Syzygy between covariants of pairs of ternary quadratic forms

Question

In the book Nonlinear Computational Geometry, Page 208 (or page 15 of the online version on the author's website: http://www.loria.fr/~petitjea/papers/imaconics.pdf), Remark 5.1, Petitjean states that

"The joint covariants $Q_S,Q_T ,Q_U$ and $G$ are not algebraically independent. They satisfy a fundamental syzygy which, when $Q_S = Q_T = 0$, reduces to $\displaystyle G^2 = Q_U^3.$"

He stated this without a clear reference. He cited something related in Grace and Young's book The Algebra of Invariants, a fairly old book. I was only able to find a much older version (1873 instead of the 1908 version cited by Petitjean) and I do not see an identity in the relevant chapter which could be the full version of the syzygy mentioned above. This could however be my inability to read the notation in that book as it is quite alien to me.

Does anyone have a more modern reference where this syzygy is written down explicitly?

Answer 1

I emailed Professor Sylvain Petitjean regarding the syzygy, and he replied with a link to Salmon's classical treatise on invariant theory. To state the result, we recall that for a given pair $(A,B)$ of ternary quadratic forms, the action of $\operatorname{GL}_3(\mathbb{Z})$ on $(A,B)$ yields four invariants $a_3, a_2, a_1, a_0$, given as the coefficients of the binary cubic form

$$\displaystyle f(x,y) = \det(Ax + By) = a_3 x^3 + a_2 x^2 y + a_1 xy^2 + a_0 y^3.$$

Then by Salmon (https://archive.org/details/treatiseonconics00salmuoft), page 362, one has:

$$\displaystyle G^2 = Q_U^3 - Q_U^2(a_2 Q_S + a_1 Q_T) + Q_U (a_0 a_2 Q_S^2 + a_1 a_3 Q_T^2) + Q_U Q_S Q_T(a_2 a_1 - 3 a_3 a_0)$$ $$\displaystyle -a_0^2 a_3 Q_S^3 - a_0 a_3^2 Q_T^3 + a_0(2a_3 a_1 - a_2^2)Q_S^2 Q_T + a_3(2a_0 a_2 - a_1^2)Q_S Q_T^2.$$