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If one has (say) five degree $3$ homogeneous polynomials $f_1,f_2,f_3,f_4,f_5$ in three variables $x,y,z$, and $f_j(x_0,y_0,z_0)=0$ for all $j$ for some fixed $(x_0,y_0,z_0)$, can we conclude that there must be some relationship between some of these polynomials? In short, is there something linear algebraic that one can exploit to show that this can't happen given certain conditions on these $f_j$? Obviously given any specific set of these guys we can use "back substitution" in order to determine whether they can have simultaneous zeros, but I would love to know if algebraic geometry and friends can offer a more general approach.

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