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In this paper we study mixed quantifier prefixes over Diophantine equationswith integer variables. For example, we prove that $\forall^2\exists^4$ over$\mathbb Z$ is undecidable, that is, there is no algorithm to determine for any$P(x_1,\ldots,x_6)\in\mathbb Z[x_1,\ldots,x_6]$ whether $$\forall x_1\forallx_2\exists x_3\exists x_4\exists x_5\exists x_6(P(x_1,\ldots,x_6)=0),$$ where$x_1,\ldots,x_6$ are integer variables. We also have some similar undecidableresults with universal quantifies bounded, for example,$\exists^2\forall^2\exists^2$ over $\mathbb Z$ with $\forall$ bounded isundecidable. We conjecture that $\forall^2\exists^2$ over $\mathbb Z$ isundecidable.

Mixed quantifier prefixes over Diophantine equations with integer variables