论文标题
有限产生的交换环中的毒液问题
The Diophantine problem in finitely generated commutative rings
论文作者
论文摘要
我们研究具有身份元素的无限生成的交换环中多项式方程的系统。对于每个这样的环$ r $，我们通过$ \ mathbb {q} $或$ \ mathbb {f} _p（t）$的有限场扩展的整数$ o $ o $的方程式进行解释。这意味着在$ o $中，$ o $中的二芬太汀问题（多项式方程式的可决定性）可降低，以$ r $中的同一问题。尤其是$ r $具有积极的特征，或者更一般而言，如果$ r $具有无限的排名，那么我们进一步通过$ r $中的环$ \ mathbb {f} _p [t] $进行解释。这意味着在这种情况下，$ r $中的二芬太丁问题是不可确定的。在剩下的情况下，$ r $具有有限的排名和零特征，我们看到$ o $是代数整数的环，然后长期以来的猜想是，$ \ mathbb {z} $始终可以由代数整数圈中的方程式可以解释。如果是真的，这意味着$ r $中的二芬太丁问题也不确定。因此，在这种情况下，每个无限产生的交换统一环中的二氧甘氨酸问题是不可确定的。 目前是我们研究不同类型的环和代数中的双磷酸问题的一系列论文中的第一篇。
We study systems of polynomial equations in infinite finitely generated commutative associative rings with an identity element. For each such ring $R$ we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that the Diophantine problem (decidability of systems of polynomial equations) in $O$ is reducible to the same problem in $R$. If, in particular, $R$ has positive characteristic or, more generally, if $R$ has infinite rank, then we further obtain an interpretation by systems of equations of the ring $\mathbb{F}_p[t]$ in $R$. This implies that the Diophantine problem in $R$ is undecidable in this case. In the remaining case where $R$ has finite rank and zero characteristic, we see that $O$ is a ring of algebraic integers, and then the long-standing conjecture that $\mathbb{Z}$ is always interpretable by systems of equations in a ring of algebraic integers carries over to $R$. If true, it implies that the Diophantine problem in $R$ is also undecidable. Thus, in this case the Diophantine problem in every infinite finitely generated commutative unitary ring is undecidable. The present is the first in a series of papers were we study the Diophantine problem in different types of rings and algebras.
