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

TL;DR

This paper improves the polynomial complexity of defining the set of integers over rationals and demonstrates the undecidability of certain quantified polynomial equations over the rationals.

Contribution

It reduces the number of variables needed to define $\

Findings

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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 $P(t,x_1,\ldots,x_{n})\in\mathbb Z[t,x_1,\ldots,x_{n}]$ 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 $\mathbb Q$.