On automorphisms of Danielewski surfaces

On Double Danielewski Surfaces and the Cancellation Problem

We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.

关于双重Danielewski簇

On double Danielewski varieties

本文研究双重Danielewski簇，描述其Makar-Limanov不变量与局部幂零导子，并预告后续将验证其为消去问题的反例。

Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces

We describe the set of all locally nilpotent derivations of the quotient ring $\mathbb{K}[X,Y,Z]/(f(X)Y - φ(X,Z))$ constructed from the defining equation $f(X)Y = φ(X,Z)$ of a generalized Danielewski surface in $\mathbb K^3$ for a specific choice of polynomials $f$ and $φ$, with $\mathbb K$ an algebraically closed field of characteristic zero. As a consequence of this description we calculate the $ML$-invariant and the Derksen invariant of this ring. We also determine a set of generators for the group of $\mathbb K$-automorphisms of $\mathbb K[X,Y,Z]/(f(X)Y - φ(Z))$ also for a specific choice …

关于双重丹杰洛斯基表面的注记

A note on double Danielewski surfaces

本文修正了[arXiv:2403.02876]中定理3.11的证明，并提供了讨论各种情况的例子。

关于任意特征域上的广义Danielewski曲面

On Generalised Danielewski Surfaces over fields of arbitrary characteristic

本文研究任意特征域上广义Danielewski曲面的指数映射，刻画其Makar-Limanov不变量与Derksen不变量，完整分类曲面间同构，并构造消去问题的反例子族。

有限域上 Danielewski 曲面和双 Danielewski 曲面的有理点计数

Rational Points on Danielewski Surfaces over Finite fields and Polygonal Numbers

利用 gcd 方法、结式、特征和、高斯和以及 König-Rados 定理，研究有限域上 Danielewski 曲面和双 Danielewski 曲面的有理点个数，给出多种结构情形下的显式公式、一般上界，并揭示与多边形数的有趣联系。