Diophantine sets over polynominal rings

Recursively enumerable sets of polynomials over a finite field

“…In what follows, we will write down Diophantine definitions with existential quantifiers ("there exists," ∃), as in formula (2). In this notation, intersections correspond with logical conjunctions ("and," ∧), and unions with logical disjunctions ("or," ∨).…”

“…Before the proof of DPRM was completed, Davis and Putnam proved that every recursively enumerable subset of Z is Diophantine over Z[W ] (see [2]). Our Main Theorem is the analogue of that, but over F q [Z] instead of Z.…”

On the Metamathematics of Rings and Integral Domains

“…It is easy to show that this property is in turn sufficient to ensure F0 \-F and hence JDx i-F. If we specialize to the very particular case where F is (x)(Fy)/(x, y)=0 we find (using a result from [1]) that there is no method to decide whether a formula F of the given form is derivable from JDx or not. 2.…”

“…A proof of GEH is given in [3, p. 448], and the notion of pure variable proof is introduced in [3, p. 451]. A prenex formula is said to have standard form or to be a standard prenex formula if (1) no variable occurs free and bound in it, (2) every bound variable occurs exactly once in the prefix,…”

On the metamathematics of rings and integral domains

“…A proof of GEH is given in [3, p. 448], and the notion of pure variable proof is introduced in [3, p. 451]. A prenex formula is said to have standard form or to be a standard prenex formula if (1) no variable occurs free and bound in it, (2) every bound variable occurs exactly once in the prefix, (3) every bound variable occurs explicitly in the quantifier-free part. We denote such a formula, e.g.…”