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arXiv 1805.04085math.GRmath.LOmath.NT
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可解群中的丢番图问题

Diophantine problems in solvable groups

Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov

备注：
New results were added (in particular, several results regarding groups of matrices). Some results were strengthened. Helpful referee suggestions were implemented. 28 pages
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摘要：

本文研究了满足某些自然"非交换性"条件的不同类型的非线性生成的可解群（幂零群、多环群、亚交换群、自由可解群等）上的丢番图问题（有限方程组的可判定性）.对于每个组$G$在这些类之一，我们证明了存在一个环的代数整数$O$是解释在$G$有限系统的方程（e-解释），因此，丢番图问题在$O$是多项式时间约简为丢番图问题在$G$。数论中的一个主要的开放式命题指出，任何这样的$O$中的丢番图问题都是不可判定的。如果为真，这将意味着丢番图问题在任何这样的$G$也是不可判定的。此外，我们证明了，对于许多特殊的群$G$如上所述，环$O$是同构的整数环$\mathbb{Z}$，所以丢番图问题在$G$确实是不可判定的。这特别适用于自由幂零或自由可解的非阿贝尔群，以及非阿贝尔广义海森堡群和单三角群$UT(n,\mathbb{Z}), n \geq 3$。然后，我们将这些结果应用到非可解群，其中包含非虚交换的极大可解生成幂零子群。例如，我们证明了丢番图问题在群$GL(3,\mathbb{Z}), SL(3,\mathbb{Z}), T(3,\mathbb{Z})$中是不可判定的。

英文摘要：

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural "non-commutativity" conditions. For each group $G$ in one of these classes, we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by finite systems of equations (e-interpretable), and hence that the Diophantine problem in $O$ is polynomial time reducible to the Diophantine problem in $G$. One of the major open conjectures in number theory states that the Diophantine problem in any such $O$ is undecidable. If true this would imply that the Diophantine problem in any such $G$ is also undecidable. Furthermore, we show that for many particular groups $G$ as above, the ring $O$ is isomorphic to the ring of integers $\mathbb{Z}$, so the Diophantine problem in $G$ is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups $UT(n,\mathbb{Z}), n \geq 3$. Then we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups $GL(3,\mathbb{Z}), SL(3,\mathbb{Z}), T(3,\mathbb{Z})$.

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