Algebraic and rational differential invariants

These notes start with an introduction to differential invariants. They continue with an algebraic treatment of the theory. The algebraic, differential algebraic and differential geometric tools that are necessary to the development of the theory are explained in detail. We expose the recent results on the topic of rational and algebraic differential invariants. Finally we give a new algebraic version of the finiteness theorem of Lie–Tresse for the case of finite dimensional algebraic groups.

Tipo de documento: Artículo - Article

Palabras clave: differential invariants, rational invariants, jet spaces, geometric invariant theory.

Source: http://www.bdigital.unal.edu.co

Teaser

Bol.
Mat.
19(2), 133–188 (2012) 133 Algebraic and rational differential invariants David Blázquez–Sanz1 Escuela de Matemáticas Universidad Nacional de Colombia Sede Medellı́n These notes start with an introduction to differential invariants.
They continue with an algebraic treatment of the theory.
The algebraic, differential algebraic and differential geometric tools that are necessary to the development of the theory are explained in detail.
We expose the recent results on the topic of rational and algebraic differential invariants.
Finally we give a new algebraic version of the finiteness theorem of Lie–Tresse for the case of finite dimensional algebraic groups. Keywords: differential invariants, rational invariants, jet spaces, geometric invariant theory. Estas notas comienzan con una introducción a los invariantes diferenciales.
Continuan con un tratamiento algebraico de los mismos.
Las herramientas de álgebra, álgebra diferencial y geometrı́a diferencial necesarias para el desarrollo de la teorı́a son explicadas en detalle.
Se exponen los resultados recientes sobre el tema de los invariantes diferenciales racionales y algebraicos, con nuevas demostraciones.
Finalmente se da una nueva versión algebraica del teorema de Lie–Tresse para el caso de grupos algebraicos de dimensión finita. Palabras claves: invariantes diferenciales, invariantes racionales, espacios de jets, teorı́a geométrica de invariantes. MSC: 14L24, 53A55, 58A20 Recibido: 12 de octubre de 2012 1 dblazquezs@unal.edu.co Aceptado: 12 de diciembre de 2012 134 1 Blázquez–Sanz, Algebraic and rational differential invariants Introduction Differential invariants appear in many problems related to differential equations or differential geometry, as the reduction of differential equations, or the equivalence problem.
They were introduced by S.
Lie [12]. The theorem of the finiteness invariant for actions of finite–dimensional groups was demonstrated by L...