JournalsjemsVol. 19, No. 7pp. 2103–2138

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

  • Kirsten Eisenträger

    The Pennsylvania State University, University Park, USA
  • Alexandra Shlapentokh

    East Carolina University, Greenville, USA
Hilbert's Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field cover
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Abstract

We prove that the existential theory of any function field KK of characteristic p>0p > 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.

Cite this article

Kirsten Eisenträger, Alexandra Shlapentokh, Hilbert's Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field. J. Eur. Math. Soc. 19 (2017), no. 7, pp. 2103–2138

DOI 10.4171/JEMS/714