Testing the equivalence of planar curves.

*(English)*Zbl 0982.13004
Eindhoven: Eindhoven University of Technology. vi, 89 p. (2001).

An important goal of invariant theory is to determine whether two mathematical objects are equivalent by comparing invariants that are associated to the objects. The objects of interest in this dissertation are complex planar projective curves, and equivalence means lying in the same orbit under SL\(_3(\mathbb{C})\). Therefore the thesis studies invariants of ternary forms (of given degree) under SL\(_3(\mathbb{C})\), which is a classical branch of invariant theory. The author takes a very computational point of view and presents methods for explicitly computing invariants. The symbolic method, Lie algebras, and covariants are the main tools. The value of this publication is enhanced by the fact that the author implemented the algorithms in Maple.

The first chapter gives a brief introduction into invariant theory. In the second and third chapter, Lie algebra methods and the symbolic method are reviewed in a very readable way. The presentation in chapter 3 is strongly influenced by B. Sturmfels’ book [“Algorithms in invariant theory” (Wien 1993; Zbl 0802.13002)]. The last three chapters deal with ternary cubics, quartics and quintics. Finally, an appendix informs the reader about implementations in Maple.

The first chapter gives a brief introduction into invariant theory. In the second and third chapter, Lie algebra methods and the symbolic method are reviewed in a very readable way. The presentation in chapter 3 is strongly influenced by B. Sturmfels’ book [“Algorithms in invariant theory” (Wien 1993; Zbl 0802.13002)]. The last three chapters deal with ternary cubics, quartics and quintics. Finally, an appendix informs the reader about implementations in Maple.

Reviewer: Gregor Kemper (Heidelberg)

##### MSC:

13A50 | Actions of groups on commutative rings; invariant theory |

13-04 | Software, source code, etc. for problems pertaining to commutative algebra |

15-04 | Software, source code, etc. for problems pertaining to linear algebra |

17B45 | Lie algebras of linear algebraic groups |

15A72 | Vector and tensor algebra, theory of invariants |

14H50 | Plane and space curves |