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Formulas for the eigendiscriminants of ternary and quaternary forms

A $d$-dimensional tensor $A$ of format $n\times n\times \cdots \times n$ defines naturally a rational map $Ψ$ from the projective space $\mathbb{P}^{n-1}$ to itself and its eigenscheme is then the subscheme of $\mathbb{P}^{n-1}$ 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 $\mathbb{P}^3$ as the ratio of some determinants of resultant matrices.