$\mathbb Q\setminus \mathbb Z$ is diophantine over $\mathbb Q$ with $32$ unknowns
Volume 70 / 2022
Abstract
In 2016 J. Koenigsmann refined a celebrated theorem of J. Robinson by proving that $\mathbb Q\setminus \mathbb Z$ is diophantine over $\mathbb Q $, i.e., there is a polynomial $P(t,x_1,\ldots ,x_{n})\in \mathbb Z [t,x_1,\ldots ,x_{n}]$ such that for any rational number $t$ we have $$t\not \in \mathbb Z \iff \exists x_1\cdots \exists x_{n} \ [P(t,x_1,\ldots ,x_{n})=0],$$ where variables range over $\mathbb Q$, equivalently $$t\in \mathbb Z \iff \forall x_1\cdots \forall x_{n}\ [P(t,x_1,\ldots ,x_{n})\not =0].$$ In this paper we prove that we may take $n=32$. Combining this with a result of Z.-W. Sun, we show that there is no algorithm to decide for any $f(x_1,\ldots ,x_{41})\in \mathbb Z [x_1,\ldots ,x_{41}]$ whether $$\forall x_1\cdots \forall x_9\exists y_1\cdots \exists y_{32}\ [f(x_1,\ldots ,x_9,y_1,\ldots ,y_{32})=0],$$ where variables range over $\mathbb Q$.
