Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Moret-Bailly, Laurent; Shlapentokh, Alexandra

Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 2103-2118.

Soit $K$ un corps de fonctions d’une variable sur un corps de caractéristique nulle. Soit $R$ un anneau d’holomorphie de $K$, distinct de $K$. Si $K$ est récursif, nous démontrons que le dixième problème de Hilbert sur $R$ est indécidable. En général, il existe $x_1,...,x_n$ dans $R$ tels qu’il n’y ait pas d’algorithme décidant si une équation polynomiale à coefficients dans $\mathbb{Q}(x_1,...,x_n)$ a une solution dans $R$.

Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: if $K$ is recursive, then Hilbert’s Tenth Problem is undecidable in $R$. In general, there exist $x_1,...,x_n\in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\mathbb{Q}(x_1,...,x_n)$ has solutions in $R$.

Classification : 11U05, 03D35, 11G05
Keywords: Hilbert’s tenth problem, elliptic curves, Diophantine undecidability
Mot clés : dixième problème de Hilbert, courbes elliptiques, indécidabilité diophantienne

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