Hilbert's Tenth Problem for fields of rational functions over finite fields

Diophantine Undecidability over Algebraic Function Fields over Finite Fields of Constants

“…In particular, J. Denef has shown that Hilbert's Tenth Problem is undecidable over rational function fields over formally real constant fields (see [8]), and Kim and Roush have extended this result to some other characteristic 0 rational function fields (see [13] and [15]). T. Pheidas has shown in [20] that Diophantine problem is undecidable over rational function fields over finite fields of constants of characteristic greater than 2. In [29] Videla has proved the analogous result for the case of the characteristic equal to 2.…”

Results on undecidability of isomorphism of forms over polynomial rings

“…Sarkisian [23] proved a decidability result for isomorphism of forms over Q in 1980. Related results are established in [12], [13], [14], [22].…”

The Work of Kim and Roush on Questions of Decidability in Algebra and Number Theory

“…Undecidability of Hilbert's Tenth Problem is known for function fields of curves over finite fields [14,[20][21][22]24], and also for several rational function fields of characteristic zero: In 1978 Denef proved the undecidability of Hilbert's Tenth Problem for rational function fields K(T ) over formally real fields K [12], and he was the first to use rank-one elliptic curves to prove undecidability. Kim and Roush [5] showed that the problem is undecidable for the purely transcendental function field C(t 1 , t 2 ) (and for F p (t 1 , t 2 ) when p = 2).…”