发表机构
Khmelnytskyi National University（赫梅利尼茨基国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对平面n点构型，通过齐次向量-余向量表示结合SL(3,ℂ)经典不变量理论，得到多项式联合一阶微分射影不变量的生成性质，补充了相关理论并提供代数基础。
AI 中文摘要
我们研究平面上n个点构型的多项式绝对及相对联合一阶微分射影不变量，主要方法基于转向齐次向量-余向量表示，将问题简化为SL(3,ℂ)群的经典不变量理论。我们描述了多项式绝对不变量的代数，证明其由循环不变量生成；对于权重为-1的多项式相对不变量，得到了分次描述，证明其作为绝对不变量代数上的模是有限生成的，并构造了显式有限生成系。当n=3时，该模为秩1的自由模；当n=4时，构造了由39个元素组成的极小分次生成系。所得结果用多项式对应补充了联合射影微分不变量的有理理论，为进一步构造射影不变积分特征量提供了代数基础。
英文摘要
We study polynomial absolute and relative joint first-order differential projective invariants for configurations of $n$ points in the plane. The main approach is based on passing to a homogeneous vector--covector representation, which reduces the problem to the classical invariant theory of the group $SL(3,\mathbb C)$. The algebra of polynomial absolute invariants is described and shown to be generated by cyclic invariants. For polynomial relative invariants of weight $-1$, a graded description is obtained, their finite generation as a module over the algebra of absolute invariants is proved, and an explicit finite generating system is constructed. For $n=3$, the corresponding module is shown to be free of rank $1$. For $n=4$, a minimal homogeneous generating system consisting of $39$ elements is constructed. The obtained results complement the rational theory of joint projective differential invariants by its polynomial counterpart and provide an algebraic foundation for the further construction of projectively invariant integral characteristics.
Comments26 pages
