5 problems

Let $p\in\mathbb{R}[\mathbf{x}]$ be a real-zero (RZ) polynomial with $p(\mathbf{0})=1$. For a polynomial $f$, write $\operatorname{rcs}(f)$ for its rigidly convex set. Generalized…

A homogeneous polynomial $h\in\mathbb{R}[x_1,\ldots,x_n]$ is hyperbolic with respect to $e\in\mathbb{R}^n$ if $h(e)>0$ and, for every $a\in\mathbb{R}^n$, the polynomial $h(te-a)$ h…

A hyperbolicity region is the connected component containing the reference point of the set where a real homogeneous polynomial is nonzero. A spectrahedron is a set defined by a li…

Let $M\in\mathbb R^{p\times q}_{\geq 0}$ have rank $k+1$ and satisfy … Let $P$ and $Q$ be the two polytopes corresponding to $M$ in the nested-polytope representation. Spectrahedro…

Let $A,B,C \in \operatorname{Sym}_r(\mathbb{R})$ be real symmetric matrices of size $r$, and set … Assume that $f$ is a polynomial of degree at least three. Singularity conjecture.…