Geodesic
Hilbert's Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field
Journal of the European Mathematical Society, Tome 19 (2017) no. 7, pp. 2103-2138 Cet article a éte moissonné depuis la source EMS Press

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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.
DOI : 10.4171/jems/714
Classification : 11-XX, 03-XX
Keywords: Undecidability, Hilbert's Tenth Problem 
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     author = {Kirsten Eisentr\"ager and Alexandra Shlapentokh},
     title = {Hilbert's {Tenth} {Problem} over function fields of positive characteristic not containing the algebraic closure of a finite field},
     journal = {Journal of the European Mathematical Society},
     pages = {2103--2138},
     year = {2017},
     volume = {19},
     number = {7},
     doi = {10.4171/jems/714},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/714/}
}
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Kirsten Eisenträger; Alexandra Shlapentokh. Hilbert's Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field. Journal of the European Mathematical Society, Tome 19 (2017) no. 7, pp. 2103-2138. doi: 10.4171/jems/714

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