Page 47 - Geometric Modeling and Algebraic Geometry

44 F. Aries et al.
translates into a system of polynomial equations of degree 12 in the coefficients of f.
The second case is isolated by remarking (by mere evaluation on the representative)
that Φ 11 vanishes identically on the orbit of (2.33), and not on the six orbits of para-
meterizations giving true quartics. This gives another system of equations of degree
12. The third case is detected by the vanishing of the maximal minors of the 4 × 6
matrix of the coefficients of the f i ’s. This is a system of equations of degree 4.
Now we evaluate the covariants of our collection on the representatives of the
six orbits in U, and find that Φ 14 makes possible the discrimination. Let us explain
how: Φ 14 (f) is a quadratic form on R . Let M(f) be its matrix. Then the inertia of
3
Φ 14 (f) is the following ordered pair: (number of positive eigenvalues of M, number
of negative eigenvalues of M(f)). The covariance property of Φ 14 can be stated as
follows:

Φ 14 (ρ ◦ f ◦ θ −1 ) (λ)=det(θ) −6 det(ρ) (Φ 14 (f)) (λ ◦ θ −1 )
2
Because the powers of the determinants involved in the formulas are even, the inertia
of Φ 14 (f) takes only one value on each orbit of F under G. As a consequence, it
defines a function on U. Table 2.3 shows its values.

Orbit of [f] inertia of Φ 14(f) equations and inequalities
Ii (0, 3) A 3 > 0 ∧ A 2 > 0 ∧ A 1 > 0
Iii (2, 1) A 3 > 0 ∧ (A 2 ≤ 0 ∨ A 1 ≤ 0)
Iiii (1, 2) A 3 < 0
IIi (1, 1) A 3 =0 ∧ A 2 < 0
IIii (0, 2) A 3 =0 ∧ A 2 > 0
III (0, 1) A 3 = A 2 =0

Table 2.3. Discrimination between the orbits.

It is already an interesting result that the inertia of one quadratic form attached
to f is enough to discriminate between the six orbits in U.
Now, we want to go further and define the orbits by equations and inequalities.
For this we introduce the characteristic polynomial of M(f):

det(t · I − M(f)) = t + A 1 (f) t + A 2 (f)t + A 3 (f). (2.34)
2
3
Any condition on the inertia can be translated into equations and inequalities involv-
ing the coefficients of A i (f). The formulas obtained are presented in the last column
of Table 2.3. They are obtained trivially, except those for discriminating between
inertias (2, 1) and (0, 3), that makes use of Descartes’ law of signs [5].
Note that A 3 (f) is a non–trivial invariant of degree 24. Thus it should be pro-
portional to ∆. One finds (by evaluation on the representative of Orbit Ii) that the
coefficient of proportionality is positive. Thus in the sign conditions above, we are
allowed to substitute A 3 with ∆.