Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and designing methods and algorithms to compute them remains an active area of ongoing research with an abundance of applications. In this incredibly vast topic, we focus on two particular themes displaying a fruitful interplay between the differential and algebraic invariant theories. First, we show how an algebraic adaptation of the moving frame method from differential geometry leads to a practical algorithm for computing a generating set of rational invariants. Then we discuss the notion of differential invariant signature, its role in solving equivalence problems in geometry and algebra, and some successes and challenges in designing algorithms based on this notion.

翻译：不变量在变换下保持不变，因此代表了对象或现象的本质。在数学中，变换通常构成群作用。自19世纪以来，研究各类不变量的结构并设计计算它们的方法与算法，始终是一个活跃的研究领域，并具有广泛的应用。在这一极其宏大的主题中，我们聚焦于两个特定方向，它们展现了微分不变量理论与代数不变量理论之间富有成果的交互。首先，我们展示了如何将微分几何中的活动标架法进行代数化改编，从而导出一个计算有理不变量生成集的实用算法。随后，我们讨论了微分不变量签名的概念，其在解决几何与代数中的等价性问题中的作用，以及基于此概念设计算法所取得的一些成功与面临的挑战。