Product vs Sum in K[x,y,z]K[x,y,z]K[x,y,z]
Let KKK be a field and R=K[x,y,z]R=K[x,y,z]R=K[x,y,z]. Prove or disprove: for any A,B,C,D∈RA,B,C,D\in RA,B,C,D∈R, A2B2C2D2∈(A3,B3,C3,D3)A^2B^2C^2D^2 \in (A^3,B^3,C^3,D^3)A2B2C2D2∈(A3,B3,C3,D3).
Updates: Known to be false in characteristic p>0p>0p>0. Still open in characteristic 000.
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