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$\mathbb Q\setminus \mathbb Z$ is diophantine over $\mathbb Q$ with $32$ unknowns

Volume 70 / 2022

Geng-Rui Zhang, Zhi-Wei Sun Bulletin Polish Acad. Sci. Math. 70 (2022), 93-106 MSC: Primary 03D35; Secondary 11U05, 03D25, 11D99, 11S99. DOI: 10.4064/ba221231-19-3 Published online: 14 April 2023


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 $\mathbb Q$.


  • Geng-Rui ZhangSchool of Mathematical Sciences
    Peking University
    Beijing 100871
    People’s Republic of China
  • Zhi-Wei SunDepartment of Mathematics
    Nanjing University
    Nanjing 210093
    People’s Republic of China

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