ADDENDUM TO: CR SINGULARITIES OF REAL THREEFOLDS IN ℂ^{4}

ADAM COFFMAN

Last update: May 27, 2018.

6. Updates

The article and its Reference [5] have been reviewed as noted in bibliographic items [C_{3}], [C_{2}], below. Reference [4] has also been indexed in MR: [C_{1}]. Reference [6] has appeared as [C_{4}], and the web address printed in this entry is now obsolete.

My contact information has changed, and my current web page can be found at:

  http://users.pfw.edu/CoffmanA/

7. Citations

The article is cited in these papers: [B_{1}], [B_{2}].

The following Sections of this addendum include some calculations which are lower-degree special cases of some of the comparisons of coefficients that appear in [C_{3}].

8. Addendum to: Section 3, Higher-order terms in the non-degenerate case

The following Subsections could be read in [C_{3}] at the beginning of Section 3, as the construction of a solution p⃗ in terms of e⃗ in the case when e⃗ has cubic terms. This is a special case of the calculation that appears in Section 3.

8.1. Stabilizer of the non-degenerate normal form.

We consider M satisfying the first non-degeneracy condition, and in a normal form:

(8.1)  y_{2} = E_{2}(z_{1}, z̄_{1}, x_{2}) = O(3)
z_{3} = z̄_{1}^{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = (z_{1} + εx_{2})z̄_{1} + e_{4}(z_{1}, z̄_{1}, x_{2}).

with ε either 1 or 0, corresponding to whether M satisfies the second non-degeneracy condition (type (I)) or not (type (II)). To see what transformations will be available to normalize some higher degree terms, we will find the group of transformations which leaves (8.1) invariant.

We start with transformations that have some effect on the quadratic terms, and whose linear part is as in (2.7):

(8.2)  z̃_{1} = c_{1}z_{1} + c_{2}z_{2}
z̃_{2} = r_{2}z_{2} + p_{2}(z_{1}, z_{2}, z_{3}, z_{4})
z̃_{3} = c_{3}z_{3} + c_{4}z_{4} + p_{3}(z_{1}, z_{2}, z_{3}, z_{4})
z̃_{4} = c_{5}z_{3} + c_{6}z_{4} + p_{4}(z_{1}, z_{2}, z_{3}, z_{4}),

where p_{2} has only weight 2 terms and p_{3} and p_{4} have only quadratic weight 2 terms. Any terms of higher weight, or any further terms in the z̃_{1} expression, would contribute only terms of degree 3 or higher in the new defining equations. The transformation corresponding to Lemma 2.3 is given by z_{1} = \frac{1}{c_{1}} z̃_{1} − \frac{c_{2}}{c_{1}r_{2}} x̃_{2} + ϕ_{1}(z̃_{1}, ¯ z̃_{1}, x̃_{2}), x_{2} = \frac{1}{r_{2}} x̃_{2} + ϕ_{3}(z̃_{1}, ¯ z̃_{1}, x̃_{2}).

Let p_{4}(z_{1}, z_{2}, z_{3}, z_{4}) = p_{4}^{20}z_{1}^{2} + p_{4}^{11}z_{1}z_{2} + p_{4}^{02}z_{2}^{2}. A stabilizing transformation should leave no quadratic terms in the following expression:

z̃_{3} − ¯ z̃_{1}^{2}

= c_{3}z_{3} + c_{4}z_{4} + p_{3} − \overline{(c_{1}z_{1} + c_{2}z_{2})}^{2}
= c_{3}(z̄_{1}^{2} + e_{3}) + c_{4}((z_{1} + εx_{2})z̄_{1} + e_{4}) + p_{3}(z_{1}, x_{2} + iH_{2}, h_{3}, h_{4})

−\overline{(c_{1}z_{1} + c_{2}z_{2})}^{2}

= (c_{3} − \overline{c_{1}}^{2})z̄_{1}^{2} + (c_{4}ε − 2\overline{c_{1}c_{2}})z̄_{1}x_{2} + c_{4}z_{1}z̄_{1} − \overline{c_{2}}^{2}x_{2}^{2} + p_{3}(z_{1}, x_{2}, 0, 0) + O(3)
= \frac{c_{3} − \overline{c_{1}}^{2}}{\overline{c_{1}}^{2}} ¯ z̃_{1}^{2} + \frac{c_{4}}{c_{1}\overline{c_{1}}} z̃_{1} ¯z̃_{1} + \frac{p_{3}^{20}}{c_{1}^{2}} z̃_{1}^{2} + \frac{εc_{1}\overline{c_{1}}c_{4} − 2c_{1}\overline{c_{2}}c_{3} − \overline{c_{1}}c_{2}c_{4}}{c_{1}\overline{c_{1}}^{2}r_{2}} ¯ z̃_{1}x̃_{2}

+\frac{p_{3}^{11}c_{1}\overline{c_{1}} − 2p_{3}^{20}\overline{c_{1}}c_{2} − c_{1}\overline{c_{2}}c_{4}}{c_{1}^{2}\overline{c_{1}}r_{2}} z̃_{1}x̃_{2}

+\frac{p_{3}^{20}\overline{c_{1}}^{2}c_{2}^{2} − p_{3}^{11}c_{1}\overline{c_{1}}^{2}c_{2} + p_{3}^{02}c_{1}^{2}\overline{c_{1}}^{2} + c_{1}^{2}\overline{c_{2}}^{2}c_{3} − εc_{1}^{2}\overline{c_{1}c_{2}}c_{4} + c_{1}\overline{c_{1}}c_{2}\overline{c_{2}}c_{4}}{c_{1}^{2}\overline{c_{1}}^{2}r_{2}^{2}} x̃_{2}^{2}

+Õ(3).

The only z̃_{1} ¯z̃_{1} term has coefficient \frac{c_{4}}{c_{1}\overline{c_{1}}} , so a stabilizing transformation must have

c_{4} = 0, and similarly p_{3}^{20} = 0. The ¯ z̃_{1}^{2} term has coefficient \frac{c_{3}−\overline{c_{1}}^{2}}{\overline{c_{1}}^{2}} , so c_{3} = \overline{c_{1}}^{2}. Making these substitutions into the remaining terms gives the quantity

\frac{−2\overline{c_{2}}}{r_{2}} ¯ z̃_{1}x̃_{2} + \frac{p_{3}^{11}}{c_{1}r_{2}} z̃_{1}x̃_{2} + \frac{p_{3}^{02}c_{1} − p_{3}^{11}c_{2} + c_{1}\overline{c_{2}}^{2}}{c_{1}r_{2}^{2}} x̃_{2}^{2} + Õ(3).

So, c_{2} = p_{3}^{11} = 0 and this implies p_{3}^{02} = 0.

ADDENDUM: CR SINGULARITIES OF REAL THREEFOLDS IN ℂ^{4} 3

For the next step, we use transformation (8.2), with c_{2} = 0, so z̃_{1} = c_{1}z_{1}.

z̃_{4} − (z̃_{1} + εx̃_{2})¯z̃_{1}

= c_{5}z_{3} + c_{6}z_{4} + p_{4} − (c_{1}z_{1} + εRe(r_{2}z_{2} + p_{2}))\overline{(c_{1}z_{1})}
= c_{5}(z̄_{1}^{2} + e_{3}) + c_{6}((z_{1} + εx_{2})z̄_{1} + e_{4}) + p_{4}(z_{1}, x_{2} + iH_{2}, h_{3}, h_{4})

−(c_{1}z_{1} + εRe(r_{2}(x_{2} + iH_{2}) + p_{2}))\overline{(c_{1}z_{1})}

= c_{5}z̄_{1}^{2} + ε(c_{6} − r_{2}\overline{c_{1}})z̄_{1}x_{2} + (c_{6} − c_{1}\overline{c_{1}})z_{1}z̄_{1} + p_{4}(z_{1}, x_{2}, 0, 0) + O(3)
= \frac{c_{5}}{\overline{c_{1}}^{2}} ¯ z̃_{1}^{2} + \frac{c_{6} − c_{1}\overline{c_{1}}}{c_{1}\overline{c_{1}}} z̃_{1} ¯z̃_{1} + \frac{ε(c_{6} − \overline{c_{1}}r_{2})}{\overline{c_{1}}r_{2}} ¯z̃_{1}x̃_{2}

+\frac{p_{4}^{20}}{c_{1}^{2}} z̃_{1}^{2} + \frac{p_{4}^{11}}{c_{1}r_{2}} z̃_{1}x̃_{2} + \frac{p_{4}^{02}}{r_{2}^{2}} x̃_{2}^{2} + Õ(3).

The only ¯ z̃_{1}^{2} term has coefficient \frac{c_{5}}{\overline{c_{1}}^{2}} , so c_{5} = 0, and similarly p_{4}^{20} = p_{4}^{11} = p_{4}^{02} = 0. The z̃_{1} ¯z̃_{1} term has coefficient \frac{c_{6}−c_{1}\overline{c_{1}}}{c_{1}\overline{c_{1}}} , so c_{6} = c_{1}\overline{c_{1}}. The only remaining ¯ z̃_{1}x_{2} term has coefficient \frac{ε(c_{1}−r_{2})}{r_{2}}, so if ε = 1, then c_{1} = r_{2}.

For the final step, consider the transformation (8.2) with c_{2} = 0 and

p_{2}(z⃗) = p_{2}^{1}z_{3} + p_{2}^{2}z_{4} + p_{2}^{3}z_{1}^{2} + p_{2}^{4}z_{1}z_{2} + p_{2}^{5}z_{2}^{2}.

Im(z̃_{2}) = Im(r_{2}z_{2} + p_{2})
= Im(r_{2}x_{2} + ir_{2}H_{2}) + Im(p_{2}(z_{1}, x_{2} + iH_{2}, h_{3}, h_{4}))
= r_{2}H_{2}(z_{1}, z̄_{1}, x_{2})

+Im(p_{2}^{1}(z̄_{1}^{2} + e_{3}) + p_{2}^{2}((z_{1} + εx_{2})z̄_{1} + e_{4}) +p_{2}^{3}z_{1}^{2} + p_{2}^{4}z_{1}(x_{2} + iH_{2}) + p_{2}^{5}(x_{2} + iH_{2})^{2})

= \frac{p_{2}^{1} − \overline{p_{2}^{3}}}{2i\overline{c_{1}}^{2}} ¯ z̃_{1}^{2} + \frac{p_{2}^{3} − \overline{p_{2}^{1}}}{2ic_{1}^{2}} z̃_{1}^{2} + \frac{p_{2}^{2} − \overline{p_{2}^{2}}}{2ic_{1}\overline{c_{1}}} z̃_{1} ¯z̃_{1}

+\frac{ε\overline{p_{2}^{2}} − p_{2}^{4}}{2i\overline{c_{1}}r_{2}} ¯ z̃_{1}x_{2} + \frac{\overline{p_{2}^{4}} − εp_{2}^{2}}{2ic_{1}r_{2}} z̃_{1}x̃_{2} + \frac{p_{2}^{5} − \overline{p_{2}^{5}}}{2ir_{2}^{2}} x̃_{2}^{2} + Õ(3).

For all the quadratic terms to be 0, it follows that p_{2}^{2} and p_{2}^{5} are real, p_{2}^{3} = \overline{p_{2}^{1}}, and p_{2}^{4} = ε\overline{p_{2}^{2}}.

We arrive at the following conclusions in the two cases. If ε = 1, the stabilizer of the quadratic normal form is the set of transformations of the form:

(8.3)  z̃_{1} = rz_{1} + p_{1}(z⃗)
z̃_{2} = rz_{2} + p_{2}^{1}z_{3} + p_{2}^{2}z_{4} + \overline{p_{2}^{1}}z_{1}^{2} + \overline{p_{2}^{2}}z_{1}z_{2} + p_{2}^{5}z_{2}^{2} + p_{2}(z⃗)
z̃_{3} = r^{2}z_{3} + p_{3}(z⃗)
z̃_{4} = r^{2}z_{4} + p_{4}(z⃗),

where r is a non-zero real number, p_{2}^{2} and p_{2}^{5} are real, p_{1}(z⃗) has weight ≥ 2, and p_{2}, p_{3}, p_{4} have weight ≥ 3. If ε = 0, the stabilizer is given by:

(8.4)  z̃_{1} = c_{1}z_{1} + p_{1}(z⃗)
z̃_{2} = r_{2}z_{2} + p_{2}^{1}z_{3} + p_{2}^{2}z_{4} + \overline{p_{2}^{1}}z_{1}^{2} + p_{2}^{5}z_{2}^{2} + p_{2}(z⃗)
z̃_{3} = \overline{c_{1}}^{2}z_{3} + p_{3}(z⃗)
z̃_{4} = c_{1}\overline{c_{1}}z_{4} + p_{4}(z⃗),

where c_{1} is a non-zero complex number, r_{2} is a non-zero real number, p_{2}^{2}, p_{2}^{5} are real, and the functions p_{1}, p_{2}, p_{3}, p_{4} have weights as in the previous case.

8.2. The cubic normal form in the non-degenerate case.

We consider M with a type (I) normal form ((8.1), with ε = 1). In analogy with (2.4), any terms of the form z_{1}^{3}, z_{1}^{2}x_{2}, z_{1}x_{2}^{2}, x_{2}^{3} can be eliminated from the h_{3} and h_{4} series, without introducing any other quadratic or cubic terms in h_{3} or h_{4}, by a weight 3 transformation of the form z̃_{3} = z_{3} + p_{3}(z_{1}, z_{2}, 0, 0), z̃_{4} = z_{4} + p_{4}(z_{1}, z_{2}, 0, 0). However, we leave these terms in the expression (8.6) since even if they are eliminated as a first step, they may re-appear when other terms are eliminated. So, to deal simultaneously with all ten cubic terms in each of h_{3}, h_{4}, we consider a coordinate transformation of the form z̃_{1} = z_{1} + p_{1}, z̃_{2} = z_{2} + p_{2}, z̃_{3} = z_{3} + p_{3}, z̃_{4} = z_{4} + p_{4}, with

(8.5) p_{1} = p_{1}^{1}z_{3} + p_{1}^{2}z_{1}^{2} + p_{1}^{3}z_{1}z_{2} + p_{1}^{4}z_{2}^{2}
p_{2} = p_{2}^{1}z_{1}z_{3} + p_{2}^{2}z_{2}z_{4} + p_{2}^{3}z_{1}^{3} + p_{2}^{4}z_{1}^{2}z_{2} + p_{2}^{5}z_{1}z_{2}^{2} + p_{2}^{6}z_{2}^{3}
p_{3} = p_{3}^{1}z_{1}z_{3} + p_{3}^{2}z_{1}z_{4} + p_{3}^{3}z_{2}z_{3} + p_{3}^{4}z_{2}z_{4} + p_{3}^{5}z_{1}^{3} + p_{3}^{6}z_{1}^{2}z_{2} + p_{3}^{7}z_{1}z_{2}^{2} + p_{3}^{8}z_{2}^{3}
p_{4} = p_{4}^{1}z_{1}z_{3} + p_{4}^{2}z_{1}z_{4} + p_{4}^{3}z_{2}z_{3} + p_{4}^{4}z_{2}z_{4} + p_{4}^{5}z_{1}^{3} + p_{4}^{6}z_{1}^{2}z_{2} + p_{4}^{7}z_{1}z_{2}^{2} + p_{4}^{8}z_{2}^{3},

so p_{1} has weight 2 and p_{2}, p_{3}, p_{4} have weight 3, and although there is a linear term in p_{1}, Lemma 2.3 applies. Since it fits the form of (8.3), such a transformation does not affect the quadratic terms of H_{2}, h_{3} or h_{4}, as considered in Section 8.1. For points on M,

z̃_{3} − ¯ z̃_{1}^{2} (8.6)

= z_{3} + p_{3}(z⃗) − \overline{(z_{1} + p_{1}(z⃗))}^{2}
= e_{3} + p_{3} − 2z̄_{1}\overline{p_{1}} − \overline{p_{1}}^{2}
= e_{3}^{210}z_{1}^{2}z̄_{1} + e_{3}^{120}z_{1}z̄_{1}^{2} + e_{3}^{111}z_{1}z̄_{1}x_{2} + e_{3}^{030}z̄_{1}^{3} + e_{3}^{021}z̄_{1}^{2}x_{2} + e_{3}^{012}z̄_{1}x_{2}^{2}
+e_{3}^{300}z_{1}^{3} + e_{3}^{201}z_{1}^{2}x_{2} + e_{3}^{102}z_{1}x_{2}^{2} + e_{3}^{003}x_{2}^{3}
+p_{3}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1}) − 2z̄_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})} + O(4)
= e_{3}^{210}z_{1}^{2}z̄_{1} + e_{3}^{120}z_{1}z̄_{1}^{2} + e_{3}^{111}z_{1}z̄_{1}x_{2} + e_{3}^{030}z̄_{1}^{3} + e_{3}^{021}z̄_{1}^{2}x_{2} + e_{3}^{012}z̄_{1}x_{2}^{2}
+e_{3}^{300}z_{1}^{3} + e_{3}^{201}z_{1}^{2}x_{2} + e_{3}^{102}z_{1}x_{2}^{2} + e_{3}^{003}x_{2}^{3}
+p_{3}^{1}z_{1}z̄_{1}^{2} + p_{3}^{2}z_{1}(z_{1} + x_{2})z̄_{1} + p_{3}^{3}x_{2}z̄_{1}^{2} + p_{3}^{4}x_{2}(z_{1} + x_{2})z̄_{1}
+p_{3}^{5}z_{1}^{3} + p_{3}^{6}z_{1}^{2}x_{2} + p_{3}^{7}z_{1}x_{2}^{2} + p_{3}^{8}x_{2}^{3}
−2z̄_{1}\overline{(p_{1}^{1}z̄_{1}^{2} + p_{1}^{2}z_{1}^{2} + p_{1}^{3}z_{1}x_{2} + p_{1}^{4}x_{2}^{2})} + O(4),

z̃_{4} − (z̃_{1} + x̃_{2})¯z̃_{1} (8.7)

= z_{4} + p_{4}(z⃗) − (z_{1} + p_{1}(z⃗) + x_{2} + Re(p_{2}(z⃗)))\overline{(z_{1} + p_{1}(z⃗))}
= e_{4} + p_{4} − (z_{1} + x_{2})\overline{p_{1}} − z̄_{1}(p_{1} + Re(p_{2})) − p_{1}\overline{p_{1}} − Re(p_{2})\overline{p_{1}}
= e_{4}^{210}z_{1}^{2}z̄_{1} + e_{4}^{120}z_{1}z̄_{1}^{2} + e_{4}^{111}z_{1}z̄_{1}x_{2} + e_{4}^{030}z̄_{1}^{3} + e_{4}^{021}z̄_{1}^{2}x_{2} + e_{4}^{012}z̄_{1}x_{2}^{2}
+e_{4}^{300}z_{1}^{3} + e_{4}^{201}z_{1}^{2}x_{2} + e_{4}^{102}z_{1}x_{2}^{2} + e_{4}^{003}x_{2}^{3}
+p_{4}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})

−(z_{1} + x_{2})\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})} − z̄_{1}p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1}) + O(4)

= e_{4}^{210}z_{1}^{2}z̄_{1} + e_{4}^{120}z_{1}z̄_{1}^{2} + e_{4}^{111}z_{1}z̄_{1}x_{2} + e_{4}^{030}z̄_{1}^{3} + e_{4}^{021}z̄_{1}^{2}x_{2} + e_{4}^{012}z̄_{1}x_{2}^{2}
+e_{4}^{300}z_{1}^{3} + e_{4}^{201}z_{1}^{2}x_{2} + e_{4}^{102}z_{1}x_{2}^{2} + e_{4}^{003}x_{2}^{3}
+p_{4}^{1}z_{1}z̄_{1}^{2} + p_{4}^{2}z_{1}(z_{1} + x_{2})z̄_{1} + p_{4}^{3}x_{2}z̄_{1}^{2} + p_{4}^{4}x_{2}(z_{1} + x_{2})z̄_{1}
+p_{4}^{5}z_{1}^{3} + p_{4}^{6}z_{1}^{2}x_{2} + p_{4}^{7}z_{1}x_{2}^{2} + p_{4}^{8}x_{2}^{3}
−(z_{1} + x_{2})\overline{(p_{1}^{1}z̄_{1}^{2} + p_{1}^{2}z_{1}^{2} + p_{1}^{3}z_{1}x_{2} + p_{1}^{4}x_{2}^{2})}
−z̄_{1}(p_{1}^{1}z̄_{1}^{2} + p_{1}^{2}z_{1}^{2} + p_{1}^{3}z_{1}x_{2} + p_{1}^{4}x_{2}^{2}) + O(4).

The last line can be converted to z̃_{1}, x̃_{2} coordinates, but this would only introduce higher order terms, without changing any of the cubic coefficients. Inspecting (8.6) and (8.7) shows that the p_{2} quantity does not contribute cubic terms to h_{3} or h_{4}. Each of the defining equations h_{3} and h_{4} has 10 complex cubic coefficients, and we have a choice of 20 complex coefficients in the three components p_{1}, p_{3}, p_{4} of the coordinate transformation. By comparing like terms in the above quantities, it turns out that we can find coefficients for p_{1}, p_{3}, p_{4} that eliminate all the cubic terms of h_{3} and h_{4}, by solving a linear system of 20 unknowns (coefficients of p⃗) given 20 constants (coefficients of e).

The twenty equations are

0 = e_{3}^{210} + p_{3}^{2} − 2\overline{p_{1}^{1}}
0 = e_{3}^{021} + p_{3}^{3} − 2\overline{p_{1}^{3}}
0 = e_{3}^{120} + p_{3}^{1}
0 = e_{3}^{111} + p_{3}^{2} + p_{3}^{4}
0 = e_{3}^{030} − 2\overline{p_{1}^{2}}
0 = e_{3}^{012} − 2\overline{p_{1}^{4}} + p_{3}^{4}
0 = e_{3}^{300} + p_{3}^{5}
0 = e_{3}^{201} + p_{3}^{6}
0 = e_{3}^{102} + p_{3}^{7}
0 = e_{3}^{003} + p_{3}^{8}
0 = e_{4}^{210} + p_{4}^{2} − p_{1}^{2}
0 = e_{4}^{021} − \overline{p_{1}^{2}} + p_{4}^{3}
0 = e_{4}^{012} − \overline{p_{1}^{3}} − p_{1}^{4} + p_{4}^{4}
0 = e_{4}^{120} + p_{4}^{1} − \overline{p_{1}^{2}}
0 = e_{4}^{030} − p_{1}^{1}
0 = e_{4}^{111} − \overline{p_{1}^{3}} − p_{1}^{3} + p_{4}^{2} + p_{4}^{4}
0 = e_{4}^{300} − \overline{p_{1}^{1}} + p_{4}^{5}
0 = e_{4}^{201} − \overline{p_{1}^{1}} + p_{4}^{6}
0 = e_{4}^{102} − \overline{p_{1}^{4}} + p_{4}^{7}
0 = e_{4}^{003} − \overline{p_{1}^{4}} + p_{4}^{8},

ADDENDUM: CR SINGULARITIES OF REAL THREEFOLDS IN ℂ^{4} 7

and the solutions are

p_{1}^{1} = e_{4}^{030}
p_{1}^{2} = \frac{1}{2}\overline{e_{3}^{030}}
p_{1}^{3} = −e_{4}^{210} + e_{4}^{111} − e_{4}^{012} − e_{4}^{030} + \frac{1}{2}\overline{e_{3}^{030}} − \frac{1}{2}\overline{e_{3}^{111}} + \frac{1}{2}\overline{e_{3}^{210}} + \frac{1}{2}\overline{e_{3}^{012}}
p_{1}^{4} = −e_{4}^{030} − \frac{1}{2}\overline{e_{3}^{111}} + \frac{1}{2}\overline{e_{3}^{210}} + \frac{1}{2}\overline{e_{3}^{012}}
p_{3}^{1} = −e_{3}^{120}
p_{3}^{2} = −e_{3}^{210} + 2\overline{e_{4}^{030}}
p_{3}^{3} = e_{3}^{030} − e_{3}^{111} + e_{3}^{210} + e_{3}^{012} − 2\overline{e_{4}^{210}} + 2\overline{e_{4}^{111}} − 2\overline{e_{4}^{012}} − 2\overline{e_{4}^{030}} − e_{3}^{021}
p_{3}^{4} = −e_{3}^{111} − 2\overline{e_{4}^{030}} + e_{3}^{210}
p_{3}^{5} = −e_{3}^{300}
p_{3}^{6} = −e_{3}^{201}
p_{3}^{7} = −e_{3}^{102}
p_{3}^{8} = −e_{3}^{003}
p_{4}^{1} = \frac{1}{2}e_{3}^{030} − e_{4}^{120}
p_{4}^{2} = \frac{1}{2}\overline{e_{3}^{030}} − e_{4}^{210}
p_{4}^{3} = \frac{1}{2}e_{3}^{030} − e_{4}^{021}
p_{4}^{4} = \frac{1}{2}e_{3}^{030} − \frac{1}{2}e_{3}^{111} + \frac{1}{2}e_{3}^{210} + \frac{1}{2}e_{3}^{012} − \overline{e_{4}^{210}} + \overline{e_{4}^{111}} − \overline{e_{4}^{012}}

−\overline{e_{4}^{030}} − e_{4}^{012} − e_{4}^{030} − \frac{1}{2}\overline{e_{3}^{111}} + \frac{1}{2}\overline{e_{3}^{210}} + \frac{1}{2}\overline{e_{3}^{012}}

p_{4}^{5} = \overline{e_{4}^{030}} − e_{4}^{300}
p_{4}^{6} = \overline{e_{4}^{030}} − e_{4}^{201}
p_{4}^{7} = −\frac{1}{2}e_{3}^{111} + \frac{1}{2}e_{3}^{210} + \frac{1}{2}e_{3}^{012} − \overline{e_{4}^{030}} − e_{4}^{102}
p_{4}^{8} = −\frac{1}{2}e_{3}^{111} + \frac{1}{2}e_{3}^{210} + \frac{1}{2}e_{3}^{012} − \overline{e_{4}^{030}} − e_{4}^{003}.

The cubic terms can similarly be eliminated from H_{2}, using only the p_{2} component of the transformation. For points on M, Im(z̃_{2}) = Im(z_{2} + p_{2}(z⃗)) (8.8)

= Im(x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2}) + p_{2}(z_{1}, x_{2} + iH_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4}))
= H_{2}(z_{1}, z̄_{1}, x_{2}) + Im(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})) + O(4)
= e_{2}^{210}z_{1}^{2}z̄_{1} + \overline{e_{2}^{210}}z_{1}z̄_{1}^{2} + e_{2}^{111}z_{1}z̄_{1}x_{2} + e_{2}^{300}z_{1}^{3} + \overline{e_{2}^{300}}z̄_{1}^{3}

+\overline{e_{2}^{201}}z̄_{1}^{2}x_{2} + \overline{e_{2}^{102}}z̄_{1}x_{2}^{2} + e_{2}^{201}z_{1}^{2}x_{2} + e_{2}^{102}z_{1}x_{2}^{2} + e_{2}^{003}x_{2}^{3}

+\frac{1}{2i}(p_{2}^{1}z_{1}z̄_{1}^{2} + p_{2}^{2}x_{2}(z_{1} + x_{2})z̄_{1} + p_{2}^{3}z_{1}^{3} + p_{2}^{4}z_{1}^{2}x_{2} + p_{2}^{5}z_{1}x_{2}^{2} + p_{2}^{6}x_{2}^{3})

−\frac{1}{2i}\overline{(p_{2}^{1}z_{1}z̄_{1}^{2} + p_{2}^{2}x_{2}(z_{1} + x_{2})z̄_{1} + p_{2}^{3}z_{1}^{3} + p_{2}^{4}z_{1}^{2}x_{2} + p_{2}^{5}z_{1}x_{2}^{2} + p_{2}^{6}x_{2}^{3})} + O(4).

The coefficients e_{2}^{111} and e_{2}^{003} are real, but the others can be complex, leading to six equations in six unknowns:

0 = e_{2}^{300} + \frac{1}{2i}p_{2}^{3}
0 = e_{2}^{201} + \frac{1}{2i}p_{2}^{4}
0 = e_{2}^{111} + \frac{1}{2i}(p_{2}^{2} − \overline{p_{2}^{2}})
0 = \overline{e_{2}^{210}} + \frac{1}{2i}p_{2}^{1}
0 = e_{2}^{003} + \frac{1}{2i}(p_{2}^{6} − \overline{p_{2}^{6}})
(8.9)  0 = e_{2}^{102} + \frac{1}{2i}p_{2}^{5} − \frac{1}{2i}\overline{p_{2}^{2}}.

The solutions are

p_{2}^{1} = −2i\overline{e_{2}^{210}}
p_{2}^{2} = −ie_{2}^{111}
p_{2}^{3} = −2ie_{2}^{300}
p_{2}^{4} = −2ie_{2}^{201}
(8.10)  p_{2}^{5} = −2ie_{2}^{102} + ie_{2}^{111}
p_{2}^{6} = −ie_{2}^{003}.

9. Addendum to: Section 4, Some degenerate cases

The following Subsections could be read in [C_{3}] at the beginning of Section 4.

9.1. Cubic normal forms in a degenerate case. We consider M with a type (II) normal form (satisfying the first non-degeneracy condition, but not the second: (8.1), with ε = 0). To deal simultaneously with all ten cubic terms in each of h_{3}, h_{4}, we consider a coordinate transformation of the form z̃_{1} = z_{1} + p_{1}, z̃_{2} = z_{2} + p_{2}, z̃_{3} = z_{3} + p_{3}, z̃_{4} = z_{4} + p_{4}, with

p_{1} = p_{1}^{1}z_{3} + p_{1}^{2}z_{1}^{2} + p_{1}^{3}z_{1}z_{2} + p_{1}^{4}z_{2}^{2} + p_{1}^{5}z_{4}
p_{2} = p_{2}^{1}z_{1}z_{3} + p_{2}^{2}z_{2}z_{4} + p_{2}^{3}z_{1}^{3} + p_{2}^{4}z_{1}^{2}z_{2} + p_{2}^{5}z_{1}z_{2}^{2} + p_{2}^{6}z_{2}^{3}
p_{3} = p_{3}^{1}z_{1}z_{3} + p_{3}^{2}z_{1}z_{4} + p_{3}^{3}z_{2}z_{3} + p_{3}^{4}z_{2}z_{4} + p_{3}^{5}z_{1}^{3} + p_{3}^{6}z_{1}^{2}z_{2} + p_{3}^{7}z_{1}z_{2}^{2} + p_{3}^{8}z_{2}^{3}
p_{4} = p_{4}^{1}z_{1}z_{3} + p_{4}^{2}z_{1}z_{4} + p_{4}^{3}z_{2}z_{3} + p_{4}^{4}z_{2}z_{4} + p_{4}^{5}z_{1}^{3} + p_{4}^{6}z_{1}^{2}z_{2} + p_{4}^{7}z_{1}z_{2}^{2} + p_{4}^{8}z_{2}^{3},

so p_{1} has weight 2 and p_{2}, p_{3}, p_{4} have weight 3. This is similar to (8.5), but the term p_{1}^{5}z_{4} is included, to give all the available weight 2 terms in p_{1}, and even with both linear terms, Lemma 2.3 applies. Since it fits the form of (8.4), such a transformation does not affect the quadratic terms of H_{2}, h_{3} or h_{4}, as considered in Section 8.1. For points on M,

z̃_{3} − ¯ z̃_{1}^{2} (9.1)

= z_{3} + p_{3}(z⃗) − \overline{(z_{1} + p_{1}(z⃗))}^{2}
= e_{3} + p_{3} − 2z̄_{1}\overline{p_{1}} − \overline{p_{1}}^{2}
= e_{3}^{210}z_{1}^{2}z̄_{1} + e_{3}^{120}z_{1}z̄_{1}^{2} + e_{3}^{111}z_{1}z̄_{1}x_{2} + e_{3}^{030}z̄_{1}^{3} + e_{3}^{021}z̄_{1}^{2}x_{2} + e_{3}^{012}z̄_{1}x_{2}^{2}
+e_{3}^{300}z_{1}^{3} + e_{3}^{201}z_{1}^{2}x_{2} + e_{3}^{102}z_{1}x_{2}^{2} + e_{3}^{003}x_{2}^{3}
+p_{3}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}) − 2z̄_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})} + O(4)
= e_{3}^{210}z_{1}^{2}z̄_{1} + e_{3}^{120}z_{1}z̄_{1}^{2} + e_{3}^{111}z_{1}z̄_{1}x_{2} + e_{3}^{030}z̄_{1}^{3} + e_{3}^{021}z̄_{1}^{2}x_{2} + e_{3}^{012}z̄_{1}x_{2}^{2}
+e_{3}^{300}z_{1}^{3} + e_{3}^{201}z_{1}^{2}x_{2} + e_{3}^{102}z_{1}x_{2}^{2} + e_{3}^{003}x_{2}^{3}
+p_{3}^{1}z_{1}z̄_{1}^{2} + p_{3}^{2}z_{1}^{2}z̄_{1} + p_{3}^{3}x_{2}z̄_{1}^{2} + p_{3}^{4}x_{2}z_{1}z̄_{1}
+p_{3}^{5}z_{1}^{3} + p_{3}^{6}z_{1}^{2}x_{2} + p_{3}^{7}z_{1}x_{2}^{2} + p_{3}^{8}x_{2}^{3}
−2z̄_{1}\overline{(p_{1}^{1}z̄_{1}^{2} + p_{1}^{2}z_{1}^{2} + p_{1}^{3}z_{1}x_{2} + p_{1}^{4}x_{2}^{2} + p_{1}^{5}z_{1}z̄_{1})} + O(4)
= (p_{3}^{1} − 2\overline{p_{1}^{5}} + e_{3}^{120})z̄_{1}^{2}z_{1} + (e_{3}^{201} + p_{3}^{6})z_{1}^{2}x_{2} + (p_{3}^{3} − 2\overline{p_{1}^{3}} + e_{3}^{021})z̄_{1}^{2}x_{2}
+(−2\overline{p_{1}^{1}} + p_{3}^{2} + e_{3}^{210})z̄_{1}z_{1}^{2} + (e_{3}^{003} + p_{3}^{8})x_{2}^{3} + (p_{3}^{4} + e_{3}^{111})z_{1}z̄_{1}x_{2}
+(p_{3}^{7} + e_{3}^{102})z_{1}x_{2}^{2} + (p_{3}^{5} + e_{3}^{300})z_{1}^{3} + (−2\overline{p_{1}^{2}} + e_{3}^{030})z̄_{1}^{3}
+(e_{3}^{012} − 2\overline{p_{1}^{4}})x_{2}^{2}z̄_{1} + O(4),

z̃_{4} − z̃_{1} ¯z̃_{1} (9.2)

= z_{4} + p_{4}(z⃗) − (z_{1} + p_{1}(z⃗))\overline{(z_{1} + p_{1}(z⃗))}
= e_{4} + p_{4} − z_{1}\overline{p_{1}} − z̄_{1}p_{1} − p_{1}\overline{p_{1}}
= e_{4}^{210}z_{1}^{2}z̄_{1} + e_{4}^{120}z_{1}z̄_{1}^{2} + e_{4}^{111}z_{1}z̄_{1}x_{2} + e_{4}^{030}z̄_{1}^{3} + e_{4}^{021}z̄_{1}^{2}x_{2} + e_{4}^{012}z̄_{1}x_{2}^{2}

+e_{4}^{300}z_{1}^{3} + e_{4}^{201}z_{1}^{2}x_{2} + e_{4}^{102}z_{1}x_{2}^{2} + e_{4}^{003}x_{2}^{3} +p_{4}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})

−z_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})} − z̄_{1}p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}) + O(4)

= e_{4}^{210}z_{1}^{2}z̄_{1} + e_{4}^{120}z_{1}z̄_{1}^{2} + e_{4}^{111}z_{1}z̄_{1}x_{2} + e_{4}^{030}z̄_{1}^{3} + e_{4}^{021}z̄_{1}^{2}x_{2} + e_{4}^{012}z̄_{1}x_{2}^{2}

+e_{4}^{300}z_{1}^{3} + e_{4}^{201}z_{1}^{2}x_{2} + e_{4}^{102}z_{1}x_{2}^{2} + e_{4}^{003}x_{2}^{3} +p_{4}^{1}z_{1}z̄_{1}^{2} + p_{4}^{2}z_{1}^{2}z̄_{1} + p_{4}^{3}x_{2}z̄_{1}^{2} + p_{4}^{4}x_{2}z_{1}z̄_{1} +p_{4}^{5}z_{1}^{3} + p_{4}^{6}z_{1}^{2}x_{2} + p_{4}^{7}z_{1}x_{2}^{2} + p_{4}^{8}x_{2}^{3}

−z_{1}\overline{(p_{1}^{1}z̄_{1}^{2} + p_{1}^{2}z_{1}^{2} + p_{1}^{3}z_{1}x_{2} + p_{1}^{4}x_{2}^{2} + p_{1}^{5}z_{1}z̄_{1})} −z̄_{1}(p_{1}^{1}z̄_{1}^{2} + p_{1}^{2}z_{1}^{2} + p_{1}^{3}z_{1}x_{2} + p_{1}^{4}x_{2}^{2} + p_{1}^{5}z_{1}z̄_{1}) + O(4)

= (−p_{1}^{1} + e_{4}^{030})z̄_{1}^{3} + (e_{4}^{120} − \overline{p_{1}^{2}} − p_{1}^{5} + p_{4}^{1})z̄_{1}^{2}z_{1} + (e_{4}^{300} + p_{4}^{5} − \overline{p_{1}^{1}})z_{1}^{3}

+(p_{4}^{3} + e_{4}^{021})z̄_{1}^{2}x_{2} + (p_{4}^{8} + e_{4}^{003})x_{2}^{3} + (p_{4}^{7} + e_{4}^{102} − \overline{p_{1}^{4}})z_{1}x_{2}^{2}

+(p_{4}^{4} − \overline{p_{1}^{3}} + e_{4}^{111} − p_{1}^{3})z̄_{1}z_{1}x_{2} + (e_{4}^{210} − \overline{p_{1}^{5}} − p_{1}^{2} + p_{4}^{2})z̄_{1}z_{1}^{2} + (p_{4}^{6} + e_{4}^{201})z_{1}^{2}x_{2} +(e_{4}^{012} − p_{1}^{4})x_{2}^{2}z̄_{1} + O(4)

The last line can be converted to z̃_{1}, x̃_{2} coordinates, but this would only introduce higher order terms, without changing any of the cubic coefficients. Inspecting (9.1) and (9.2) shows that the p_{2} quantity does not contribute cubic terms to h_{3} or h_{4}. Each of the defining equations h_{3} and h_{4} has 10 complex cubic coefficients, and we have a choice of 21 complex coefficients in the three components p_{1}, p_{3}, p_{4} of the coordinate transformation. By inspecting the z̄_{1}x_{2}^{2} terms, we see that the only coefficient which transforms these coefficients is p_{1}^{4}, but it is unlikely to cancel both e_{3} and e_{4}. However, if we choose to eliminate the z̄_{1}x_{2}^{2} term from e_{3} only, it turns out that we can find coefficients for p⃗ that also eliminate the remaining 18 cubic terms of h_{3} and h_{4}, and in fact this can be done while choosing p_{1}^{3} = p_{1}^{5} = 0.

All the cubic terms can be eliminated from E_{2} using p_{2}, in a way similar to (but even simpler than) the calculation from Section 8.2. The six equations are the same except that (8.9) is just 0 = e_{2}^{102} + \frac{1}{2i}p_{2}^{5}, and the six solutions are the same, except that (8.10) is just p_{2}^{5} = −2ie_{2}^{102}. In fact, a real-valued series in z_{1}, z̄_{1}, x_{2} can always be written, formally, as the real or imaginary part of some series in z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}, so it is easy to find p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}) that eliminates the lowest-degree terms in E_{2}, without the rearrangements that were used in Section 3.

So, after the transformation, the defining equations become

(9.3)  y_{2} = E_{2}(z_{1}, z̄_{1}, x_{2})
z_{3} = z̄_{1}^{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = z_{1}z̄_{1} + e_{4}^{012}z̄_{1}x_{2}^{2} + e_{4}(z_{1}, z̄_{1}, x_{2}),

where E_{2}, e_{3}, e_{4} have degree 4. A transformation of the form z̃_{1} = c_{1}z_{1}, z̃_{2} = z_{2}, z̃_{3} = \overline{c_{1}}^{2}z_{3}, z̃_{4} = c_{1}\overline{c_{1}}z_{4}, as in (8.4), changes the coefficient from e_{4}^{012} to c_{1}e_{4}^{012}, so we get two cubic normal forms, corresponding to scaling the z̄_{1}x_{2}^{2} coefficient to be either 1 or 0.

9.2. Stabilizer of the cubic degenerate normal form.

Without going through all the calculations as in Section 8.1, we can state the subgroup of the stabilizer group (8.4), consisting of transformations which preserve the normal form (9.3) with e_{4}^{012} = 1:

(9.4)  z̃_{1} = r^{2}z_{1} + p_{1A} + p_{1B}
p_{1A} = p_{1}^{11}z_{1}z_{2} + p_{1}^{0}z_{4}
z̃_{2} = rz_{2} + p_{2A} + p_{2B} + p_{2C}
p_{2A} = p_{2}^{1}z_{3} + p_{2}^{2}z_{4} + \overline{p_{2}^{1}}z_{1}^{2} + p_{2}^{5}z_{2}^{2}
p_{2B} = p_{2}^{101}z_{1}z_{3} + \overline{p_{2}^{101}}z_{1}z_{4} + p_{2}^{011}z_{2}z_{3} + p_{2}^{010}z_{2}z_{4}
+\overline{p_{2}^{011}}z_{1}^{2}z_{2} + p_{2}^{2}z_{1}z_{2}^{2} + p_{2}^{030}z_{2}^{3}
z̃_{3} = r^{4}z_{3} + p_{3A} + p_{3B}
p_{3A} = 2r^{2}\overline{p_{1}^{0}}z_{1}z_{3} + 2r^{2}\overline{p_{1}^{11}}z_{2}z_{3}
z̃_{4} = r^{4}z_{4} + p_{4A} + p_{4B}
p_{4A} = r^{2}p_{1}^{0}z_{1}z_{3} + r^{2}\overline{p_{1}^{0}}z_{1}z_{4} + r^{2}(p_{1}^{11} + \overline{p_{1}^{11}})z_{2}z_{4},

where r is a non-zero real number, p_{2}^{2}, p_{2}^{5}, p_{2}^{010}, p_{2}^{030} are real, p_{1B}(z⃗) has weight 3, and the functions p_{2C}(z⃗), p_{3B}(z⃗), p_{4B}(z⃗) have weight 4.

9.3. The quartic normal form for the degenerate case.

Again considering the normal form (9.3) with e_{4}^{012} = 1, we can in fact eliminate all the degree 4 terms from E_{2}, e_{3}, e_{4} by a polynomial coordinate change. We begin by considering all transformations of the form (9.4), with r = 1.

As in (9.1), for points on M,

z̃_{3} − ¯ z̃_{1}^{2}
= z_{3} + p_{3}(z⃗) − \overline{(z_{1} + p_{1}(z⃗))}^{2}
= e_{3} + p_{3} − 2z̄_{1}\overline{p_{1}} − \overline{p_{1}}^{2}
= e_{3} + p_{3A} + p_{3B} − 2z̄_{1}\overline{(p_{1A} + p_{1B})} − \overline{(p_{1A} + p_{1B})}^{2}
(9.5)  = e_{3}^{4} + p_{3A} + p_{3B} − 2z̄_{1}\overline{(p_{1A} + p_{1B})} − \overline{p_{1A}}^{2} + O(5),

where in line (9.5), e_{3}^{4} represents the degree 4 terms of e_{4} and all the cubic terms cancel, by the assumption that the transformation stabilizes the cubic normal form. Similarly, as in (9.2),

z̃_{4} − (z̃_{1} + x̃_{2}^{2})¯z̃_{1}
= z_{4} + p_{4}(z⃗) − (z_{1} + p_{1}(z⃗) + (x_{2} + Re(p_{2}(z⃗)))^{2})\overline{(z_{1} + p_{1}(z⃗))}
= e_{4} + p_{4} − (z_{1} + x_{2}^{2})\overline{p_{1}} − (z̄_{1} + \overline{p_{1}})(p_{1} + 2x_{2}Re(p_{2}) + (Re(p_{2}))^{2})
= e_{4}^{4} + p_{4A} + p_{4B} − (p_{1A} + p_{1B} + 2x_{2}Re(p_{2A}))z̄_{1} − p_{1A}\overline{p_{1A}}

−z_{1}\overline{(p_{1A} + p_{1B})} − x_{2}^{2}\overline{p_{1A}} + O(5). (9.6)

Note that the quantities (9.5) and (9.6) are non-linear in p_{1A}. Rather than give the whole expansion of these expressions, we can consider just the z̄_{1}x_{2}^{3} terms first. If the coefficients of p_{1A} and p_{2A} are as in (9.4), the z̄_{2}^{3} term of p_{1B} has coefficient p_{1}^{030}, and the z̄_{1}x_{2}^{3} terms of e_{3} and e_{4} are e_{3}^{013} and e_{4}^{013}, then the coefficient of the z̄_{1}x_{2}^{3} term in (9.5) is e_{3}^{013} − 2\overline{p_{1}^{030}} and in (9.6) is e_{4}^{013} − p_{1}^{030} − 2Re(p_{2}^{5}) − \overline{p_{1}^{11}}. We can choose p_{1}^{030} to eliminate the term from e_{3}, but to simultaneously eliminate it from e_{4}, the real number Re(p_{2}^{5}) may not be enough, so we would need to use p_{1}^{11}.

However, all the other terms in e_{3}^{4} and e_{4}^{4} can be eliminated by p_{1B}, p_{3B}, p_{4B}. Specifically, for k = 3, 4,

e_{k}^{4} = e_{k}^{400}z_{1}^{4} + e_{k}^{310}z_{1}^{3}z̄_{1} + e_{k}^{301}z_{1}^{3}x_{2} + e_{k}^{220}z_{1}^{2}z̄_{1}^{2} + e_{k}^{211}z_{1}^{2}z̄_{1}x_{2}
+e_{k}^{202}z_{1}^{2}x_{2}^{2} + e_{k}^{130}z_{1}z̄_{1}^{3} + e_{k}^{121}z_{1}z̄_{1}^{2}x_{2} + e_{k}^{112}z_{1}z̄_{1}x_{2}^{2} + e_{k}^{103}z_{1}x_{2}^{3}
+e_{k}^{040}z̄_{1}^{4} + e_{k}^{031}z̄_{1}^{3}x_{2} + e_{k}^{022}z̄_{1}^{2}x_{2}^{2} + e_{k}^{013}z̄_{1}x_{2}^{3} + e_{k}^{004}x_{2}^{4},

so there are 30 coefficients, which can be eliminated by a transformation of the following form, a special case of (9.4):

(9.7) z̃_{1} = z_{1} + p_{1}^{11}z_{1}x_{2} + p_{1}
p_{1} = p_{1}^{210}z_{1}^{2}z_{2} + p_{1}^{030}z_{2}^{3} + p_{1}^{011}z_{2}z_{3}
z̃_{2} = z_{2} + p_{2}(z_{1}, z_{2}, z_{3}, z_{4})
z̃_{3} = z_{3} + 2\overline{p_{1}^{11}}z_{2}z_{3} + p_{3}
p_{3} = p_{3}^{001}z_{3}z_{4} + p_{3}^{200}z_{1}^{2}z_{4} + p_{3}^{110}z_{1}z_{2}z_{4} + p_{3}^{020}z_{2}^{2}z_{4} + p_{3}^{400}z_{1}^{4}
+p_{3}^{310}z_{1}^{3}z_{2} + p_{3}^{220}z_{1}^{2}z_{2}^{2} + p_{3}^{201}z_{1}^{2}z_{3} + p_{3}^{111}z_{1}z_{2}z_{3} + p_{3}^{021}z_{2}^{2}z_{3}
+p_{3}^{130}z_{1}z_{2}^{3} + p_{3}^{040}z_{2}^{4} + p_{3}^{002}z_{3}^{2}
z̃_{4} = z_{4} + (p_{1}^{11} + \overline{p_{1}^{11}})z_{2}z_{4} + p_{4}
p_{4} = p_{4}^{001}z_{3}z_{4} + p_{4}^{200}z_{1}^{2}z_{4} + p_{4}^{110}z_{1}z_{2}z_{4} + p_{4}^{020}z_{2}^{2}z_{4} + p_{4}^{400}z_{1}^{4}
+p_{4}^{310}z_{1}^{3}z_{2} + p_{4}^{220}z_{1}^{2}z_{2}^{2} + p_{4}^{201}z_{1}^{2}z_{3} + p_{4}^{111}z_{1}z_{2}z_{3} + p_{4}^{021}z_{2}^{2}z_{3}
+p_{4}^{130}z_{1}z_{2}^{3} + p_{4}^{040}z_{2}^{4} + p_{4}^{002}z_{3}^{2}.

The quantity p_{1} does not depend on z_{4} and has three out of six weight 3 terms, omitting z_{1}^{3}, z_{1}z_{2}^{2}, and z_{1}z_{3}. The quantities p_{3} and p_{4} have all 13 weight 4 terms. Together with the previously mentioned p_{1}^{11}, this gives exactly 30 unknowns, enough to find the following solution that eliminates all the quartic terms from (9.5), (9.6):

(9.8)  p_{1}^{11} = \frac{1}{2}\overline{e_{3}^{013}} − e_{4}^{013}
p_{1}^{011} = e_{4}^{031}
p_{1}^{210} = \frac{1}{2}\overline{e_{3}^{031}}
p_{1}^{030} = \frac{1}{2}\overline{e_{3}^{013}}
p_{3}^{201} = −e_{3}^{220}
p_{3}^{111} = −e_{3}^{121}
p_{3}^{001} = −e_{3}^{130}
p_{3}^{020} = −e_{3}^{112}
p_{3}^{040} = −e_{3}^{004}
p_{3}^{200} = −e_{3}^{310}
p_{3}^{002} = −e_{3}^{040}
p_{3}^{400} = −e_{3}^{400}
p_{3}^{110} = −e_{3}^{211} + 2\overline{e_{4}^{031}}
p_{3}^{130} = −e_{3}^{103}
p_{3}^{310} = −e_{3}^{301}
p_{3}^{220} = −e_{3}^{202}
p_{3}^{021} = −e_{3}^{022} + \frac{1}{4}(e_{3}^{013})^{2} − e_{3}^{013}\overline{e_{4}^{013}} + (\overline{e_{4}^{013}})^{2}
p_{4}^{220} = −e_{4}^{202}
p_{4}^{200} = −e_{4}^{310}
p_{4}^{001} = −e_{4}^{130}
p_{4}^{310} = −e_{4}^{301} + \overline{e_{4}^{031}}
p_{4}^{002} = −e_{4}^{040}
p_{4}^{400} = −e_{4}^{400}
p_{4}^{110} = \frac{1}{2}\overline{e_{3}^{031}} − e_{4}^{211}
p_{4}^{040} = −e_{4}^{004}
p_{4}^{111} = −e_{4}^{121} + \frac{1}{2}e_{3}^{031}
p_{4}^{201} = −e_{4}^{220}
p_{4}^{130} = −e_{4}^{103} + \frac{1}{2}e_{3}^{013}
p_{4}^{021} = −e_{4}^{022}
p_{4}^{020} = −\frac{1}{2}e_{4}^{013}e_{3}^{013} + e_{4}^{013}\overline{e_{4}^{013}} + \frac{1}{4}\overline{e_{3}^{013}}e_{3}^{013} − \frac{1}{2}\overline{e_{3}^{013}}\overline{e_{4}^{013}} − e_{4}^{112}.

The non-linear solutions for p_{3}^{021} and p_{4}^{020} reflect the appearance of non-linear terms in (9.5) and (9.6). The quartic terms can also be eliminated from E_{2} using a weight 4 function p_{2}, as in Section 8.2.

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Department of Mathematical Sciences, Purdue University Fort Wayne, 2101 E. Coliseum Blvd., Fort Wayne, IN 46805-1499

   E-mail address: CoffmanA@pfw.edu