For q-bounded algebraic extensions of F_p(t), rings of S-integral functions are first-order definable, and if the constant field is infinite, the first-order theory of the field is undecidable.
MR4647281 ↑1, 2, 28 [GFP20] Natalia Garcia-Fritz and Hector Pasten, Towards Hilbert’s tenth problem for rings of integers throu gh Iwasawa theory and Heegner points , Math
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First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability
For q-bounded algebraic extensions of F_p(t), rings of S-integral functions are first-order definable, and if the constant field is infinite, the first-order theory of the field is undecidable.