$\mathbb{Q}\setminus \mathbb{Z}$ is diophantine over $\mathbb{Q}$ with $7$ unknowns

Abstract

In 2016 J. Koenigsmann proved that $\mathbb{Q}\setminus \mathbb{Z}$ is diophantine over $\mathbb{Q}$, i.e., there is a polynomial $P(t,x_1,\dots ,x_n)\in \mathbb{Z}[t,x_1,\dots ,x_n]$ such that for any rational number $t$ we have $$t\notin \mathbb{Z}⟺\exists x_1,\dots ,x_n\in \mathbb{Q}[P(t,x_1,\dots ,x_n)=0].$$ In this paper we show that we may take $n=7$ which improves the previous record $n=10$ obtained by Daans in 2024. (Actually we even extend this to any global field.) This, together with a previous result of Z.-W. Sun, implies that there is no algorithm to decide for any $F(x_1,\dots ,x_{16})\in \mathbb{Z}[x_1,\dots ,x_{16}]$ whether $$\forall x_1,\dots ,x_9\in \mathbb{Q}\exists y_1,\dots ,y_7\in \mathbb{Q}[F(x_1,\dots ,x_9,y_1,\dots ,y_7)=0].$$

19 pages