Spectral linear matrix inequalities (Q2237373)

A homogeneous polynomial \(h\in\mathbb{R}[x_1, \cdots,x_n]\) is said to be \textit{hyperbolic } with respect to \(e\in\mathbb{R}^n\) if \(h(e)>0\) and if for every \(a\in\mathbb{R}^n\) the univariant polynomial \(h(te-a)\) in \(t\) has only real roots. The \textit{hyperbolicity cone} \(C(h, e)\) of \(h\) at \(e\) is the set of all \(a\in\mathbb{R}^n\) such that all zeros of \(h(te-a)\) are nonnegative. Assume that \(A_i~(i=1,\dots, n)\) are real symmetric matrices with the property that \(A(e)\) is positive definite, where \[ A(x)=x_1A_1+\cdots+x_nA_n\,. \] An instructive example of a polynomial that is hyperbolic with respect to \(e\) is given by \(\det A(x)\). Then the hyperbolicity cone is defined by a linear matrix inequality \[ C(\det A(x), e)=\{a\in\mathbb{R}^n~:~A(a) ~\text{is positive definite}\}\,. \] Such sets are called \textit{spectrahedral cones}. The generalized Lax conjecture states that hyperbolicity cones are spectrahedral. There is positive and negative evidence for this conjecture. Using Rolle's theorem, \[ D_e^kh:=\left(\sum_{i=1}^{n}e_i\cdot \frac{\partial}{\partial x_i}\right)^kh \] is hyperbolic with respect to \(e\) for all \(k\le \deg(h)\) if \(h\) is. These hyperbolic polynomials are said to be \textit{Renegar derivatives}. In some special case, the generalized Lax conjecture implies that the hyperbolicity cone \(C\left(D_e^k\det A(x), e\right)\) is spectrahedral. Many mathematicians proved that if \(A(x)\) is a diagonal matrix and \(k=1\), then the Lax conjecture holds true and then proved this result for non-diagonal \(A(x)\). In this paper the author extends this result to arbitrary \(k\).