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.
Videla, Hilbert’s tenth problem for rational function fields in char acteristic 2, Proc
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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.