Repository containing the computations of the paper Inverting catalecticant matrices of ternary quartics by Laura Brustenga i Moncusí, Elisa Cazzador and Roser Homs.
The code we present here involves two essential steps to understand the relation between the degree of the reciprocal variety of Cat(2,3) and the ML-degree of the model represented by Cat(2,3).
Computing the ML-degree (Section 4)
Julia code in MLDegreeTernaryQuartics.jl, using the package LinearCovarianceModels.jl, provides an efficient procedure to compute the ML-degree of Cat(2,3). This value is 36. In fact, this can be used to compute the ML-degree of any space Cat(k,n+1) of catalecticant matrices of (n+1)-ary forms of degree 2k.
The Macaulay2 code in MLdegree.m2 presents a symbolic aproach to the same calculation. However, it only provides an actual answer for spaces of catalecticant matrices associated to binary forms of small degree.
Understanding the locus of rank 2 matrices (Lemma 3.4 and Proposition 4.4)
Macaulay2 computations in Rank2Locus.m2 allow us to prove that the dimension of the rank 2 locus of the inversion map is exactly 12 (Lemma 3.4) and that it intersects the orthogonal of Cat(2,3). Therefore, the degree of the reciprocal variety of Cat(2,3) has 36 as a strict lower bound.