Mixed quantifier prefixes over Diophantine equations with integer variables

In this paper we study mixed quantifier prefixes over Diophantine equations with integer variables. For example, we prove that ∀ 2 ∃ 4 over Z is undecidable, that is, there is no algorithm to determine for any P (x1, . . . , x6) ∈ Z[x1, . . . , x6] whether ∀x1∀x2∃x3∃x4∃x5∃x6(P (x1, . . . , x6) = 0), where x1, . . . , x6 are integer variables. We also have some similar undecidable results with universal quantifies bounded, for example, ∃∀∃ over Z with ∀ bounded is undecidable. We conjecture that ∀∃ over Z is undecidable.