Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
Annales de l'Institut Fourier, Volume 59 (2009) no. 5, p. 2103-2118

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 $ℚ\left({x}_{1},...,{x}_{n}\right)$ has solutions in $R$.

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 $ℚ\left({x}_{1},...,{x}_{n}\right)$ a une solution dans $R$.

DOI : https://doi.org/10.5802/aif.2484
Classification:  11U05,  03D35,  11G05
Keywords: Hilbert’s tenth problem, elliptic curves, Diophantine undecidability
@article{AIF_2009__59_5_2103_0,
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},
publisher = {Association des Annales de l'institut Fourier},
volume = {59},
number = {5},
year = {2009},
pages = {2103-2118},
doi = {10.5802/aif.2484},
zbl = {1226.11131},
mrnumber = {2573198},
zbl = {pre05641409},
language = {en},
url = {http://www.numdam.org/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, Volume 59 (2009) no. 5, pp. 2103-2118. doi : 10.5802/aif.2484. http://www.numdam.org/item/AIF_2009__59_5_2103_0/

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