Distributions admitting a local basis of homogeneous polynomials

Abstract

The paper is a survey of several results by the authors, the main one of them being the following characterization of homogeneous algebraic distributions: Let us consider a vertical distribution D on the vector bundle p:E→M locally spanned by vertical vector fields X1,⋯,Xr. Let χ be the Liouville vector field of the vector bundle. Then there exists an r×r invertible matrix with smooth entries (cij) such that the vector fields Yj=∑ri=1cijXi, 1≤j≤r, are homogeneous algebraic of degree d if and only if there exists an r×r matrix A=(aij) of smooth functions given by [χ,Xj]=∑ri=1aijXi such that A restricted to the zero section of E is(d−1) times the identity matrix. Examples and applications are given.