 The divisibility condition for type "A"≥4 is that "x"1 divides "P"1.If "x"1 does not divide "P"1 then we have type exactly "A"3 (the zero-level-set here is a tacnode).If "x"1 divides "P"1 we complete the square on "x"12 + "P"1 and change coordinates so that we have "x"22 + "P"2 where "P"2 is quintic (order five) in "x"2 and "y"2.