# Formulas for the eigendiscriminants of ternary and quaternary forms

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
Abstract : 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 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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.inria.fr/hal-02881339
Contributor : Laurent Busé <>
Submitted on : Thursday, June 25, 2020 - 3:55:29 PM
Last modification on : Thursday, November 26, 2020 - 3:50:03 PM

### Identifiers

• HAL Id : hal-02881339, version 1
• ARXIV : 2006.10975

### Citation

Laurent Busé. Formulas for the eigendiscriminants of ternary and quaternary forms. 2020. ⟨hal-02881339⟩

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