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

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 $P(t,x_1,\dots ,x_n)\in \mathbb{Z}[t,x_1,\dots ,x_n]$ such that for any rational number $t$ we have $$t\notin \mathbb{Z}⟺\exists x_1\cdots \exists x_n[P(t,x_1,\dots ,x_n)=0]$$ where variables range over $\mathbb{Q}$, equivalently $$t\in \mathbb{Z}⟺\forall x_1\cdots \forall x_n[P(t,x_1,\dots ,x_n)\ne 0].$$ 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 $f(x_1,\dots ,x_{41})\in \mathbb{Z}[x_1,\dots ,x_{41}]$ whether $$\forall x_1\cdots \forall x_9\exists y_1\cdots \exists y_{32}[f(x_1,\dots ,x_9,y_1,\dots ,y_{32})=0],$$ where variables range over $\mathbb{Q}$.

arXiv comment13 pages

References in corpus (1)

• Existential rank and essential dimension of diophantine sets