[Paper Review] Inductive Approach to Cartan's Moving Frame Method with Applications to Classical Invariant Theory

Irina A. Kogan|arXiv (Cornell University)|Sep 4, 2019

Nonlinear Waves and Solitons27 references12 citations

TL;DR

This paper presents an inductive algorithmic approach to Cartan's moving frame method for solving the equivalence problem of submanifolds under Lie group actions, particularly applied to classical invariant theory. By leveraging group factorizations and recursive normalization, it enables efficient computation of differential invariants and symmetry groups for polynomials, yielding explicit classifications of ternary cubics and conditions for equivalence to $x^n + y^n + z^n$, with computational verification via Gröbner bases.

ABSTRACT

This thesis is devoted to algorithmic aspects of the implementation of Cartan's moving frame method to the problem of the equivalence of submanifolds under a Lie group action. We adopt a general definition of a moving frame as an equivariant map from the space of submanifolds to the group itself and introduce two algorithms, which simplify the construction of such maps. The first algorithm is applicable when the group factors as a product of two subgroups $G=BA$, allowing us to use moving frames and differential invariants for the groups $A$ and $B$ in order to construct a moving frame and differential invariants for $G$. This approach produces the relations among the invariants of $G$ and its subgroups. We use the groups of the projective, the affine and the Euclidean transformations on the plane to illustrate the algorithm. We also introduce a recursive algorithm, allowing, provided the group action satisfies certain conditions, to construct differential invariants order by order, at each step normalizing more and more of the group parameters, at the end obtaining a moving frame for the entire group. The development of this algorithm has been motivated by the applications of the moving frame method to the problems of the equivalence and symmetry of polynomials under linear changes of variables. In the complex or real case these problems can be reduced and, in theory, completely solved as the problem of the equivalence of submanifolds. Its solution however involves algorithms based on the Gröbner basis computations, which due to their complexity, are not always feasible. Nevertheless, some interesting new results were obtained, such as a classification of ternary cubics and their groups of symmetries, and the necessary and sufficient conditions for a homogeneous polynomial in three variables to be equivalent to $x^n+y^n+z^n.$

Motivation & Objective

• To develop algorithmic methods for constructing moving frames and differential invariants under Lie group actions, especially for problems in classical invariant theory.
• To address the computational complexity of Gröbner basis methods in high-order jet prolongations by introducing a recursive, inductive construction of moving frames.
• To classify ternary cubic forms and determine conditions under which a homogeneous polynomial is equivalent to $x^n + y^n + z^n$.
• To provide a systematic, algorithmic framework that combines the advantages of Cartan's method of equivalence and Fels-Olver's moving frame approach for practical computation.

Proposed method

• Introduces a recursive algorithm to construct moving frames order by order, normalizing group parameters step-by-step to build a full moving frame for the entire group.
• Proposes a product-based inductive method for groups $G = BA$, using moving frames and invariants of subgroups $A$ and $B$ to construct those for $G$, with explicit relations among invariants.
• Applies the method to Euclidean, affine, and projective groups on the plane, demonstrating its simplification of computations through subgroup decomposition.
• Employs Gröbner basis computations (with input from Schost and Lecerf) to derive invariants and symmetries, particularly for binary and ternary forms.
• Uses the signature manifold concept to encode equivalence classes and compute invariants via rational functions in jet variables.
• Implements the method in Maple, with code for computing symmetries and invariants of binary and ternary forms, including verification of symmetry counts (e.g., 36 or 18 symmetries).

Experimental results

Research questions

• RQ1 How can the moving frame method be systematically applied to compute differential invariants for polynomial equivalence under linear group actions?
• RQ2 What is the structure of the symmetry group of a ternary cubic form, and how can it be algorithmically determined?
• RQ3 Under what conditions is a homogeneous polynomial in three variables equivalent to $x^n + y^n + z^n$?
• RQ4 Can the moving frame construction be made recursive and inductive to avoid high-order jet prolongations?
• RQ5 How can the functional relations among invariants be computed efficiently using group factorization and recursive normalization?

Key findings

• The recursive algorithm successfully constructs moving frames and differential invariants order by order, avoiding the need for high-order jet prolongations.
• For the group $G = BA$, the method computes moving frames and invariants for $G$ using those of $A$ and $B$, with explicit relations among the invariants.
• The method classifies ternary cubic forms, identifying 36 symmetries for certain forms and 18 for others, with explicit invariants computed via Gröbner basis techniques.
• The signature manifold for $x^n + y^n + z^n$ is shown to be defined by $[1 + 6I_1]$, and the method confirms equivalence of elliptic curves of the form $q^2 = p^3 + ap$ for all $a$.
• The algorithm computes explicit rational invariants for binary and ternary forms, such as $Pinv(f,3)$ for $f(p,q) = p^3 + p - q^2$, with verified symmetry counts (e.g., 36 or 18 symmetries).
• The method confirms that $f(p,q) = -q^2 + 2p + p^3$ has 36 symmetries, and similar forms with constant shifts (e.g., $+4$, $+1$) also yield 36 or 18 symmetries depending on the case.