  A $d$-dimensional tensor $A$ of format $n\times n\times \cdots \times n$
defines naturally a rational map $\Psi$ 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 $\Psi$. 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.
