no. 5
Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
[Indécidabilité diophantienne des anneaux d’holomorphie de corps de fonctions de caractéristique nulle]
Moret-Bailly, Laurent1 ; Shlapentokh, Alexandra2
1 IRMAR Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex (France)
2 East Carolina University Department of Mathematics Greenville, NC 27858 (U.S.A.)
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 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 ∈R such that there is no algorithm to tell whether a polynomial equation with coefficients in ℚ(x 1 ,...,x n ) has solutions in R.

DOI : 10.5802/aif.2484
Classification : 11U05, 03D35, 11G05
Keywords: Hilbert’s tenth problem, elliptic curves, Diophantine undecidability
Mot clés : dixième problème de Hilbert, courbes elliptiques, indécidabilité diophantienne

Moret-Bailly, Laurent 1 ; Shlapentokh, Alexandra 2

1 IRMAR Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex (France)
2 East Carolina University Department of Mathematics Greenville, NC 27858 (U.S.A.) 
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Moret-Bailly, Laurent; Shlapentokh, Alexandra. Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0. Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 2103-2118. doi : 10.5802/aif.2484. https://aif.centre-mersenne.org/articles/10.5802/aif.2484/

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