@Carl Love I said over complex numbers, this excludes the possibility c_1 = -x^3\in C[x].

@Carl Love I mean there are no c_i such that  c_1[x^2, 0, 0]+c_2[0, y^2, 0]+c_3[x^5, 0, 0]=0 (because the first coordinate cannot vanish in this case).

@dharr I thought there's a special function for this. Thank you very much!!