If F in k[x_0,...,x_n] is homogeneous and F=x^alpha_0 G (where x_0 does not divide G) then beta*alpha(F)=G.

rialsv 2022-10-02 Answered
If F k [ x 0 , . . . , x n ] is homogeneous and F = x 0 α G (where x 0 does not divide G ) then β α ( F ) = G
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trutdelamodej0
Answered 2022-10-03 Author has 11 answers
Let's write G = | α | = d ( G ) a α x α , we can do so as G is homogeneous. Now we have
x 0 d ( G ) G ( 1 , x 1 x 0 , , x n x 0 ) = x 0 d ( G ) | α | = d ( G ) a α 1 α 0 x 1 α 1 x 0 α 1 x n α n x 0 α n   = x 0 d ( G ) | α | = d ( G ) a α 1 α 0 x 1 α 1 x n α n x 0 α 1 + + α n   = x 0 d ( G ) | α | = d ( G ) a α x 1 α 1 x n α n x 0 d ( G ) α 0   = | α | = d ( G ) a α x 0 α 0 x 1 α 1 x n α n   = | α | = d ( G ) a α x α   = G ( x 0 , , x n ) .
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