Diophantine undecidability over algebraic function fields over finite fields of constants
Let \(F(t)\) be a function field over a finite field \(F\). Let \(K\) be a finite extension of \(F(t)\). One can ask the following analogue of Hilbert's Tenth problem for \(K\): Is the existential theory of \(F(t)\) in the language \(\{0,1,t,+,\cdot\}\) decidable, that is, is there an algorithm which checks the existence of solutions \((x_1,\dots,x_n)\) of an arbitrary polynomial equation \(f=0\) in \(K\) with \(f\in F[t,X_1,\dots, X_n]\)? The author shows that the answer to this question is ``No''. This is the first result for algebraic function fields of arbitrary degree over \(F(t)\) and may be an evidence for the similar question for number fields, an outstanding open problem. The similar question for arbitrary algebraic function fields over any \(F\) remains open.
scientific article; zbMATH DE number 894610