Automorphisms of hyperbolic polynomials

\(\)The pair \( (p,e) \) is hyperbolic if \( p : \mathbb{R}^{n} \to \mathbb{R} \) is a homogeneous polynomial, if \( e \in \mathbb{R}^{n} \), if \( p(e) > 0 \), and if the roots of \( t \mapsto p(te – x) \) are real for all \( x \in \mathbb{R}^{n} \). In that case, the \( x \) for whom these roots are nonnegative form a closed convex cone \( \Lambda_{p,e} \) called a hyperbolicity cone. Many cones used in optimization are hyperbolicity cones. For example, all homogeneous and symmetric cones are hyperbolicity cones.

In this setting we borrow a definition of “automorphism” wherein every automorphism of \( (p,e) \) is an automorphism of \( \Lambda_{p,e} \) satisfying some additional properties. When \( \Lambda_{p,e} \) is pointed and \( p,e \) are chosen judiciously, these automorphisms are characterized by

\(
\begin{equation*}
\text{Aut}(p,e)
=
\text{Aut}(\Lambda_{p,e})_{e}
=
\text{Aut}(\Lambda_{p,e}) \cap \text{Isom}(p,e).
\end{equation*}
\)

Here \( \text{Aut}(\Lambda_{p,e})_{e} \) is the subgroup of \( \text{Aut}(\Lambda_{p,e}) \) that fixes \( e \), and \( \text{Isom}(p,e) \) is the isometry group with respect to a particular norm. This generalizes an important result for symmetric cones, and specializes to homogeneous ones. Subsequently we clarify the relationship between two constructions of homogeneous cones, and find the degree of the hyperbolic polynomial \( p \) most commonly used with them.

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