Diophantine Sets Over Polynomial Rings

Martin Davis

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E. X e TT, Definition . A predicate y ( u, v ) is called a Julia Robinson predicate if ; (1) /(u, v) -^ V \/ [P(t^, ... , t, ^, u, v, x) = /^- f(u, v)] t-i J • . . ) t, ,U, V where P is a polynomial and f{u, v) is any Julia Robinson predicate. Suppose that j^(u, v) y ip T II V U IT ■i- ■)>♦•• J ^-^rt '-' > ^ > '^■^> * • » > ^^ [P(T^, ... , \, U, V, x) = /\ AQ(U, V, U^, ... , U^) = A T^ e P A, . . . , A T^^ e P U^ e P A, . . . , A U^ e P /\ U e P A, V e P], Lem^ia 3. 3. Let p(u, v) v = 2 A u ...> 3, Then P(u, v) is a Julia Robinson predicate and is Diophantine over J[t|].
Proof. By [i].

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