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8 W. Plesken, D. RobertzProof. 2) We show that B generates X as a K-vector space. Let u ∈ X. Then there arer g ∈ R with u = ∑ g∈J r gg. If all r g ∈ K[M(g)], we are done. In general, r g is a K-linearcombination of monomials m 1 m 2 with m 1 ∈ Mon(K[M(g)]) and m 2 ∈ Mon(K[M(g)]).Call the degree of m 2 the defect d(m 1 m 2 ) of m 1 m 2 in r g and note that d(m 1 m 2 ) = 0for all monomials of r g means r g ∈ K[M(g)]. If the defect of some monomial m in r g ispositive, say equal to d, then m = m ′ x for some x ∈ M(g). By passivity, mg = m ′ xgcan be replaced by m ′ L J (xg) in the above representation of u, which has only terms ofdefect at most d −1. Repeating this process until no monomials of positive defect are left,finally produces an expression u = ∑ g∈J s gg with s g ∈ K[M(g)], i. e. u ∈ 〈B〉 K , proving〈B〉 K = X. The bijectivity of Lt from B to S(X,
Janet’s approach to presentations and resolutions for polynomials and linear pdes. 9Definition 20. Let X ≤ R q be a submodule of R q . The Janet graph Γ(X,
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