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 R such that there is no algorithm to tell whether a polynomial equation with coefficients in (x 1 ,...,x n ) 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 (x 1 ,...,x n ) 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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