Invariants: Computation and Applications

TL;DR

This paper explores the computation of invariants through algebraic and differential methods, highlighting algorithms for generating rational invariants and the use of differential invariant signatures in solving geometric equivalence problems.

Contribution

It introduces an algebraic adaptation of the moving frame method for computing rational invariants and discusses the role of differential invariant signatures in geometric and algebraic equivalence.

Findings

Algorithm for generating rational invariants developed

Differential invariant signatures aid in solving equivalence problems

Challenges identified in designing algorithms based on invariant signatures

Abstract

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.