Algebraic and Differential Invariants

This article highlights a coherent series of algorithmic tools to compute and work with algebraic and differential invariants. Introduction Group actions are ubiquitous in mathematics and arise in diverse fields of science and engineering, including physics, mechanics, and computer vision. Invariants of these group actions typically arise to reduce a problem or to decide if two objects, geometric or abstract, are obtained from one another by the action of a group element. [8, 9, 10, 11, 13, 15, 17, 39, 40, 42, 43, 45, 46, 52, 59] are a few recent references of applications. Both algebraic and differential invariant theories have become in recent years the subject of computational mathematics [13, 14, 17, 40, 60]. Algebraic invariant theory studies polynomial or rational invariants of algebraic group actions [18, 22, 23, 54]. A typical example is the discriminant of a quadratic binary form as an invariant of an action of the special linear group. The differential invariants appearing in differential geometry are smooth functions on a jet bundle that are invariant under a prolonged action of a Lie group [4, 16, 34, 48, 53]. A typical example is the curvature of a plane curve, invariant under the action of the group of the isometries on the plane. Curvature is not a rational function, but an algebraic function. Concomitantly the classical Lie groups are linear algebraic groups. This article reviews results of [14, 28, 29, 30, 31, 32] in order to show their coherence in addressing algorithmically an algebraic description of the differential invariants of a group action. In the first section we show how to compute the rational invariants of a group action and give concrete expressions to a set of algebraic invariants that are of fundamental importance in the differential context. The second section addresses the question of finite representation of differential invariants with invariant derivations, a set of generating differential invariants and the differential relationships among them. In the last section we describe the algebraic structure that better serves the representation of differential invariants.