my forty years on his shoulders - Department of Mathematics

13(Davis 1973), (Matiyasevich 1993). The MRDP theorem was shown to be provablein the weak fragment of arithmetic, EFA = IΣ 0 (exp), in (Dimitracopoulus,Gaifman 1982).We can use (Dimitracopoulus, Gaifman 1982) to obtain the following.THEOREM 3.3. Let T be a consistent extension of EFA in many sorted predicatecalculus whose relational type and axioms are recursively enumerable. There is apurely existential equation (∃x 1 ,...,x n )(s = t) in L(Q) that is neither provable norrefutable in T.It is not clear whether EFA can be replaced by a weaker system inTheorem 3.3 such as Q.An important issue is whether there is a “reasonable” existential equation(∃x 1 ,...,x n )(s = t) that can be used in Theorem 3.3 for, say, T = PA or T = ZFC. Notethat (∃x 1 ,...,x n )(s = t) corresponds to the Diophantine problem “does thepolynomial s-t with integer coefficients have a solution in the nonnegativeintegers?”Let us see what can be done on the purely recursion theoretic side withregards to the complexity of polynomials with integer coefficients. The mostobvious criteria area. The number of unknowns.b. The degree of the polynomial.c. The number of operations (additions and multiplications).In 1992, Matiyasevich showed that nine unknowns over the nonnegativeintegers suffices for recursive unsolvability. One form of the result (not the