problem remains open. 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 Z[t] in F [t]. Let R beasubringofF (t), containing the natural image of Z[t] inF (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 positiveexistentially definable over the ring R, in the language Lt. Hence the positiveexistential theory of R in Lt is decidable if and only if the positiveexistential ringtheory of R in the language of rings, augmented by a constantsymbol for t, is decidable. 1.
AN EXTENSION OF BÜCHI’S PROBLEM FOR POLYNOMIAL RINGS IN ZERO CHARACTERISTIC
"... Abstract. We prove a strong form of the “n Squares Problem ” over polynomial rings with characteristic zero constant field. In particular we prove: for all r ≥ 2 there exists an integer M = M(r) depending only on r such that, if z1,z2,..., zM are M distinct elements of F and we have polynomials f,g, ..."
Abstract
 Add to MetaCart
Abstract. We prove a strong form of the “n Squares Problem ” over polynomial rings with characteristic zero constant field. In particular we prove: for all r ≥ 2 there exists an integer M = M(r) depending only on r such that, if z1,z2,..., zM are M distinct elements of F and we have polynomials f,g,x1,x2,...,xM ∈ F [t], with some xi nonconstant, satisfiying the equations xr i =(zi+ f) r + g for each i, thengis the zero polynomial. 1.