Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Laurent Moret-Bailly[1]; Alexandra Shlapentokh[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.)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 5, page 2103-2118
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Moret-Bailly, Laurent, and Shlapentokh, Alexandra. "Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0." Annales de l’institut Fourier 59.5 (2009): 2103-2118. <http://eudml.org/doc/10448>.

@article{Moret2009,
abstract = {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 $x_1,\ldots ,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\mathbb\{Q\}(x_1,\ldots ,x_n)$ has solutions in $R$.},
affiliation = {IRMAR Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex (France); East Carolina University Department of Mathematics Greenville, NC 27858 (U.S.A.)},
author = {Moret-Bailly, Laurent, Shlapentokh, Alexandra},
journal = {Annales de l’institut Fourier},
keywords = {Hilbert’s tenth problem; elliptic curves; Diophantine undecidability; Hilbert's tenth problem; undecidability},
language = {eng},
number = {5},
pages = {2103-2118},
publisher = {Association des Annales de l’institut Fourier},
title = {Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0},
url = {http://eudml.org/doc/10448},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Moret-Bailly, Laurent
AU - Shlapentokh, Alexandra
TI - Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 5
SP - 2103
EP - 2118
AB - 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 $x_1,\ldots ,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\mathbb{Q}(x_1,\ldots ,x_n)$ has solutions in $R$.
LA - eng
KW - Hilbert’s tenth problem; elliptic curves; Diophantine undecidability; Hilbert's tenth problem; undecidability
UR - http://eudml.org/doc/10448
ER -

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