The Hilbert Property for integral points of affine smooth cubic surfaces

具有无限离散自同构群的三次曲面

Cubic surfaces with infinite, discrete automorphism group

本文证明了方程为$xyz=g(x,y)$的仿射三次曲面的自同构群包含${\mathbb Z}$作为有限指数子群，此类方程最早由Mordell研究。

Affine equivalence of cubic homogeneous rotation symmetric Boolean functions

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. This paper studies the much more complicated cubic case for such functions. A new concept of \emph{patterns} is introduced, by means of which the structure of the smallest group G_n, whose action on t…

An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields

We give a brief report on our computations of linear determinantal representations of smooth plane cubics over finite fields. After recalling a classical interpretation of linear determinantal representations as rational points on the affine part of Jacobian varieties, we give an algorithm to obtain all linear determinantal representations up to equivalence. We also report our recent study on computations of linear determinantal representations of twisted Fermat cubics defined over the field of rational numbers. This paper is a summary of the author's talk at the JSIAM JANT workshop on algorit…

余类细化的迹统计、节点特征和立方范数托里 affine 分支

Coset-refined trace statistics, nodal characters, and affine branches in cubic norm tori

本文研究了立方范数托里的余类细化迹统计、节点特征以及立方范数托里的affine分支，证明了余类细化迹定理，并探讨了迹和范数曲线上的几何输入及其对局部分支理论的影响。

Integral points on Markoff type cubic surfaces

For integers $k$, we consider the affine cubic surface $V_{k}$ given by $M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. We show that for almost all $k$ the Hasse Principle holds, namely that $V_{k}(\mathbb{Z})$ is non-empty if $V_{k}(\mathbb{Z}_p)$ is non-empty for all primes $p$, and that there are infinitely many $k$'s for which it fails. The Markoff morphisms act on $V_{k}(\mathbb{Z})$ with finitely many orbits and a numerical study points to some basic conjectures about these "class numbers" and Hasse failures. Some of the analysis may be extended to less special affine cubic s…