Evaluate Salmon's invariants

# Setup

Let $V \subset \mathbb{C}[x,y,z,w]$ be the space of all homogeneous cubic polynomials. The action of $\mathrm{SL}(4)$ on $x,y,z,w$ extends to an action on $V$. For each cubic monomial $m$ define the function $c_m \colon V \to \mathbb{C}$ such that $c_m(f)$ is the coefficient of the monomial $m$ appearing in $f$. The coordinate ring of $V$ is $R = \mathbb{C} [c_m \mid \text{m is a cubic monomial}]$.

Salmon (1860) determined 6 fundamental polynomials (of degrees 8,16,24,32,40,100) in $R$ that are invariant with respect to the action of $\mathrm{SL}(4)$. The square of the last is a polynomial in the first five. It was later shown by Beklemishev (1982) that the ring of invariants $R^{\mathrm{SL}(4)}$ is generated by these 6 polynomials.

# Problem

Write the 5 invariants of Salmon explicitly as polynomials in $c_m$. Otherwise, given a cubic form $f \in V$ devise an algorithm to evaluate these 6 polynomials on $f$. Same problem for the tropicalizations.

The moduli space (of isomorphism classes of) cubics is isomorphic to the weighted projective space $\mathbb{P}(1,2,3,4,5)$, see Reinecke's bachelor thesis for a comprehensive proof of this classical fact. The map $V \to \mathbb{P}(1,2,3,4,5)$ is obtained by evaluating the first 5 of Salmon's invariants. The value of the sixth invariant is, in principle, determined by the first five and therefore it does not play a role here.

Since a general cubic surface can be brought into the pentahedral form

$\{a_0x_0^3+a_1x_1^3+a_2x_2^3+a_3x_3^3+a_4x_4^3=0\} \cap \{ x_0+x_1+x_2+x_3+x_4 = 0\}$,

one can use the algorithm 3.1 in Decomposition to compute the coefficients $a_0,...,a_4$. Let $\sigma_1,...,\sigma_5$ be the first five elementary symmetric polynomials in a_i's. The Salmon's invariants are given by $I_8= \sigma_4^2-4 \sigma_3 \sigma_5, I_{16}=\sigma_5^3\sigma_1, I_{24}=\sigma_5^4\sigma_4, I_{32}=\sigma_5^6\sigma_2, I_{40}=\sigma_5^8$.

# Major progress

Stephen Elsenhans wrote a Magma code to evaluate the invariants of Salmon. The function is built-in to Magma with instructions to use available here. The code itself is readable by all Magma users in the local Magma directory:

magma/package/Geometry/SrfDP/inv_cub_surf.m

This solves the problem of identifying if two cubics are isomorphic.

People who are interested in this problem can still try to determine the invariants explicitly as polynomials in terms of the (indeterminate) coefficients of cubics. A more accessible goal might be to figure out the Newton polytope of each of these polynomials.

The explicit determination of the invariants might be possible using the strategy outlined by Elsenhans , preprint available here. There is a comprehensive study of invariants of cubic surfaces written by Elsenhans and Jahnel , with preprint available here.

Bibliography
1. A.-S. Elsenhans: Explicit computations of invariants of plane quartic curves, Journal of Symbolic Computation 68, Part 2 (2015) 109-115.
2. A.-S. Elsenhans and J. Jahnel: Moduli spaces and the inverse Galois problem for cubic surfaces, Transactions of the AMS 367 (2015) 7837-7861.