Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariants: Computation and Applications

Irina A. Kogan

TL;DR

The paper addresses the problem of computing and applying invariants under group actions by bridging algebraic and differential invariant theories. It develops two main contributions: (i) an algebraic moving-frame approach that uses cross-sections and elimination (Gröbner-basis) methods to compute generating sets of rational invariants $\\mathbb{K}(\\mathcal{Z})^{\\mathcal{G}}$, and (ii) the differential invariant signature framework, which prolongs group actions to jet spaces to yield a small set of classifying invariants (e.g., curvature $\\kappa$ and its derivative $\\kappa_s$) that define a signature for equivalence testing of curves and binary forms. The work provides both practical algorithms and conceptual insights into how algebraic and differential invariant theories interact, with detailed treatments of smooth vs algebraic settings and explicit examples. These tools enable efficient symmetry reduction, orbit classification, and robust equivalence testing in geometry and algebraic geometry, offering a path from classical invariant theory to modern computational frameworks.

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.

Invariants: Computation and Applications

TL;DR

The paper addresses the problem of computing and applying invariants under group actions by bridging algebraic and differential invariant theories. It develops two main contributions: (i) an algebraic moving-frame approach that uses cross-sections and elimination (Gröbner-basis) methods to compute generating sets of rational invariants , and (ii) the differential invariant signature framework, which prolongs group actions to jet spaces to yield a small set of classifying invariants (e.g., curvature and its derivative ) that define a signature for equivalence testing of curves and binary forms. The work provides both practical algorithms and conceptual insights into how algebraic and differential invariant theories interact, with detailed treatments of smooth vs algebraic settings and explicit examples. These tools enable efficient symmetry reduction, orbit classification, and robust equivalence testing in geometry and algebraic geometry, offering a path from classical invariant theory to modern computational frameworks.

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.

Paper Structure

This paper contains 6 sections, 1 theorem, 65 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Group actions, orbits, invariants
  3. Cross-section and invariantization
  4. Smooth setting
  5. Algebraic setting
  6. Differential Signatures

Key Result

theorem 3

For an action $\alpha\colon\mathcal{G}\times\mathcal{Z}\to\mathcal{Z}$ and a cross-section $\mathcal{K}$, the coefficients of the reduced Gröbner basis $Q$ of the ideal $I^e\subset\mathbb {K}(z)[Z]$, relative to any monomial order on the variables $Z$, is a generating set of the field $\mathbb {K}(\

Figures (5)

  • Figure 1: Orbits under the actions in Example \ref{['ex-act-orb']}.
  • Figure 2: A moving frame map $\rho$.
  • Figure 3: The Frenet frame.
  • Figure 4: $SE_2$-signatures of curves $\Gamma_1$ and $\Gamma_2$ in Example \ref{['euc-sig']}.
  • Figure 5: A signature as a projection onto a cross-section.

Theorems & Definitions (10)

  • definition 1: action
  • definition 2: orbit
  • definition 3: stabilizer
  • definition 4: invariants
  • definition 5: separating set
  • definition 6: Local
  • definition 7: Smooth cross-section
  • Remark 2: Moving frames
  • definition 8: Algebraic cross-section
  • theorem 3: Generators of $\mathbb {K}(\mathcal{Z})^\mathcal{G}$