Mixed Quantifier Prefixes over Diophantine Equations with Integer Variables

In this paper, the author first reviews the history of Hilbert’s Tenth Problem, and then study mixed quantifier prefixes over Diophantine equations with integer variables. For example, the author proves that ∀²∃⁴ over ℤ is undecidable, that is, there is no algorithm to determine for any P(x₁, ⋯, x₆) ∈ ℤ[x₁, ⋯, x₆] whether \(\forall{x}_1\forall{x}_2\exists{x}_3\exists{x}_4\exists{x}_5\exists{x}_6(P(x_1,\cdot\cdot\cdot,x_6)=0),\) $\mathrm{\forall }{x}_1\mathrm{\forall }{x}_2\mathrm{\exists }{x}_3\mathrm{\exists }{x}_4\mathrm{\exists }{x}_5\mathrm{\exists }{x}_6(P(x_1,\cdot \cdot \cdot ,x_6)=0),$ where x₁, ⋯, x₆ are integer variables. The author also has some similar undecidable results with universal quantifies bounded, for example, ∃²∀²∃² over ℤ with ∀ bounded is undecidable. The author conjectures that ∀²∃² over ℤ is undecidable.