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The intrinsic definition yields the nonhomogeneous polynomials. One can then define the subspace of homogeneous polynomials of degree q to be composed of the degree q polynomials that satisfy the equation ℰ(f) = -qf, where ℰ is the Euler vector field on T*M defined by θ=-ℰ⌋dθ, and θ is the ℝ-valued canonical one-form on T*M.
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M. Gotay, "A multisymplectic framework for classical field theory and the calculus of variations, I: Covariant hamiltonian formalism," in Mechanics, Analysis and Geometry: 200 Years after Lagrange, edited by M. Francaviglia (North Holland, Amsterdam, 1991), pp. 203-235.
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