 It follows that the third order taylor series of "f" is given by "L"2 ± "LQ" where "Q" is quadratic in "x" and "y".We can complete the square to show that "L"2 ± "LQ" = ("L" ± ½"Q")2 – ¼"Q"4.We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that ("L" ± ½"Q")2 − ¼"Q"4 → "x"12 + "P"1 where "P"1 is quartic (order four) in "x"1 and "y"1.