TY - JOUR

T1 - Hilbert's Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field

AU - Eisenträger, Kirsten

AU - Shlapentokh, Alexandra

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

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U2 - 10.4171/JEMS/714

DO - 10.4171/JEMS/714

M3 - Article

AN - SCOPUS:85020042548

VL - 19

SP - 2103

EP - 2138

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 7

ER -