AI 中文总结
该研究改进了有理数集减去整数集在有理数域上为丢番图集的未知量个数，从10个降至7个并推广到任意整体域，结合前期结果证明了相关判定问题不存在算法。
AI 中文摘要
2016年，J. Koenigsmann证明了$\boldsymbol{\text{Q}\backslash\text{Z}}$（有理数集减去整数集）在$\boldsymbol{\text{Q}}$（有理数域）上是丢番图集，即存在多项式$P(t,x_1,\boldsymbol{\text{…}},x_n)\boldsymbol{\text{∈}}\boldsymbol{\text{Z}}[t,x_1,\boldsymbol{\text{…}},x_n]$，使得对任意有理数$t$，$t\boldsymbol{\text{∉}}\boldsymbol{\text{Z}}$等价于存在有理数$x_1,\boldsymbol{\text{…}},x_n$满足$P(t,x_1,\boldsymbol{\text{…}},x_n)=0$。本文证明可取$n=7，将Daans在2024年得到的此前记录$n=10$进行了改进，实际上该结果可推广到任意整体域。结合Z.-W. Sun的前期结果，意味着不存在算法可对任意$F(x_1,\boldsymbol{\text{…}},x_{16})\boldsymbol{\text{∈}}\boldsymbol{\text{Z}}[x_1,\boldsymbol{\text{…}},x_{16}]$判定是否满足对任意9个有理数$x_1,\boldsymbol{\text{…}},x_9$，存在7个有理数$y_1,\boldsymbol{\text{…}},y_7$使得$F(x_1,\boldsymbol{\text{…}},x_9,y_1,\boldsymbol{\text{…}},y_7)=0$。
英文摘要
In 2016 J. Koenigsmann proved 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 $$t\not\in\mathbb Z\iff \exists x_1,\ldots,x_{n}\in\mathbb Q\,[P(t,x_1,\ldots,x_{n})=0].$$ In this paper we show that we may take $n=7$ which improves the previous record $n=10$ obtained by Daans in 2024. (Actually we even extend this to any global field.) This, together with a previous result of Z.-W. Sun, implies that there is no algorithm to decide for any $F(x_1,\ldots,x_{16})\in\mathbb Z[x_1,\ldots,x_{16}]$ whether $$\forall x_1,\ldots,x_9\in\mathbb Q\exists y_1,\ldots,y_{7}\in\mathbb Q\,[F(x_1,\ldots,x_9,y_1,\ldots,y_{7})=0].$$
Comments19 pages
