Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Moret-Bailly, Laurent; Shlapentokh, Alexandra

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, Laurent ; Shlapentokh, Alexandra

Annales de l'Institut Fourier, Tome 59 (2009), p. 2103-2118 / Harvested from Numdam

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Abstract

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 $x_{1}, ..., x_{n} \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $ℚ (x_{1}, ..., x_{n})$ has solutions in $R$ .

Publié le : 2009-01-01

MR 2573198 | Zbl pre05641409 | Zbl 1226.11131

DOI : https://doi.org/10.5802/aif.2484
Classification: 11U05, 03D35, 11G05
Mots clés: dixième problème de Hilbert, courbes elliptiques, indécidabilité diophantienne

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     author = {Moret-Bailly, Laurent and Shlapentokh, Alexandra},
     title = {Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic~0},
     journal = {Annales de l'Institut Fourier},
     volume = {59},
     year = {2009},
     pages = {2103-2118},
     doi = {10.5802/aif.2484},
     zbl = {pre05641409},
     mrnumber = {2573198},
     zbl = {1226.11131},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2009__59_5_2103_0}
}

Moret-Bailly, Laurent; Shlapentokh, Alexandra. Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0. Annales de l'Institut Fourier, Tome 59 (2009) pp. 2103-2118. doi : 10.5802/aif.2484. http://gdmltest.u-ga.fr/item/AIF_2009__59_5_2103_0/

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