$\mathbb Z$ is diophantine over $\mathbb Q$ with 26 unknowns
$\mathbb Z$ is diophantine over $\mathbb Q$ with 26 unknowns
摘要
Recently, OpenAI proved that Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers has a negative solution. Motivated by this work, G. Conant et al. showed that $\mathbb Z$ is diophantine over $\mathbb Q$, i.e., there is an integer-coefficient polynomial $F(x,y_1,\ldots,y_n)$ such that for any $x\in\mathbb Q$ we have $$ x\in\mathbb Z\quad\iff \exists\,y_1,\ldots,y_{n}\in\mathbb Q\ [F(x,y_1,\ldots,y_{n})=0].$$They constructed such a polynomial $F$ with $\deg F=4$ and $n=221$. In this paper, we prove the existence of such a polynomial $F$ with $n=26$.
Abstract
Recently, OpenAI proved that Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers has a negative solution. Motivated by this work, G. Conant et al. showed that $\mathbb Z$ is diophantine over $\mathbb Q$, i.e., there is an integer-coefficient polynomial $F(x,y_1,\ldots,y_n)$ such that for any $x\in\mathbb Q$ we have $$ x\in\mathbb Z\quad\iff \exists\,y_1,\ldots,y_{n}\in\mathbb Q\ [F(x,y_1,\ldots,y_{n})=0].$$They constructed such a polynomial $F$ with $\deg F=4$ and $n=221$. In this paper, we prove the existence of such a polynomial $F$ with $n=26$.关键词
Hilbert's tenth problem/existential definition of integers/rational numbers/Lucas sequences/elliptic formal groups/pole predicates.Key words
Hilbert's tenth problem/existential definition of integers/rational numbers/Lucas sequences/elliptic formal groups/pole predicates.引用本文复制引用
Zhi-Wei Sun.$\mathbb Z$ is diophantine over $\mathbb Q$ with 26 unknowns[EB/OL].(2026-11-13)[2026-11-14].https://chinaxiv.org/abs/202611.00103.学科分类
数学
