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

Research paper by **Geng-Rui Zhang, Zhi-Wei Sun**

Indexed on: **07 Apr '21**Published on: **06 Apr '21**Published in: **arXiv - Mathematics - Number Theory**

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#### 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 $P(t,x_1,\ldots,x_{n})\in\mathbb Q[t,x_1,\ldots,x_{n}]$ 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 further that we
may even take $n=32$ and require deg$\,P<6\times10^{11}$, which provides the
best record in this direction. Combining this with a result of Sun, we get that
there is no algorithm to decide for any $P(x_1,\ldots,x_{41})\in\mathbb
Z[x_1,\ldots,x_{41}]$ whether $$\forall x_1\cdots\forall x_9\exists
y_1\cdots\exists y_{32}[P(x_1,\ldots,x_9,y_1,\ldots,y_{32})=0].$$