Formulas for the eigendiscriminants of ternary and quaternary forms

Abstract

A d-dimensional tensor A of format defines naturally a rational map Ψ from the projective space to itself and its eigenscheme is then the subscheme of of fixed points of Ψ. The eigendiscriminant is an irreducible polynomial in the coefficients of A that vanishes for a given tensor if and only if its eigenscheme is singular. In this paper, we contribute two formulas for the computation of eigendiscriminants in the cases n = 3 and n = 4. In particular, by restriction to symmetric tensors, we obtain closed formulas for the eigendiscriminants of plane curves and surfaces in as the ratio of some determinants of resultant matrices.