By Serge Lang
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Z k ), ˜ 0 , . . , X k )(−T0 Z 1 + Z˜ 1 , Z 2 , . . , Z k )]. T0 f (X 0 , . . , X k )(Z 1 , . . , Z k ) + f (X The comparison of the (2,1) blocks in the two block column expressions for ˜ 0 , X 1 , . . , X k )(col [Z 1 , Z˜ 1 ], Z 2 , . . , Z k ) f (X 0 ⊕ X gives ˜ 0 , X 1 , . . , X k )(Z˜ 1 , Z 2 , . . , Z k ) f (X ˜ 0 , X 1 , . . , X k )(−T0 Z 1 + Z˜ 1 , Z 2 , . . , Z k ). = T0 f (X 0 , . . , X k )(Z 1 , . . , Z k ) + f (X Using linearity in Z 1 of f (X 0 , . . , X k )(Z 1 , .
X k−1 , X k,0 )(Z 1 , . . , Z k−1 , Z k,0 ), ΔR f (X 0 , . . , X k−1 , X k,0 , X k,1 )(Z 1 , . . , Z k−1 , Z k,0 , Z k+1,1 ), . . , ΔR f (X 0 , . . , X k−1 , X k,0 , . . , X k, )(Z 1 , . . , Z k−1 , Z k,0 , Z k+1,1 , . . , Z k+1, ) . 12, we will need the following lemma. 14. , Ω = Ω; see Appendix A. Then for all n0 , . . , n , X 0 ∈ Ωn0 , . . , X ∈ Ωn , Z 1 ∈ Mn0 ×n1 , . . , Z ∈ Mn −1 ×n one has ⎡ j ⎤ Z j+1 0 ··· 0 X .. ⎥ ⎢ .. ⎢ 0 X j+1 . . . ⎥ ⎢ ⎥ ⎢ .. ⎥ .. .. ∈ Ωnj +···+n , j = 0, .
L,j = ΔL,ej ) where ej is the j-th standard basis vector in Rd , j = 1, . . , d. 34) ΔL f (X, Y )(Z) = ΔL,j f (X, Y )(Zj ). 5 have exactly the same form for the directional nc diﬀerence-diﬀerential operators. 3, ΔR,μ l(X, Y )(A) = Al(μ) and ΔL,μ l(X, Y )(A) = Al(μ); in particular, in the case where M = Rd we have ΔR,i lj (X, Y )(A) = δij A and ΔL,i lj (X, Y )(A) = δij A. 6 for the right and left partial nc diﬀerence-diﬀerential operators in the case where M = Rd and N = R is ΔR,i (g ◦ f )(X, Y )(A) = ΔR,j g(f (X), f (Y ))(ΔR,i fj (X, Y )(A)) j=1 and ΔL,i (g ◦ f )(X, Y )(A) = ΔL,j g(f (X), f (Y ))(ΔL,i fj (X, Y )(A)) j=1 for i = 1, .
Introduction to algebraic geometry by Serge Lang