Computation of canonical forms for ternary cubics

ABSTRACT

In this paper we conduct a careful study of the equivalence classes of ternary cubics under general complex linear changes of variables. Our new results are based on the method of moving frames and involve triangular decompositions of algebraic varieties. We provide a computationally efficient algorithm that matches an arbitrary ternary cubic with its canonical form and explicitly computes a corresponding linear change of coordinates. We also describe a classification of the symmetry groups of ternary cubics.

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Analysis of algorithms

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)
      G.1.5 Roots of Nonlinear Equations
          Subjects: Polynomials, methods for

General Terms:
Algorithms, Documentation, Experimentation

Keywords:
classical invariant theory, elliptic curves, method of moving frames, systems of algebraic equations, ternary cubics, triangular decompositions