The analogue of Büchi's problem for rational functions

• Autores: Thanases Pheidas, Xavier Vidaux
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 74, Nº 3, 2006, págs. 545-565
• Idioma: inglés
• DOI: 10.1112/s0024610706023283
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• Resumen

  • Büchi's problem asked whether there exists an integer $M$ such that the surface defined by a system of equations of the form $$x_{n}^2+x_{n-2}^2=2x_{n-1}^2+2,\quad n=2,\dotsc, M-1,$$ has no integer points other than those that satisfy $\pm x_n=\pm x_0+n$ (the $\pm$ signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system $Q=(q_1,\dotsc,q_r)$ of integral quadratic forms and an arbitrary $r$-tuple $B=(b_1,\dotsc,b_r)$ 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 positive-existential 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: an analogue of Büchi's problem in rings of polynomials of characteristic either 0 or $p\geq17$ and for fields of rational functions of characteristic 0; and an analogue of Büchi's problem in fields of rational functions of characteristic $p\geq19$, 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 $L_{t}$ 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 $\mathbb{Z}[t]$ in $F[t]$. Let $R$ be a subring of $F(t)$, containing the natural image of $\mathbb{Z}[t]$ in $F(t)$. Assume that one of the following is true: $R\subset F[t]$; the characteristic of $F$ is either 0 or $p\geq19$. Then multiplication is positive-existentially definable over the ring $R$, in the language $L_t$. Hence the positive-existential theory of $R$ in $L_{t}$ 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