babylonpolice.com

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 $\mathbb Q$.