Mixed quantifier prefixes over Diophantine equations with integer variables

Zhi-Wei Sun

Published 2021-03-09Version 1

In this paper we study mixed quantifiers over Diophantine equations with 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 $$\forall x_1\forall x_2\exists x_3\exists x_4\exists x_5\exists x_6(P(x_1,\ldots,x_6)=0),$$ where $x_1,\ldots,x_6$ are integer variables. We also have some similar undecidable results with universal quantifies bounded, for example, $\exists^2\forall^2\exists^2$ over $\mathbb Z$ with $\forall$ bounded is undecidable. We conjecture that $\forall^2\exists^2$ over $\mathbb Z$ is undecidable.