A new technique for obtaining diophantine representations via elimination of bounded universal quantifiers

Hilbert's Tenth Problem in Coq (Extended Version)

“…4 For instance, these operations are assumed as Diophantine producing operators in e.g. [JM84,Mat70,Mat97,Mat00]. Sometimes Diophantine relations are restricted to a single polynomial equation.…”

“…This map can be compared with map 3 of Proposition 2.2 and allows to recognise bounded universal quantification as a legitimate Diophantine shape. We have implemented the direct proof of Matiyasevich [Mat97] which does not involve a detour through a model of computation. Notice that the bound f ν in ∀u, u < f ν → .…”

“…Admissibility of Bounded Universal Quantification (Theorem 2.8). As hinted earlier, we provide an implementation of the algorithm for the elimination of bounded universal quantification described in [Mat97]. It does not involve the use of a model of computation, hence does not create a chicken-and-egg problem when used for the proof of the DPRM theorem.…”

“…It does not involve the use of a model of computation, hence does not create a chicken-and-egg problem when used for the proof of the DPRM theorem. The technique of [Mat97] uses the exponential function and thus Theorem 2.7 (a lot), and a combination of arithmetic and bitwise operations over N through base 2 and base 2 q representations of natural numbers.…”

“…The section explains why we slightly modified the original proof of the elimination of bounded universal quantification [Mat97] to avoid overflows when dealing with parallel multiplications, in search for a Diophantine encoding of the simultaneous equations a 1 = b 1 c 1 , . .…”