Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

2008/05/22 by Laurent Moret-Bailly, Moret-Bailly, Laurent, Alexandra Shlapentokh +1
Computer Science · Mathematics · #03D35 #11G05 #11U05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.LO #math.NT #msc:03D35 #msc:11G05 #msc:11U05

paper · pdf · doi:10.48550/arxiv.0805.3458

This version contains minor revisions and will appear in Annales de l Institut Fourier

openalex publication_date 2008/05/22 · arxiv created 2009/01/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 x₁,...,x_n ∈ R such that there is no algorithm to tell whether a polynomial equation with coefficients in \Q(x₁,...,x_n) has solutions in R.

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