Waring's Problem For Locally Nilpotent Groups: The Case of Discrete Heisenberg Groups

群中的多项式映射与多项式序列

Polynomial maps and polynomial sequences in groups

本文发展了从交换半群到任意群的多项式映射理论，证明了其在目标群局部幂零时的形式性质，并将其应用于解决 Heisenberg 群的 Waring 问题。

二次丢番图方程、Heisenberg群与形式语言

Quadratic Diophantine equations, the Heisenberg group and formal languages

该研究利用EDT0L语言刻画整数环中的二元二次方程解，并通过归约到整数环方程，给出Heisenberg群中一元方程解的形式语言描述。

Diophantine problems in solvable groups

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 st…

二阶幂零李群上的格点计数问题

Lattice point counting problems on step-two nilpotent Lie groups

本文发展了二阶连通单连通幂零李群上格点计数理论，采用抛物型膨胀和齐次范数族，推导了所有维数和α>0下的显式计数偏差估计，并改进了海森堡群上的已知结果。