## Abstract

We prove that the existential theory of any function field K of characteristic p > 0 is undecidable in the language of rings augmented by constant symbols for the elements of a suitable recursive subfield, provided that the constant field does not contain the algebraic closure of a finite field. This theorem is the natural generalization of a theorem of Kim and Roush from 1992. We also extend our previous undecidability proof for function fields of higher transcendence degree to characteristic 2 and show that the first-order theory of any function field of positive characteristic is undecidable in the language of rings without parameters.

Original language | English (US) |
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Pages (from-to) | 2103-2138 |

Number of pages | 36 |

Journal | Journal of the European Mathematical Society |

Volume | 19 |

Issue number | 7 |

DOIs | |

State | Published - 2017 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics