Kippenhahn's Theorem for joint numerical ranges and quantum states

TL;DR

This paper extends Kippenhahn's Theorem to joint numerical ranges of Hermitian matrices, providing a geometric framework for analyzing quantum states through convex hulls of semi-algebraic sets.

Contribution

It introduces a new geometric approach to study joint numerical ranges and dual cones, linking algebraic curves with quantum state analysis.

Findings

Joint numerical range is the convex hull of a semi-algebraic set.

Bases of dual cones are closed under linear operations.

Provides a new geometric method for quantum state analysis.

Abstract

Kippenhahn's Theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many Hermitian matrices is similarly the convex hull of a semi-algebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove that the class of bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states.

Equations

$$\mathcal{B}={\mathcal{B}}_d=\{\rho \in H_d:\rho ⪰0,\mathrm{tr}(\rho )=1\},$$

$$W=W_{A_1,A_2,\dots ,A_n}=\{(⟨\rho ,A_1⟩,\dots ,⟨\rho ,A_n⟩):\rho \in {\mathcal{B}}_d\},$$

$$W^{\sim }=W_{A_1,A_2,\dots ,A_n}^{\sim }=\{(⟨\psi |A_1\psi ⟩,\dots ,⟨\psi |A_n\psi ⟩):\psi \in {\mathbb{C}}^d,⟨\psi |\psi ⟩=1\},$$

$$p=\det (x_0\text{mathds}1+x_1{A}_1+\cdots +x_n{A}_n)\in \mathbb{R}[x_0,x_1,\dots ,x_n],$$

$$\mathcal{V}(p)=\{x\in {\mathbb{P}}^n\mid p(x)=0\}.$$

$$T=\{y_1+i{y}_2\mid y_1,y_2\in \mathbb{R},(1:y_1:y_2)\in \mathcal{V}(p{)}^{\ast }\}\subset \mathbb{C},$$

$$T_i=\left\{(y_1,\dots ,y_n)\in ({\mathbb{R}}^n{)}^{\ast }\mid (1:y_1:\cdots :y_n)\in (X_i^{\ast }{)}_{\mathrm{reg}}\right\},i=1,\dots ,r,$$

$${\rho }_x=\frac{e^{x_1{A}_1+\cdots +x_n{A}_n}}{\mathrm{tr}{e}^{x_1{A}_1+\cdots +x_n{A}_n}}\in {\mathcal{B}}_d,x=(x_1,\dots ,x_n{)}^T\in {\mathbb{R}}^n.$$

$$\int da{\mathcal{W}}_{\rho }(a_1,\dots ,a_n)f(x_1{a}_1+\cdots +x_n{a}_n)=⟨\rho ,f(x_1{A}_1+\cdots +x_n{A}_n)⟩$$

$$\Pi =\{⟨\psi \otimes \phi |A_i\psi \otimes \phi {⟩}_{i=1}^n\mid \psi ,\phi \in {\mathbb{C}}^2,⟨\psi |\psi ⟩=⟨\phi |\phi ⟩=1\},$$

$${\Pi }_{A_1,A_2,\dots ,A_n}^{\mathrm{sym}}=\{⟨\psi \otimes \psi |A_i\psi \otimes \psi {⟩}_{i=1}^n\mid \psi \in {\mathbb{C}}^2,⟨\psi |\psi ⟩=1\}$$

$$S^{\perp }=\{\ell \in V^{\ast }:\ell (x)=0\text{ for all }x\in S\},$$

$$S^{\vee }=\{\ell \in V^{\ast }:\ell (x)\ge 0\text{ for all }x\in S\},$$

$$S^{\circ }=\{\ell \in V^{\ast }:1+\ell (x)\ge 0\text{ for all }x\in S\}.$$

$$h_{\ell }(S)=\{x\in S:\ell (x)=1\},\ell \in V^{\ast },\ell \ne 0.$$

$$\delta^{\ast}(\ell|K)=\sup_{x\in K}\ell(x)=\left\{\begin{array}[]{ll}0&\text{if $\ell\in-K^{\vee}$,}\\ \infty&\text{if $\ell\not\in-K^{\vee}$,}\end{array}\right.\qquad\ell\in V^{\ast},$$

$$h_x(\mathrm{cc}(\mathrm{clos}(C)))=\mathrm{conv}(h_x(\mathrm{clos}(C)))=\mathrm{conv}(\mathrm{clos}(h_x(C)))$$

$$\displaystyle K^{\vee}=\operatorname{cc}\big{(}\operatorname{clos}(C_{1}\cup C_{2}\cup\dots\cup C_{r})\big{)}$$

$$\displaystyle\operatorname{h}_{e}(K^{\vee})=\operatorname{conv}\Big{(}\operatorname{clos}\big{(}\operatorname{h}_{e}(C_{1})\cup\operatorname{h}_{e}(C_{2})\cup\dots\cup\operatorname{h}_{e}(C_{r})\big{)}\Big{)}.$$

$$K_i=\mathrm{clos}(\mathrm{cc}(C_i))=\mathrm{cc}(\mathrm{clos}(C_i)),i=1,\dots ,r.$$

$$N_C:\mathcal{F}(C)\to \mathcal{F}(C^{\vee }),F\mapsto F^{\perp }\cap C^{\vee }.$$

$$N_C(F)=\{\ell \in V^{\ast }:\ell (y-x)\ge 0\forall y\in C\forall x\in F\},$$

$$N_C(F)=x^{\perp }\cap C^{\vee }=\{\ell \in V^{\ast }:\ell (y-x)\ge 0\forall y\in C\},$$

$$H=\mathcal{V}(\ell )\subset {\mathbb{P}}^n$$

$$\mathcal{V}(x)=\{y\in ({\mathbb{P}}^n{)}^{\ast }:x_0{y}_0+x_1{y}_1+\cdots +x_n{y}_n=0\}.$$

$$\{[H]\in ({\mathbb{P}}^n{)}^{\ast }:T_{x}X\subset H\text{ for some }x\in X_{\mathrm{reg}}\}.$$

$$q=(x_0,x_1,\dots ,x_n)M_q(x_0,x_1,\dots ,x_n{)}^T.$$

$$C{N}_0(X)=\{(x,[H])\in {\mathbb{P}}^n\times ({\mathbb{P}}^n{)}^{\ast }:x\in X_{\mathrm{reg}},T_{x}X\subset H\}.$$

$$U=\{(x,[H])\in C{N}_0(X):[H]\in (X^{\ast }{)}_{\mathrm{reg}}\}$$

$$U=C{N}_0(X)\cap C{N}_0(X^{\ast }).$$

Full text

1. Introduction

Let $H_{d}$ be the real vector space of Hermitian $d\times d$ matrices. We denote the set of density matrices by

[TABLE]

where $A\succeq 0$ means that $A\in H_{d}$ is positive semi-definite. The letter $\mathcal{B}$ underlines that the set $\mathcal{B}$ is a base of the cone of positive semi-definite matrices (see Section 5). We use the bilinear form $\langle A,B\rangle=\operatorname{tr}(AB)$, $A,B\in H_{d}$, to identify $H_{d}$ and its dual space. Let $A_{1},A_{2},\dots,A_{n}\in H_{d}$, $n\in\mathbb{N}$, and define

[TABLE]

a convex compact subset of the dual space $(\mathbb{R}^{n})^{\ast}$ to $\mathbb{R}^{n}$. The set $W$ has been called joint numerical range in operator theory, see Section 2 of [8] (also joint algebraic numerical range [43]).

Our motivation for this paper is matrix theory and quantum mechanics. Physicists call density matrices quantum states, as density matrices describe the physical states of a quantum system including all statistical properties [33]. Hence, our results contribute to the geometry of quantum states [2, 5]. Section 2 presents numerical range methods in quantum mechanics.

Perhaps more commonly, the term joint numerical range refers to

[TABLE]

where $\langle\varphi|\psi\rangle$ is the standard inner product of $\varphi,\psi\in\mathbb{C}^{d}$, see [14, 41]. A pure state is a projection $\rho$ onto the span of a unit vector $\psi\in\mathbb{C}^{d}$. Since $\langle\rho,A\rangle=\langle\psi|A\psi\rangle$ holds for all $A\in H_{d}$, the set $W^{\sim}$ is a linear image of the set of pure states. As the pure states are the extreme points of the set of density matrices $\mathcal{B}$, the joint numerical range $W$ is the convex hull of $W^{\sim}$.

Thus, $W=W^{\sim}$ holds when $W^{\sim}$ is convex. This is the case if $n=2$ by the Toeplitz-Hausdorff theorem [28, 59], whose 100th anniversary we celebrated at the time of writing. Note that $W_{A_{1},A_{2}}^{\sim}$ is the standard numerical range $W(A)=\{\langle\psi|A\psi\rangle:\psi\in\mathbb{C}^{d},\langle\psi|\psi\rangle=1\}\subset\mathbb{C}$ of $A=A_{1}+\operatorname{i}A_{2}$. Also, $W^{\sim}$ is convex if $n=3$ and $d\geq 3$ [1]. The convexity of $W^{\sim}$ is an open problem for $n>3$, see [26, 41]. Numerical ranges are generally nonconvex if the complex field is replaced with the skew field of the quaternions [50, p. 39].

Algebraic geometry has been employed to study numerical ranges since the 1930s. We consider the determinant

[TABLE]

where $\mathds{1}$ denotes the $d\times d$ identity matrix, and the complex projective hypersurface

[TABLE]

If $n=2$, then $\mathcal{V}(p)\subset\mathbb{P}^{2}$ is an algebraic curve. Murnaghan [45] showed that the eigenvalues of the matrix $A_{1}+\operatorname{i}A_{2}$ are the foci of the curve

[TABLE]

where $X^{\ast}\subset(\mathbb{P}^{n})^{\ast}$ denotes the dual variety parametrizing hyperplanes tangent to a variety $X\subset\mathbb{P}^{n}$ (cf. Section 4 for more details). Kippenhahn recognized the meaning of the convex hull of the curve $T$.

Theorem 1.1** (Kippenhahn [38]).**

The numerical range of $A_{1}+\operatorname{i}A_{2}$ is the convex hull of the curve $T$, in other words, $W(A_{1}+\operatorname{i}A_{2})=\operatorname{conv}(T)$.

1.1 is a well-known tool in matrix analysis [4, 19, 23, 34]. The curve $T\subset\mathbb{C}$ is called the Kippenhahn curve or the boundary generating curve of the numerical range $W(A_{1}+\operatorname{i}A_{2})$. The curve $T$ has proven useful in classifications of numerical ranges of $3$-by-$3$ matrices [35, 38] and $4$-by-$4$ matrices [10, 15], and it has been studied for special matrices [16, 21, 22, 42].

We sketch a proof of 1.1 in a manner that may help to explain the geometry behind the proof of 4.5 later on: Consider the convex set $S=\{(x_{1},x_{2})^{T}\in\mathbb{R}^{2}\mid\mathds{1}+x_{1}A_{1}+x_{2}A_{2}\succeq 0\}$ (a spectrahedron). Assume for simplicity that $S$ is compact, $X=\mathcal{V}(p)$ is smooth, and that the degree $d$ of $p$ is even. The curve $X$ is hyperbolic, i.e. its real points consist of $\frac{d}{2}$ nested ovals in the real projective plane. The innermost oval is the boundary of $S$. All but finitely many points of the dual curve $X^{\ast}$ correspond to simple tangent lines to $X$. The set of real points of the dual curve $X^{\ast}$ (of degree $d(d-1)$) again consists of $\frac{d}{2}$ nested connected components, together with at most finitely many isolated real (singular) points. The tangent lines to the boundary of $S$ now correspond to the outermost oval of $X^{\ast}$, since all other tangent lines to $X$ do not pass through $S$ (see 4.3). The outermost oval therefore bounds the convex dual $S^{\circ}$ of $S$, which is exactly the numerical range $W(A_{1}+\operatorname{i}A_{2})$. The claim of Kippenhahn’s theorem follows if we can show that none of the isolated real singularities of $X^{\ast}$ lie outside of $W$; see 4.9.

Chien and Nakazato [14] provided a more rigorous proof of 1.1 compared to Kippenhahn’s. They also found a triple of Hermitian $3\times 3$-matrices for which the literal analogue of 1.1 fails in dimension $n=3$. We will see that the last part of the above sketch in the plane case, the position of singular points of the dual curve, is exactly what causes the failure of the theorem in higher dimensions. This can also be seen in the counterexample by Chien and Nakazato, 6.4.

By removing all singular points from the projective dual variety $\mathcal{V}(p)^{\ast}$ and taking Euclidean closure, we will prove that a modified version of 1.1 is valid in all dimensions. Let $X_{1},\dots,X_{r}$ denote the irreducible components of the hypersurface $\mathcal{V}(p)$. We consider the set $(X_{i}^{\ast})_{\mathrm{reg}}$ of the regular points of the dual variety $X_{i}^{\ast}$, the set

[TABLE]

and the Euclidean closure $T^{\sim}=\operatorname{clos}(T_{1}\cup\cdots\cup T_{r})$ of the union $T_{1}\cup\cdots\cup T_{r}$. Our main result is as follows.

Theorem 1.2**.**

The joint numerical range $W$ is the convex hull of $T^{\sim}$.

Remark 1.3**.**

1.2 implies that the set $T^{\sim}$ contains all extreme points of the compact, convex set $W$. We show in the proof of 4.5 that $T^{\sim}$ contains the exposed points of $W$, and hence all of the extreme points by a limiting argument (Straszewicz’s Theorem). But just as in Kippenhahn’s original theorem, $T^{\sim}$ is not necessarily contained in the boundary of $W$​, only in $W$. 2. 2)

We point out that 4.5, which implies the theorem stated here, holds more generally for hyperbolic hypersurfaces rather than just determinantal hypersurfaces. While this makes no difference in the plane, by the Helton-Vinnikov theorem, the statement is indeed more general in higher dimensions, and the proof relies purely on the real geometry of hyperbolic polynomials. 3. 3)

The joint numerical range $W$ is a semi-algebraic set as it is a linear image of the semi-algebraic set $\mathcal{B}$ by quantifier elimination (see e.g. [7, Thm. 2.2.1]). The set $T_{1}\cup\cdots\cup T_{r}$ and hence its Euclidean closure $T^{\sim}$ are semi-algebraic sets as well. 4. 4)

1.2 not only describes a semi-algebraic set that contains the extreme points of $W$ but more precisely the Zariski closure of the set of extreme points: The union of the dual varieties $X_{i}^{\ast}$ of the irreducible components $X_{i}$ of the algebraic boundary of the hyperbolicity cone of $p$ is the Zariski-closure of the set of extreme points of $W$, see 4.15 for details.

We organize the article as follows. Section 3 collects preliminaries from convex geometry and real algebraic geometry. Section 4 presents a detailed discussion of the remarkable fact, proved by the second author in [53], that the dual convex cone $C^{\vee}$ to a hyperbolicity cone $C$ is the closed convex cone generated by a particular semi-algebraic set. This implies that every base of $C^{\vee}$ is the closed convex hull of a section of that semi-algebraic set. The same is true for linear images of the bases as we show in Section 5, because (up to isomorphism) they are bases of dual convex cones to sections of $C$, which are hyperbolicity cones themselves. Returning to the cone of positive semi-definite matrices in Section 6, we obtain a proof of 1.2 and discuss examples. We analyze the case $n=2$ separately in 4.9, which yields a proof of Kippenhahn’s original result as stated in 1.1.

2. Connections to Quantum Mechanics

Physicists refer to linear images of certain subsets of the set of quantum states $\mathcal{B}_{d}$ as numerical ranges. Often they consider images under a map $\mathcal{B}_{d}\to\mathbb{R}^{n}$, $\rho\mapsto\langle\rho,A_{i}\rangle_{i=1}^{n}$, where $A_{1},A_{2},\dots,A_{n}\in H_{d}$ are Hermitian matrices. We discuss examples where algebraic geometry could help solving problems of quantum mechanics in the context of numerical ranges. In this paper, we ignore numerical ranges outside the pattern of linear images of subsets of $\mathcal{B}_{d}$, for example higher-rank numerical ranges [17].

2.1. Linear images of the set of all quantum states

The joint numerical range $W_{A_{1},A_{2},\dots,A_{n}}$, as defined in Equation 2, appears in problems of experimental and theoretical physics.

The geometry of the joint numerical range has been of direct interest to the experimentalists Xie et al. [61]. Their drawings of data from photonic experiments show ellipses on the boundary of the joint numerical range $W_{A_{1},A_{2},A_{3}}$ of three $3$-by-$3$ matrices $A_{1},A_{2},A_{3}\in H_{3}$, clearly in agreement with the classification: the exposed faces of positive dimensions are ellipses and segments assembling one of ten configurations [56]. The possibility to carry out experiments with $4$-level quantum systems calls for a similar classification of $W_{A_{1},A_{2},A_{3}}$ for $4$-by-$4$ matrices $A_{1},A_{2},A_{3}\in H_{4}$.

Using 1.1 and analyzing the Kippenhahn curve, Kippenhahn [38] obtained a classification of the numerical range $W(A_{1}+\operatorname{i}A_{2})=W_{A_{1},A_{2}}$ in terms of flat portions on the boundary of $W_{A_{1},A_{2}}$ for all Hermitian $3$-by-$3$ matrices $A_{1},A_{2}\in H_{3}$, see also [35]. A similar approach has been taken for $4$-by-$4$ matrices [10, 15]. We are invited to describe the exposed faces of positive dimensions of the joint numerical range $W_{A_{1},A_{2},A_{3}}$, employing 1.2 and studying the semi-algebraic set $T^{\sim}$, whose convex hull is $W_{A_{1},A_{2},A_{3}}$. This should reproduce the classification of the set $W_{A_{1},A_{2},A_{3}}$ for $3$-by-$3$ matrices [56] and lead to a classification for $4$-by-$4$ matrices.

Quantum thermodynamics [62] describes equilibrium states with multiple conserved quantities $A_{1},A_{2},\dots,A_{n}$ in terms of generalized Gibbs states

[TABLE]

The “boundary at infinity” ($|x|\to\infty$) to the manifold $\{\rho_{x}:x\in\mathbb{R}^{n}\}$ may not be closed in the Euclidean topology if the matrices $A_{1},A_{2},\dots,A_{n}$ fail to commute [60]. The physical meaning of this topological problem remains mysterious [12]. Mathematically, the discontinuity depends on the geometry of the joint numerical range $W_{A_{1},A_{2},\dots,A_{n}}=\operatorname{conv}(T^{\sim})$ [12, 51, 54]. Hence, the semi-algebraic set $T^{\sim}$ explains infinitesimal properties of the manifold of generalized Gibbs states near the boundary at infinity.

A connection between functional analysis and algebraic geometry awaits further investigation. The Wigner distribution of a quantum state $\rho\in\mathcal{B}_{d}$ with respect to $A_{1},A_{2},\dots,A_{n}$ is the tempered distribution ${\mathcal{W}}_{\rho}$ on $\mathbb{R}^{n}$ that satisfies

[TABLE]

for all $x\in\mathbb{R}^{n}$ and all infinitely differentiable functions $f:\mathbb{R}\to\mathbb{C}$. The Wigner distribution is a common tool in quantum optics and theoretical physics. Schwonnek and Werner [52] showed that the distribution ${\mathcal{W}}_{\rho}$ is compactly supported on the joint numerical range $W_{A_{1},A_{2},\dots,A_{n}}$ and that the singularities of ${\mathcal{W}}_{\rho}$ lie in the semi-algebraic set $T^{\sim}$.

2.2. Linear images of subsets of the set of quantum states

Linear images of semi-algebraic subsets of $\mathcal{B}_{d}$ are amenable to algebraic geometry as well. We mention examples relevant to quantum mechanics.

Algebraic geometry [18] has proven helpful in the theory of pure state tomography [29], the reconstruction of a pure state $\rho\in\mathcal{B}_{d}$ from its expected value tuple $\langle\rho,A_{i}\rangle_{i=1}^{n}\in W^{\sim}$ in the linear image $W^{\sim}$ of the set of pure states defined in Equation 3. A different topic, for example, in density functional theory [11], is describing the extreme points of the joint numerical range $W$. Both $W^{\sim}$ and $T^{\sim}$ are semi-algebraic subsets of $W$ that contain the extreme points of $W$. The set $T^{\sim}$ is especially suitable to study the extreme points of $W$, see 4.15.

Many-particle systems are fascinating due to interaction and correlation between the units. The simplest example in the quantum domain is the two-qubit system with Hilbert space $\mathbb{C}^{4}=\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Physicists [13, 24] have studied the joint product numerical range of $A_{1},A_{2},\dots,A_{n}\in H_{4}$,

[TABLE]

a linear image of the set of product states $\sigma\otimes\tau$, where $\sigma,\tau\in\mathcal{B}_{2}$ are pure states. The convex hull of the product states is the set of separable states, the states that lack the genuine quantum correlation called entanglement [2, 5]. Hence, the set $\Pi$ and its convex hull allow us to study quantum correlations.

Two-qubit density matrices offer insights into statistical mechanics. As per the quantum de Finetti theorem [40, 55], the two-particle marginals of an infinite bosonic qubit-system are convex combinations of symmetric product states $\sigma\otimes\sigma$, where $\sigma\in\mathcal{B}_{2}$ is a pure state. The ground state energy of an energy operator with two-party interactions on an infinite bosonic qubit-system is the distance of the origin from a supporting hyperplane to the set

[TABLE]

for suitable matrices $A_{1},A_{2},\dots,A_{n}\in H_{4}$. Notably, ruled surfaces on the boundary of the convex hull of $\Pi^{\mathrm{sym}}_{A_{1},A_{2},A_{3}}\subset\mathbb{R}^{3}$ are expressions of phase transitions [13, 63]. An analogue to 1.2 regarding the set $\Pi^{\mathrm{sym}}_{A_{1},A_{2},A_{3}}$ would be helpful for the analysis of bosonic qubit-systems.

3. Preliminaries

We collect terms, basic results, and references to the literature regarding convex geometry and (real) algebraic geometry.

3.1. Convex Geometry

We discuss various notions of cones in a finite-dimensional real vector space $V$: cones (which may be nonconvex), convex cones, and normal cones. Additionally, we define affine cones over complex projective varieties in Section 3.2. As a general reference for convex geometry, we recommend [3, 49].

A subset $C$ of $V$ is a cone if $\lambda x\in C$ whenever $\lambda>0$ and $x\in C$. A subset $K$ of $V$ is a convex cone if $K$ is a nonempty convex set and if $\lambda x\in K$ whenever $\lambda\geq 0$ and $x\in K$. The affine hull $\operatorname{aff}(S)$, convex hull $\operatorname{conv}(S)$, cone, convex cone $\operatorname{cc}(S)$, and closed convex cone generated by a subset $S\subset V$ is the smallest affine space, convex set, cone, convex cone, and closed convex cone, respectively, containing $S$.

A subset $B\subset V$ is a base of a cone $C\subset V$ if $B$ is the intersection of $C$ with an affine hyperplane, $0\not\in\operatorname{aff}(B)$, and for all nonzero points $x\in C$ there exists $y\in B$ and $\lambda>0$ such that $x=\lambda y$. A convex cone $K\subset V$ is pointed if $K\cap(-K)=\{0\}$, i.e. if $K$ contains no lines.

We denote the dual vector space of $V$ by $V^{\ast}$. The annihilator of a subset $S\subset V$ is

[TABLE]

the dual convex cone to $S$ is

[TABLE]

and the dual convex set to $S$ is

[TABLE]

We denote intersections of $S$ with affine hyperplanes avoiding the origin by

[TABLE]

Lemma 3.1**.**

Let $C\subset V$ be a cone and let $B\neq\emptyset$ be a base of $C$. Then there exists a nonzero functional $\ell\in V^{\ast}$ such that $B=\operatorname{h}_{\ell}(C)$. If $C$ admits a compact base, then $C\cup\{0\}$ is closed.

Proof.

Let $B$ be a nonempty base of the cone $C$ and let $X$ be the linear span of $C$. It follows from the definition of a base that there is a nonzero linear functional $\hat{\ell}\in X^{\ast}$ such that $B\subset\operatorname{h}_{\hat{\ell}}(X)$, hence $B=\operatorname{h}_{\hat{\ell}}(C)$. Identifying the dual space $X^{\ast}$ with any subspace complementary to the annihilator $X^{\perp}$ in $V^{\ast}$ and extending $\hat{\ell}$ to a functional $\ell\in V^{\ast}$ by the Hahn-Banach theorem, we obtain $B=\operatorname{h}_{\ell}(C)$.

Let $(x_{i})\subset C$ be a sequence converging to a nonzero point $x\in V$. If the base $\operatorname{h}_{\ell}(C)$ of $C$ is compact, then the sequence $(\tfrac{x_{i}}{\ell(x_{i})})\subset\operatorname{h}_{\ell}(C)$ has a converging subsequence with limit $y\in\operatorname{h}_{\ell}(C)$. As $\ell(x)=\lim_{i\to\infty}\ell(x_{i})\geq 0$ and as $x=\lim_{i\to\infty}\ell(x_{i})\tfrac{x_{i}}{\ell(x_{i})}=\ell(x)y$, the point $x$ lies in $C$. ∎

The biduality theorem for closed convex cones follows from the separation theorem in convex geometry, see for example [49, Theorem 14.1].

Theorem 3.2**.**

Let $K\subset V$ be a closed convex cone. Then $(K^{\vee})^{\vee}=K$.

We describe the family of bases of a closed convex cone.

Lemma 3.3**.**

Let $K\subset V$ be a closed convex cone. The following assertions are equivalent for all nonzero points $x\in V$.

The point $x$ is an interior point of $K$. 2. 2)

The set $\operatorname{h}_{x}(K^{\vee})$ is a base of $K^{\vee}$.

If one of these equivalent assertions is true, then the set $\operatorname{h}_{x}(K^{\vee})$ is compact.

Proof.

Let $x\in V$ be nonzero. By Theorem 13.1 in [49] the point $x$ is an interior point of $K$ if and only if $\ell(x)<\delta^{\ast}(\ell|K)$ holds for all $\ell\in V^{\ast}\setminus\{0\}$, where

[TABLE]

is the support function of $K$. Hence the assertion (1) is equivalent to $\ell(x)>0$ for all $\ell\in K^{\vee}\setminus\{0\}$, which is equivalent to $\operatorname{h}_{x}(K^{\vee})$ being a base of $K^{\vee}$.

The compactness follows from properties of the recession cone $0^{+}(\operatorname{h}_{x}(K^{\vee}))$, the set of vectors $\ell^{\prime}\in V^{\ast}$ such that $\ell+\lambda\ell^{\prime}$ lies in $\operatorname{h}_{x}(K^{\vee})$ for all $\ell\in\operatorname{h}_{x}(K^{\vee})$ and $\lambda\geq 0$. We can assume that the set $\operatorname{h}_{x}(K^{\vee})$ is nonempty. In this case we have $0^{+}(\operatorname{h}_{x}(K^{\vee}))=x^{\perp}\cap K^{\vee}$ by Coro. 8.3.2 and 8.3.3 of [49] as the recession cone of $\operatorname{h}_{x}(V^{\ast})$ is $x^{\perp}$. Since $\operatorname{h}_{x}(K^{\vee})$ is a base of $K^{\vee}$, we have $\ell(x)>0$ for all $\ell\in K^{\vee}\setminus\{0\}$, hence $x^{\perp}\cap K^{\vee}=\{0\}$. Now [49, Thm. 8.4] shows that $\operatorname{h}_{x}(K^{\vee})$ is bounded, hence compact. ∎

Cones that are contained in a pointed closed convex cone behave nicely.

Lemma 3.4**.**

Let $K\subset V$ be a closed convex cone with nonempty interior. Let $C\subset K^{\vee}$ be a cone and let $x$ be a nonzero interior point of $K$. Then

[TABLE]

is a compact base of the pointed, closed convex cone $\operatorname{cc}(\operatorname{clos}(C))=\operatorname{clos}(\operatorname{cc}(C))$.

Proof.

Let $x\neq 0$ be an interior point of $K$. Then $\operatorname{h}_{x}(K^{\vee})$ is a compact base of $K^{\vee}$ by 3.3. A fortiori, $\operatorname{h}_{x}(C)$ is a base of the cone $C\subset K^{\vee}$; thus $\operatorname{h}_{x}(\operatorname{cc}(C))\subset\operatorname{conv}(\operatorname{h}_{x}(C))$. The converse inclusion is clear and proves the first equality sign in Equation 5 after replacing $C$ with $\operatorname{clos}(C)$. The inclusion $\operatorname{h}_{x}(\operatorname{clos}(C))\subset\operatorname{clos}(\operatorname{h}_{x}(C))$ holds, again as $\operatorname{h}_{x}(C)$ is a base of $C$. The converse inclusion is clear and proves the second equality sign in Equation 5. As $\operatorname{h}_{x}(\operatorname{clos}(C))$ is compact, its convex hull is compact by Theorem 17.2 in [49]. Hence $\operatorname{h}_{x}(\operatorname{cc}(\operatorname{clos}(C)))$ is a compact base of the convex cone $\operatorname{cc}(\operatorname{clos}(C))$, which is closed by 3.1. This proves $\operatorname{clos}(\operatorname{cc}(C))\subset\operatorname{cc}(\operatorname{clos}(C))$; the converse inclusion is clear. ∎

3.5 allows us to focus on irreducible varieties in Section 4.

Proposition 3.5**.**

Let $K_{1},\ldots,K_{r}\subset V$ be convex cones and let $e\neq 0$ be an interior point of $K=K_{1}\cap K_{2}\cap\dots\cap K_{r}$. Let $C_{i}\subset V^{\ast}$ be a cone such that $K_{i}^{\vee}=\operatorname{clos}(\operatorname{cc}(C_{i}))$, that is to say, the dual convex cone to $K_{i}$ is the closed convex cone generated by $C_{i}$ for $i=1,\ldots,r$. Then

[TABLE]

and

[TABLE]

Proof.

Corollary 16.4.2 of [49] shows $K^{\vee}=K_{1}^{\vee}+K_{2}^{\vee}+\cdots+K_{r}^{\vee}$ as $e$ is an interior point of $K_{i}$ for all $i=1,\ldots,r$, hence $K^{\vee}=\operatorname{cc}(K_{1}^{\vee}\cup K_{2}^{\vee}\cup\cdots\cup K_{r}^{\vee})$. This proves Equation 6, as we have, by assumption and by 3.4,

[TABLE]

Equation (7) follows from the equation $\operatorname{h}_{e}(\operatorname{cc}(\operatorname{clos}(C)))=\operatorname{conv}(\operatorname{clos}(\operatorname{h}_{e}(C)))$ provided in 3.4 using $C=\bigcup_{i=1}^{r}C_{i}$ and from Equation 6. ∎

We discuss faces and normal cones. Let $C\subset V$ be a convex subset. A subset $F\subset C$ is a face of $C$ if $F$ is convex and whenever $(1-\lambda)x+\lambda y\in F$ for some $\lambda\in(0,1)$ and $x,y\in C$, then $x$ and $y$ are also in $F$. A subset $F\subset C$ is an exposed face of $C$ if there is a linear functional $\ell\in V^{\ast}$ such that $x\in F$ if and only if $\ell(x)=\inf_{y\in C}\ell(y)$ for all $x\in C$. A point $x\in C$ is an extreme point (resp. exposed point) of $C$ if $\{x\}$ is a face (resp. exposed face) of $C$. Let $\mathbb{R}_{+}=\{\lambda\in\mathbb{R}:\lambda\geq 0\}$. If $x\in C$ is a nonzero point, and $\mathbb{R}_{+}x$ is a face (resp. exposed face) of $C$, then the set $\mathbb{R}_{+}x$ is called an extreme ray (resp. exposed ray) of $C$.

We denote the set of all faces of $C$ by $\mathcal{F}(C)$. The family $\mathcal{F}(C)$ is a complete lattice of finite length under the partial ordering of set inclusion [57]. The duality operator of the closed convex cone $C$ is the map

[TABLE]

The map $N_{C}$ is antitone, that is to say, $F\subset G$ implies $N_{C}(F)\supset N_{C}(G)$ for all faces $F,G\in\mathcal{F}(C)$. The image of $N_{C}$ is the set of nonempty exposed faces of $C^{\vee}$ and [57, Prop. 2.4] shows $F\subset N_{C^{\vee}}\circ N_{C}(F)$ for all faces $F\in\mathcal{F}(C)$. We have just confirmed that the pair of duality operators $N_{C},N_{C^{\vee}}$ defines a Galois connection between $\mathcal{F}(C)$ and $\mathcal{F}(C^{\vee})$. Theorem 20 in Section V.8 of [6] then proves the following assertion, which we use in 4.5.

Lemma 3.6**.**

Let $C\subset V$ be a closed convex cone. The duality operator $N_{C}$ restricts to an antitone lattice isomorphism from the set of nonempty exposed faces of $C$ to the set of nonempty exposed faces of $C^{\vee}$. The inverse isomorphism is the restricted duality operator $N_{C^{\vee}}$.

It is easy to see that the exposed face $N_{C}(F)$ of $C^{\vee}$ is the (inner) normal cone to the closed convex cone $C$ at $F$,

[TABLE]

for all nonempty faces $F\subset C$. In addition, for every relative interior point $x$ of $F$, we have

[TABLE]

hence we also refer to $N_{C}(F)$ as the (inner) normal cone to $C$ at $x$.

3.2. Real Algebraic Geometry

We are working in the setup of real algebraic geometry. A (real) affine variety for us is a subset of $\mathbb{C}^{n}$ (for some $n\in\mathbb{N}$) that is defined by a finite number of polynomial equations $p_{1}=p_{2}=\cdots=p_{r}=0$, $r\in\mathbb{N}$, with real coefficients $p_{i}\in\mathbb{R}[x_{1},x_{2},\dots,x_{n}]$, $i=1,\dots,r$. The affine varieties in $\mathbb{C}^{n}$ are the closed sets of the Zariski topology on $\mathbb{C}^{n}$. So the Zariski closure of a set $S\subset\mathbb{C}^{n}$ is the smallest real affine variety containing $S$. A (real) projective variety for us is a subset of projective space $\mathbb{P}^{n}$ that is defined by a finite number of homogeneous polynomial equations $p_{1}=p_{2}=\cdots=p_{r}=0$, $r\in\mathbb{N}$, with real coefficients $p_{i}\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$, $i=1,\dots,r$. The projective varieties in $\mathbb{P}^{n}$ are the closed sets of the Zariski topology on $\mathbb{P}^{n}$. Identifying points in $\mathbb{P}^{n}$ with lines in $\mathbb{C}^{n+1}$, a projective variety can be seen as an affine variety in $\mathbb{C}^{n+1}$ which is an algebraic cone. The affine cone $\widehat{S}$ over a subset $S\subset\mathbb{P}^{n}$ is the union of all lines in $\mathbb{C}^{n+1}$ spanned by a vector $(x_{0},x_{1},\dots,x_{n})^{T}$ such that $(x_{0}:x_{1}:\dots:x_{n})\in S$. Conversely, the projective variety $\mathbb{P}(X)\subset\mathbb{P}^{n}$ associated with an algebraic cone $X\subset\mathbb{C}^{n+1}$ consists of the points $(x_{0}:x_{1}:\dots:x_{n})\in\mathbb{P}^{n}$ for which the vector $(x_{0},x_{1},\dots,x_{n})^{T}$ is included in $X$. These notions are explained in introductory textbooks on algebraic geometry like [27] with the caveat that affine and projective varieties are usually complex varieties, i.e. defined by finitely many polynomial equations with complex coefficients. A point $x\in\mathbb{P}^{n}$ is real if the line $\widehat{\{x\}}\subset\mathbb{C}^{n+1}$ contains a nonzero real point. We denote the set of real points of a subset $S\subset\mathbb{C}^{n}$ or $S\subset\mathbb{P}^{n}$ by $S(\mathbb{R})$. The dual projective space $(\mathbb{P}^{n})^{\ast}=\mathbb{P}((\mathbb{C}^{n+1})^{\ast})$ is the projective space over the dual vector space so that the hyperplanes in $\mathbb{P}^{n}$ are in one-to-one correspondence with points in $(\mathbb{P}^{n})^{\ast}$. We specify a functional $\ell\in(\mathbb{P}^{n})^{\ast}$ in terms of its hyperplane

[TABLE]

by writing $\ell=[H]$. Identifying $((\mathbb{P}^{n})^{\ast})^{\ast}=\mathbb{P}^{n}$, a point $x\in\mathbb{P}^{n}$ defines a hyperplane in $(\mathbb{P}^{n})^{\ast}$ which we denote by

[TABLE]

Definition 3.7**.**

Let $X\subset\mathbb{P}^{n}$ be an irreducible projective variety. We define the projective dual variety $X^{\ast}$ of $X$ as the Zariski closure of the set of hyperplanes that are tangent to $X$ at some regular point, i.e. the closure of

[TABLE]

An instructive and especially nice case of duality occurs for hypersurfaces defined by quadratic forms of full rank.

Example 3.8**.**

Let $q\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ be a quadratic form and let $M_{q}$ be the real symmetric $(n+1)\times(n+1)$ matrix representing $q$, i.e. with

[TABLE]

The projective variety $X=\mathcal{V}(q)\subset\mathbb{P}^{n}$ is smooth if and only if the rank of $M_{q}$ is $n+1$. We compute the dual variety of $X$ under the assumption that $X$ is smooth. Let $x=(x_{0}:x_{1}:\dots:x_{n})\in X$ be a point. The differential $\ell_{x}=2x^{T}M_{q}\in(\mathbb{P}^{n})^{\ast}$ of $q$ at $x$ defines the tangent hyperplane $T_{x}X=\{y\in\mathbb{P}^{n}\colon\ell_{x}(y)=0\}$ to $X$ at $x$. In other words, the dual variety to $X$ is the Zariski closure of the set $\{\ell_{x}\colon x\in X\}\subset(\mathbb{P}^{n})^{\ast}$. The condition $x\in X$ is $0=x^{T}M_{q}x=(\ell_{x}^{T}M_{q}^{-1})M_{q}(M_{q}^{-1}\ell_{x})$. We conclude that $X^{\ast}$ is the quadratic hypersurface defined by $M_{q}^{-1}$.

For irreducible algebraic varieties, the famous biduality theorem holds.

Theorem 3.9** (Biduality Theorem; see [25, Ch. 1, Thm. 1.1]).**

If $X\subset\mathbb{P}^{n}$ is an irreducible projective variety, then $(X^{\ast})^{\ast}=X$ under the canonical identification of the bidual of $\mathbb{P}^{n}$ with $\mathbb{P}^{n}$ itself.

This theorem has several useful consequences like the following.

Remark 3.10**.**

Let $X\subset\mathbb{P}^{n}$ be an irreducible projective variety. For all points $x$ of $X$ in a dense subset in the Euclidean topology of $X$, the point $x$ is regular, the hyperplane $\mathcal{V}(x)\subset(\mathbb{P}^{n})^{\ast}$ is tangent to $X^{\ast}$ at a regular point $\ell$, and the hyperplane $\mathcal{V}(\ell)\subset\mathbb{P}^{n}$ is tangent to $X$ at $x$. This is an application of the conormal variety $CN(X)$, defined as the Zariski closure of

[TABLE]

The projection $\pi_{1}\colon CN_{0}(X)\to X_{\rm reg}$ is the conormal bundle of $X$, which shows that $CN_{0}(X)$ is an irreducible and smooth variety. The biduality theorem is often proven as a consequence of the fact that $CN(X)=CN(X^{\ast})$, see [25, Chapter 1]. The biduality theorem implies that the subset

[TABLE]

is a non-empty Zariski open subset of $CN(X)$. If $x\in X_{\rm reg}$ and $[H]\in(X^{\ast})_{\rm reg}$ are regular points, then $H$ is tangent to $X$ at $x$ if and only if $\mathcal{V}(x)$ is tangent to $X^{\ast}$ at $[H]$, see [58, Thm. 1.7(b)]. This shows

[TABLE]

By definition, the right-hand side consists of pairs $(x,[H])$ of regular points $x\in X_{\rm reg}$ and $[H]\in(X^{\ast})_{\rm reg}$ such that $\mathcal{V}(x)$ is tangent to $X^{\ast}$ at $[H]$ and $H$ is tangent to $X$ at $x$. Since $U$ is dense in $CN(X)$ in the Euclidean topology [44, Thm. 2.33], the claim follows as the projection from $CN(X)$ to the first factor $X$ is continuous and surjective.

When passing to real points, the direct analogue of Remark 3.10 fails: The set of regular real points $X_{\mathrm{reg}}(\mathbb{R})$ of an irreducible projective variety $X\subset\mathbb{P}^{n}$ may not be dense in $X(\mathbb{R})$ with respect to the Euclidean topology, even if it is non-empty, see for example [7, Section 3.1] or 6.4 below. This is addressed in the following remark.

Remark 3.11**.**

Let $X\subset\mathbb{P}^{n}$ be an irreducible projective variety. For all real regular points $x$ of $X$ in a dense subset of $X_{\mathrm{reg}}(\mathbb{R})$ in the Euclidean topology, the hyperplane $\mathcal{V}(x)\subset(\mathbb{P}^{n})^{\ast}$ is tangent to $X^{\ast}$ at a real regular point $\ell$ and the hyperplane $\mathcal{V}(\ell)\subset\mathbb{P}^{n}$ is tangent to $X$ at $x$.

This claim is trivial if $X$ has no regular real points. We resume the discussion from 3.10 assuming that $X$ does contain a regular real point. The variety $CN_{0}(X)$ is smooth and contains real points, since $X$ contains smooth real points. Since $CN_{0}(X)\setminus CN_{0}(X^{\ast})=CN_{0}(X)\setminus U$ is a Zariski closed proper subset relative to $CN_{0}(X)$, it is of lower dimension. As $CN_{0}(X)(\mathbb{R})$ is a real analytic manifold of dimension $\dim(CN_{0}(X))$, see [7, Prop. 3.3.11], the set $U(\mathbb{R})$ is dense in $CN_{0}(X)(\mathbb{R})$ in the Euclidean topology. This proves the claim, because the projection of $CN_{0}(X)(\mathbb{R})$ onto the first factor $X$ is onto $X_{\mathrm{reg}}(\mathbb{R})$.

Definition 3.12**.**

We call a real point $x$ of an algebraic variety $X\subset\mathbb{C}^{n}$ central if it is in the Euclidean closure of the set of regular and real points of $X$, i.e. if $x$ is in the Euclidean closure of $X_{\mathrm{reg}}(\mathbb{R})$.

Remark 3.13**.**

For an irreducible algebraic variety $X\subset\mathbb{C}^{n}$, a point $x\in X(\mathbb{R})$ is central if and only if the local dimension of $x$ in $X(\mathbb{R})$ is equal to $\dim(X)$, see [7, Prop. 7.6.2].

4. Dual Hyperbolicity Cones

We discuss a result by the second author [53] more explicitly. The result is that the dual convex cone to a hyperbolicity cone is the convex cone generated by a particular semi-algebraic cone. This semi-algebraic cone is the Euclidean closure of the cone of regular real points on the dual variety to the hyperbolic hypersurface that lie in the right half-space. The algebraic boundary of the hyperbolicity cone allows us to simplify this semi-algebraic cone. We prove a stronger result for three-dimensional hyperbolicity cones.

A homogeneous polynomial $p\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ of degree $d$ is called hyperbolic with respect to a fixed point $e\in\mathbb{R}^{n+1}$ if $p(e)\neq 0$ and the polynomial $p(te-a)$ in one variable $t$ has only real roots for every point $a\in\mathbb{R}^{n+1}$. Without loss of generality, we fix the sign at $e$ and always assume $p(e)>0$. The roots of $p(te-a)$ are sometimes called the eigenvalues of $a$ with respect to $p$ and $e$, in analogy with characteristic polynomials of Hermitian matrices. Given any such polynomial $p$, the set

[TABLE]

is a closed convex cone called the (closed) hyperbolicity cone of $p$ with respect to $e$, and $e$ is an interior point of $C_{e}(p)$, see [48]. Our goal is to describe the dual convex cone

[TABLE]

An essential technique is projective duality. A general approach is described in the paper [53] by the second author. The goal of this section is to explain this method more explicitly for the special case of hyperbolicity cones.

An important example of a hyperbolic polynomial is the determinant of a matrix pencil, i.e. $p=\det(x_{0}A_{0}+\cdots+x_{n}A_{n})$ for Hermitian $d\times d$ matrices $A_{0},\dots,A_{n}\in H_{d}$, which is hyperbolic with respect to $e=(e_{0},\dots,e_{n})^{T}$ provided the matrix $e_{0}A_{0}+\cdots+e_{n}A_{n}$ is positive definite. In this case, $C_{e}(p)$ is the spectrahedral cone defined by $A_{0},\dots,A_{n}$, see Section 6. However, the discussion in this section does not require such a determinantal representation and we consider general hyperbolic polynomials.

The proof of our main result, 4.5, makes use of the following 4.2 on hyperbolic polynomials. For the sake of completeness, we include a short proof based on the Helton-Vinnikov theorem on determinantal representations of hyperbolic curves; see [47, Lemma 2.4] for a direct proof of a special case.

Definition 4.1**.**

Let $f\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ be a polynomial and $x\in\mathbb{C}^{n+1}$ be a point. The multiplicity of $x$ on $\mathcal{V}(f)\subset\mathbb{C}^{n+1}$ is the smallest degree of a non-zero homogeneous term in the Taylor expansion of $f$ around $x$.

Lemma 4.2**.**

Let $p\in\mathbb{R}[x_{0},\dots,x_{n}]$ be hyperbolic with respect to $e$ and let $x\in\mathbb{R}^{n+1}$. If $x$ has multiplicity $m$ on $\mathcal{V}(p)$, the hyperbolic hypersurface defined by $p$, then $t=0$ is a root of multiplicity $m$ of $p(x+te)\in\mathbb{R}[t]$. Moreover, $t=0$ is also a root of multiplicity $m$ of $p(x+t(e-x))\in\mathbb{R}[t]$.

Proof.

If the multiplicity of $x$ on $\mathcal{V}(p)$ is $m$, then $\frac{\partial^{m}}{\partial s^{m}}p(x+sy)|_{s=0}\neq 0$ for generic $y\in\mathbb{R}^{n+1}$. Fix such $y$ in the interior of $C_{e}(p)$ and consider the hyperbolic polynomial $p(rx+sy+te)$ in three variables $r,s,t$. By the Helton-Vinnikov theorem ([31, Thm. 2.2]), this polynomial has a determinantal representation

[TABLE]

with real symmetric matrices $A,B,C$, where $B$ and $C$ are positive definite, hence factor as $B=UU^{T}$ and $C=VV^{T}$, with $U$ and $V$ invertible. Now $s=0$ is a root of $p(x+sy)=\det(A+sB)$ of multiplicity $m$, which means that $U^{-1}A(U^{T})^{-1}$ has $m$-dimensional kernel. But then so does $V^{-1}A(V^{T})^{-1}$, hence the root $t=0$ of $p(x+te)=\det(A+tC)$ has multiplicity $m$ as well.

The second part of the claim follows from the part we have just proved. Indeed, write $f(s,t)=p(sx+te)\in\mathbb{R}[s,t]$ which is the homogenization of $p(x+te)$ because $p(e)\neq 0$. Since $t=0$ is a root of multiplicity $m$ of $p(x+te)$, we can write $p(x+te)=t^{m}q(t)$ with $q(0)\neq 0$. Since $p$ is hyperbolic with respect to $e$, the polynomial $q$ factors into linear terms over $\mathbb{R}$, say $q=c(1-\lambda_{1}t)\cdot\ldots\cdot(1-\lambda_{d-m}t)$ for some nonzero $c\in\mathbb{R}$. Here we used $q(0)\neq 0$ because the $\lambda_{i}$ are the reciprocals of the roots of $q$. So we get that $f(s,t)=c\cdot t^{m}\prod(s-\lambda_{i}t)$. The polynomial $p(sx+t(e-x))$ is the same as $f(s-t,t)=c\cdot t^{m}\prod(s-(\lambda_{i}+1)t)$. Dehomogenizing this again shows that $t=0$ is a root of multiplicity $m$ of $p(x+t(e-x))$. ∎

Corollary 4.3**.**

If $x$ is a regular real point of a hyperbolic hypersurface $\mathcal{V}(p)$, then the line incident with $x$ and the hyperbolicity direction $e$ is not tangent to $\mathcal{V}(p)$ at $x$, i.e. is not contained in T_{x}\bigl{(}\mathcal{V}(p)\bigr{)}.

Proof.

If a line $L$ is tangent to $\mathcal{V}(p)$ at $x$, then the multiplicity of $x$ in $L\cap\mathcal{V}(p)$ is greater than the multiplicity of $x$ in $\mathcal{V}(p)$. This is impossible if $e\in L$, by the previous 4.2. ∎

We prove basics from differential and convex geometry regarding hyperbolicity cones.

Lemma 4.4**.**

Let $p\in\mathbb{R}[x_{0},\dots,x_{n}]$ be an irreducible hyperbolic polynomial with respect to $e\in\mathbb{R}^{n+1}$. Then $M=\partial C_{e}(p)\cap\mathcal{V}(p)_{\mathrm{reg}}$ is an $n$-dimensional real analytic manifold, which is open and dense in the Euclidean boundary $\partial C_{e}(p)$ of $C_{e}(p)$ in the Euclidean topology. The (inner) normal cone to $C_{e}(p)$ at any point $x\in M$ is the ray $\mathbb{R}_{+}\ell$, where $\ell=\nabla p(x)^{T}\in C_{e}(p)^{\vee}\subset(\mathbb{R}^{n+1})^{\ast}$.

Proof.

Since $\partial C_{e}(p)\subset\mathcal{V}(p)(\mathbb{R})$ and since the set of singular points of $\mathcal{V}(p)(\mathbb{R})$ is a variety of dimension at most $n-1$, see [7, Prop. 3.3.14], the complement $M=\partial C_{e}(p)\cap\mathcal{V}(p)_{\mathrm{reg}}$ is open and dense in $\partial C_{e}(p)$ in the Euclidean topology. As $\mathcal{V}(p)_{\mathrm{reg}}(\mathbb{R})$ is an analytic manifold of dimension $n$, see [7, Prop. 3.3.11], and as the eigenvalues depend continuously on $x\in\mathbb{R}^{n+1}$, see [48], the set $M$ is an analytic manifold of dimension $n$.

Let $x\in M$. As $x$ is a regular point of $\mathcal{V}(p)$, the functional $\ell=\nabla p(x)^{T}$ is non-zero. Hence, 4.2, case $m=1$, shows $\ell(e)\neq 0$. Since $M$ is an $n$-dimensional analytic manifold included in $C_{e}(p)$, the normal cone $N(x)$ of $C_{e}(p)$ at $x$ is a subset of the line $\mathbb{R}\ell$, hence $N(x)=\mathbb{R}_{+}\ell$ or $N(x)=-\mathbb{R}_{+}\ell$ as $x\in\partial C_{e}(p)$. The derivative polynomial $p^{\prime}\in\mathbb{R}[x_{0},\dots,x_{n}]$, defined by

[TABLE]

is hyperbolic with respect to $e$ and $C_{e}(p)\subset C_{e}(p^{\prime})$ holds, see [48]. This proves $\ell(e)=p^{\prime}(x)>0$ and rules out $N(x)=-\mathbb{R}_{+}\ell$, as $N(x)\subset C_{e}(p)^{\vee}$ by Equation 8. This proves the claim. ∎

The second author obtained 4.5 and 4.14 below in [53, Example 3.15]. We slightly abuse notation in the following statements. If $X\subset\mathbb{C}^{n+1}$ is an algebraic cone, then we write $X^{\ast}\subset(\mathbb{C}^{n+1})^{\ast}$ for the affine cone over the projective dual variety $\mathbb{P}(X)^{\ast}$ of $\mathbb{P}(X)$. Let $H_{e,+}\subset(\mathbb{R}^{n+1})^{\ast}$ denote the half-space $H_{e,+}=\{\ell\in(\mathbb{R}^{n+1})^{\ast}\colon\ell(e)\geq 0\}$ for nonzero $e\in\mathbb{R}^{n+1}$.

Theorem 4.5** (Sinn [53]).**

Let $p\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ be an irreducible hyperbolic polynomial with respect to $e\in\mathbb{R}^{n+1}$. Then we have

[TABLE]

Proof.

We prove the inclusion “$\supset$”. Since $C_{e}(p)^{\vee}$ is a closed convex cone, it is enough to show that $S=(\mathcal{V}(p)^{\ast})_{\mathrm{reg}}(\mathbb{R})\cap H_{e,+}$ is contained in $C_{e}(p)^{\vee}$. By 3.11, it is enough to prove $\ell\in C_{e}(p)^{\vee}$ for all $\ell\in S$ such that the hyperplane $\mathcal{V}(\ell)$ is tangent to $\mathcal{V}(p)$ at a regular real point $x$, i.e. $\ell=\nabla p(x)^{T}\in(\mathbb{R}^{n+1})^{\ast}$. By 4.2 above, case $m=1$, we have $0\neq\frac{\partial}{\partial t}p(x+te)|_{t=0}=\ell(e)$ for such an $\ell$ and hence $e\not\in\mathcal{V}(\ell)$. As $p$ is hyperbolic with respect to every interior point of the hyperbolicity cone [48], it follows that the interior of $C_{e}(p)$ is disjoint from $\mathcal{V}(\ell)$. This means that $\ell$ has constant sign on $C_{e}(p)$. Since $\ell\in H_{e,+}$, it follows that $\ell\in C_{e}(p)^{\vee}$.

The inclusion “$\subset$” follows from [53, Coro. 3.14]. We repeat the argument, adapted to our setup, for completeness. Since $C_{e}(p)^{\vee}$ is the convex cone generated by its extreme rays and the right-hand side is a convex cone, it suffices to prove that every extreme ray of $C_{e}(p)^{\vee}$ is contained in the right-hand side. By Straszewicz’s Theorem [49, Thm. 18.6], which says that every extreme ray is a limit of exposed rays, it suffices to prove the claim for every exposed ray of $C_{e}(p)^{\vee}$, because the right-hand side is closed.

Let $\mathbb{R}_{+}\ell$ be an exposed ray of the convex cone $C_{e}(p)^{\vee}$. It is enough to show that $\ell$ is a central point of $\mathcal{V}(p)^{\ast}(\mathbb{R})$, i.e. $\ell$ lies in the (Euclidean) closure of $(\mathcal{V}(p)^{\ast})_{\rm reg}(\mathbb{R})$. Let $F_{\ell}=\{x\in C_{e}(p)\colon\ell(x)=0\}$ be the exposed face of $C_{e}(p)$ corresponding to $\ell$ and let $x$ be a point in the relative interior of $F_{\ell}$. As proven in 4.4, the analytic manifold $M=\partial C_{e}(p)\cap\mathcal{V}(p)_{\mathrm{reg}}$ is dense in the Euclidean boundary $\partial C_{e}(p)$ of the hyperbolicity cone $C_{e}(p)$. Hence, there is a sequence $(y_{j})\subset M$ converging to $x$. 3.11 shows that after slightly moving the members of the sequence $(y_{j})$ within $M$ without changing the limit $x$, the hyperplane $\mathcal{V}(y_{j})$ is tangent to $\mathcal{V}(p)^{\ast}$ at a regular real point $\ell_{j}\in(\mathcal{V}(p)^{\ast})_{\mathrm{reg}}(\mathbb{R})$ and $\mathcal{V}(\ell_{j})$ is tangent to $\mathcal{V}(p)$ at $y_{j}$ for all $j\in\mathbb{N}$.

After scaling $\ell_{j}$ with a nonzero real number, we have $\ell_{j}=\nabla p(y_{j})^{T}$. Hence, the ray $\mathbb{R}_{+}\ell_{j}$ lies in the dual convex cone $C_{e}(p)^{\vee}$ by 4.4 for all $j\in\mathbb{N}$. After normalizing and passing to a subsequence, the sequence $(\ell_{j})$ converges to a point $\ell^{\prime}$ in the compact unit sphere of $(\mathbb{R}^{n+1})^{\ast}$. We have $\ell^{\prime}\in C_{e}(p)^{\vee}$ and $\ell^{\prime}(x)=0$, the latter as

[TABLE]

holds for all $j\in\mathbb{N}$ and since $\ell_{j}\to\ell^{\prime}$ and $y_{j}\to x$ as $j\to\infty$. Since $\mathbb{R}_{+}\ell$ is an exposed ray of $C_{e}(p)^{\vee}$, the lattice isomorphism of 3.6 and Equation 8 show $\mathbb{R}_{+}\ell=F_{\ell}^{\perp}\cap C_{e}(p)^{\vee}=x^{\perp}\cap C_{e}(p)^{\vee}$. This proves $\mathbb{R}_{+}\ell=\mathbb{R}_{+}\ell^{\prime}$. Hence $\ell$ is a central point of $\mathcal{V}(p)^{\ast}(\mathbb{R})$. ∎

In other words, 4.5 says that the dual convex cone to the hyperbolicity cone $C_{e}(p)$ is the closed convex cone generated by the regular real points of the dual variety $\mathcal{V}(p)^{\ast}$ lying in the appropriate half-space $H_{e,+}$.

The well-known Steiner surface explains why singular points of the dual variety $\mathcal{V}(p)^{\ast}$ have to be excluded from the statement of 4.5.

Example 4.6**.**

The Cayley cubic is the cubic hypersurface in $\mathbb{P}^{3}$ defined by the polynomial

[TABLE]

This polynomial is irreducible and hyperbolic with respect to the point $(1,0,0,0)$ in $\mathbb{R}^{4}=\{(x_{0},x_{1},x_{2},x_{3})\}$. Its hyperbolicity cone is the homogenization of the elliptope $\mathcal{E}_{3}$, which is the feasible set of the Goemans-Williamson semidefinite relaxation of the MAX-CUT problem (for graphs with three vertices).

The dual convex cone is the closed convex cone generated by the regular real points with $y_{0}>0$ on the Steiner surface given by the equation

[TABLE]

The singular locus of this quartic surface is the union of three real lines in $(\mathbb{P}^{3})^{\ast}$, which are not contained in the dual convex cone. See Figure 1, where we draw the real affine parts of the varieties $\mathcal{V}(p)$ and $\mathcal{V}(q)$, that is to say, the set of points $(x_{1},x_{2},x_{3})^{T}\in\mathbb{R}^{3}$ for which $(1:x_{1}:x_{2}:x_{3})$ lies in $\mathcal{V}(p)$ in drawing a) and the set of points $(y_{1},y_{2},y_{3})\in(\mathbb{R}^{3})^{\ast}$ for which $(1:y_{1}:y_{2}:y_{3})$ lies in $\mathcal{V}(q)$ in drawing b).

Convex geometry suffices to generalize 4.5 from irreducible polynomials to arbitrary hyperbolic polynomials.

Corollary 4.7**.**

Let $p\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ be a hyperbolic polynomial with respect to $e\in\mathbb{R}^{n+1}$. Let $X_{1},X_{2},\dots,X_{r}$ be the irreducible components of the hyperbolic hypersurface $\mathcal{V}(p)$. Then we have

[TABLE]

where $S_{i}=(X_{i}^{\ast})_{\mathrm{reg}}(\mathbb{R})\cap H_{e,+}$ for $i=1,\dots,r$.

Proof.

If $p=p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{r}^{m_{r}}$ is a factorization of $p$ into irreducible factors, then $C_{e}(p)=C_{e}(p_{1})\cap C_{e}(p_{2})\cap\dots\cap C_{e}(p_{r})$. The claim follows from 4.5 and from Equation 6 in 3.5. ∎

Remark 4.8**.**

The cones $S_{i}$ in 4.7, their union $S_{1}\cup S_{2}\cup\dots\cup S_{r}$, and the Euclidean closure $\operatorname{clos}(S_{1}\cup S_{2}\cup\dots\cup S_{r})$, see [7, Prop. 2.2.2], are semi-algebraic sets. Hence, the closed convex cone $C_{e}(p)^{\vee}$ is the convex cone generated by the semi-algebraic cone $\;\operatorname{clos}(S_{1}\cup S_{2}\cup\dots\cup S_{r})$. 2. 2)

The Euclidean closure of the set $S_{i}$ in 4.7 is the set of central points of $X_{i}^{\ast}(\mathbb{R})$ lying in $H_{e,+}$ for all $i=1,2,\dots,r$. Hence, writing $\mathrm{cent}(X)$ for the set of central real points of $X$, we can rephrase Equation 9 as

[TABLE]

We can strengthen 4.5 to the homogeneous version of Kippenhahn’s Theorem, the statement of 4.9, if the hyperbolicity cone has dimension three. Chien and Nakazato [14] observed that this stronger version is false in higher dimensions, see 6.4.

Theorem 4.9**.**

Let $p\in\mathbb{R}[x_{0},x_{1},x_{2}]$ be an irreducible hyperbolic polynomial with respect to $e\in\mathbb{R}^{3}$. Then C_{e}(p)^{\vee}=\operatorname{cc}\big{(}\mathcal{V}(p)^{\ast}(\mathbb{R})\cap H_{e,+}\big{)}.

Proof.

The inclusion C_{e}(p)^{\vee}\subset\operatorname{cc}\big{(}\mathcal{V}(p)^{\ast}(\mathbb{R})\cap H_{e,+}\big{)} follows from 4.5 and 3.4. We prove the opposite inclusion by contradiction based on two observations. Let $\ell$ be a nonzero functional that lies in \operatorname{cc}\big{(}\mathcal{V}(p)^{\ast}(\mathbb{R})\cap H_{e,+}\big{)} but not in $C_{e}(p)^{\vee}$.

First, the hyperplane $\mathcal{V}(\ell)\subset\mathbb{C}^{3}$ intersects the interior of the hyperbolicity cone $C_{e}(p)$, as it holds for all functionals $\hat{\ell}\in H_{e,+}\setminus C_{e}(p)^{\vee}$. Since $\hat{\ell}\in H_{e,+}$, we have $\hat{\ell}(e)\geq 0$, and since $\hat{\ell}\not\in C_{e}(p)^{\vee}$ there is a point $x\in C_{e}(p)$ such that $\hat{\ell}(x)<0$. Hence, there is $\lambda\in[0,1)$ such that the point $y=(1-\lambda)e+\lambda x$ lies on $\mathcal{V}(\hat{\ell})$. As $e$ is an interior point of $C_{e}(p)$ so is $y$ [49, Thm. 6.1].

Secondly, the line $\mathcal{V}(\ell)\subset\mathbb{P}^{2}$ is tangent to $\mathcal{V}(p)$ at a point $x\in\mathcal{V}(p)$. Applying 3.10 to the projective variety $X=\mathcal{V}(p)^{\ast}$, we can choose a sequence of regular points $(\ell_{j})$ in $\mathcal{V}(p)^{\ast}$ converging to $\ell$ and a sequence of regular points $(x_{j})$ of $\mathcal{V}(p)$ such that the line $\mathcal{V}(\ell_{j})$ is tangent to $\mathcal{V}(p)$ at $x_{j}$ for all $j\in\mathbb{N}$. Because of the compactness of the projective space $\mathbb{P}^{2}$, we can assume $(x_{j})$ converges to a point $x\in\mathcal{V}(p)$. Since $\mathcal{V}(\ell_{j})$ is tangent to $\mathcal{V}(p)$ at $x_{j}$, the line $\mathcal{V}(\ell)$ is tangent to $\mathcal{V}(p)$ at $x$, see [20, Sec. 8.2].

The real line $\mathcal{V}(\ell)\subset\mathbb{P}^{2}$ intersects the hyperbolicity cone $C_{e}(p)$ in an interior point by the first observation. Since the polynomial $p$ is hyperbolic with respect to this interior point and since the point $x\in\mathcal{V}(p)$ constructed above lies on the line $\mathcal{V}(\ell)$, it follows that $x$ is a real point. 4.2 then shows that $\mathcal{V}(\ell)$ is not tangent to $\mathcal{V}(p)$ at $x$, which contradicts the second observation. ∎

Again, convex geometry suffices to generalize 4.9 from irreducible polynomials to arbitrary hyperbolic polynomials.

Corollary 4.10**.**

Let $p\in\mathbb{R}[x_{0},x_{1},x_{2}]$ be a hyperbolic polynomial with respect to $e\in\mathbb{R}^{3}$. Then $C_{e}(p)^{\vee}=\operatorname{cc}(\mathcal{V}(p)^{\ast}(\mathbb{R})\cap H_{e,+})$.

Proof.

If $p=p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{r}^{m_{r}}$ is a factorization of $p$ into irreducible factors, then

[TABLE]

and

[TABLE]

The claim follows from 4.9 and Equation 6 in 3.5. ∎

Not all components in the union $\mathcal{V}(p)^{\ast}=\mathcal{V}(p_{1})^{\ast}\cup\mathcal{V}(p_{2})^{\ast}\cup\dots\cup\mathcal{V}(p_{r})^{\ast}$ are needed in the statements of 4.10 and 4.7. The selection can be described as follows.

Definition 4.11**.**

Let $S\subset\mathbb{R}^{n}$ be a semi-algebraic set. The algebraic boundary of $S$, denoted $\partial_{a}S$, is the Zariski closure in $\mathbb{C}^{n}$ of the Euclidean boundary $\partial S$ of $S$.

Determining the algebraic boundary of the hyperbolicity cone of a hyperbolic polynomial $p\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ amounts to computing a factorization of $p$ into irreducible factors and picking the correct subset of the factors.

Remark 4.12**.**

Let $p\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ be irreducible and hyperbolic with respect to $e$. The algebraic boundary of the hyperbolicity cone $C_{e}(p)$ is the algebraic hypersurface $\mathcal{V}(p)=\{x\in\mathbb{C}^{n+1}\colon p(x)=0\}$. 2. 2)

If $p$ is hyperbolic with respect to $e$, but factors as $p=p_{1}p_{2}\dots p_{r}$ into irreducible factors, then $\mathcal{V}(p)=\mathcal{V}(p_{1})\cup\mathcal{V}(p_{2})\cup\dots\cup\mathcal{V}(p_{r})$ is the decomposition of the hypersurface $\mathcal{V}(p)$ into its irreducible components. The algebraic boundary of $C_{e}(p)$ is a union of some, not necessarily all, of the irreducible hypersurfaces $\mathcal{V}(p_{i})$. The hypersurfaces in this union are the irreducible components of $\partial_{a}C_{e}(p)$. 3. 3)

If $p$ is squarefree, i.e. $p_{j}\neq p_{i}$ for $i\neq j$, then the factors $p_{i}$ that contribute to the algebraic boundary of $C_{e}(p)$ are exactly those with the property that the hyperbolicity cone of $\prod_{j\neq i}p_{j}$ is strictly larger than $C_{e}(p)$, because $C_{e}(p)=\bigcap_{i}C_{e}(p_{i})$.

Example 4.13**.**

Consider the four ellipses depicted in Figure 2 a). The intersection of the filled ellipses is an area with nonempty interior, which is isometric to the base of a hyperbolicity cone via the embedding $\mathbb{R}^{2}\to\mathbb{R}^{3}$, $(x_{1},x_{2})^{T}\mapsto(1,x_{1},x_{2})^{T}$. Note that the red ellipse does not contribute to the algebraic boundary of the hyperbolicity cone, but the other ellipses do. Homogenizing the polynomials, we obtain four conics in $\mathbb{P}^{2}$ from the ellipses. The real affine parts of the dual conics, via the embedding $(\mathbb{R}^{2})^{\ast}\to(\mathbb{P}^{2})^{\ast}$, $(y_{1},y_{2})\mapsto(1:y_{1}:y_{2})$, are depicted in Figure 2 b). Their convex hull is a base of the dual convex cone to the hyperbolicity cone. We return to these bases in Section 5. Note that the red ellipse in Figure 2 b) is redundant, as the other three ellipses generate the same convex hull as all four together.

Corollary 4.14** (Sinn [53]).**

Let $p\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ be a hyperbolic polynomial with respect to $e\in\mathbb{R}^{n+1}$. Let $X_{1},X_{2},\dots,X_{r}$ be the irreducible components of the algebraic boundary $\partial_{a}C_{e}(p)$. Then we have

[TABLE]

where $S_{i}=(X_{i}^{\ast})_{\mathrm{reg}}(\mathbb{R})\cap H_{e,+}$ for $i=1,\dots,r$.

Proof.

This follows from Part 3) of 4.12 and 4.7. ∎

Remark 4.15**.**

The algebraic boundary of the hyperbolicity cone $C_{e}(p)$ is algebraically the optimal description of the set of extreme rays of $C_{e}(p)^{\vee}$. Using the same techniques and ideas that are presented in this paper, one can argue that the Zariski closure of the set of extreme rays of $C_{e}(p)^{\vee}$ is the union of the dual varieties $X_{i}^{\ast}$ for which the varieties $X_{i}$ belong to the algebraic boundary $\partial_{a}C_{e}(p)$, see [53, Corollary 3.5].

The statements above in this section require employing the duality theory of real algebraic geometry, which is more subtle than the duality theory of algebraic geometry. The reason is that the set of Hermitian matrices is just a real vector space, not a complex vector space.

Remark 4.16**.**

The image of the set $H_{d}^{(1)}$ of Hermitian matrices of rank $1$ under the projection map

[TABLE]

from $H_{d}$ to $\mathbb{R}^{n}$ is of interest to us (see Section 2), as its convex hull is the joint numerical range. The complexification of the real vector space $H_{d}$ is the complex vector space of all complex $d\times d$ matrices. The Zariski closure of $H_{d}^{(1)}$ is the variety $R_{d}^{(1)}$ of complex $d\times d$ matrices of rank at most $1$. Therefore, the images of $H_{d}^{(1)}$ and $R_{d}^{(1)}$ under the map $\pi$ have the same Zariski closure. To evaluate $\langle M,A_{i}\rangle$ at a complex matrix $M$, we write $M={\rm Re}(M)+\operatorname{i}{\rm Im}(M)$, where ${\rm Re}(M)$ and ${\rm Im}(M)$ are Hermitian, and define $\langle M,A_{i}\rangle=\operatorname{tr}({\rm Re}(M)A_{i})+\operatorname{i}\operatorname{tr}({\rm Im}(M)A_{i})$ (which is $\mathbb{C}$-linear). If the map $\pi$ restricted to $R_{d}^{(1)}$ is an isomorphism, then Prop. 4.1 in [25] implies roughly speaking that $\pi(R_{d}^{(1)})$ is projectively dual to the intersection of the orthogonal complement of the kernel of $\pi$ and the dual variety of $R_{d}^{(1)}$. This dual variety is the determinantal hypersurface in the space of complex $d\times d$ matrices. The assumption that $\pi$ restricted to $R_{d}^{(1)}$ is an isomorphism is generically satisfied provided that $n$ is sufficiently large relative to the size $d$ of the matrices; specifically the kernel of $\pi$ must not intersect the secant variety of $R_{d}^{(1)}$, which is the variety of matrices of rank at most $2$.

5. Bases of Dual Hyperbolicity Cones

The results of Section 4 carry over from the dual convex cone of a hyperbolicity cone to all its bases, by replacing the convex cone generated by a semi-algebraic cone with the convex hull of a base of this semi-algebraic cone. Remarkably, the results also apply to all linear images of these bases, because we can interpret these linear images as the bases of the dual convex cones to linear sections of the original hyperbolicity cone.

Before returning to hyperbolicity cones at the end of the section, we discuss the necessary convex geometry. See Figure 3 for a summary of our setup.

Definition 5.1**.**

Let $V$ be a finite-dimensional real vector space. Let $\mathcal{K}\subset V$ be a closed convex cone and let $e\neq 0$ be an interior point of $\mathcal{K}$. The set

[TABLE]

introduced in Equation 4 is a compact convex base of the dual convex cone $\mathcal{K}^{\vee}$ by 3.3. We study the image of $\mathcal{B}$ under an arbitrary linear map

[TABLE]

Let $\pi_{0}\colon V^{\ast}\to(\mathbb{R}\oplus\mathbb{R}^{n})^{\ast}$, $\ell\mapsto(\ell(e),\pi(\ell))$ and let $\pi_{2}\colon(\mathbb{R}\oplus\mathbb{R}^{n})^{\ast}\to(\mathbb{R}^{n})^{\ast}$, $(y_{0},y)\mapsto y$ denote the projection onto the second summand. We denote the dual map to $\pi$, $\pi_{0}$, and $\pi_{2}$ by $\phi:\mathbb{R}^{n}\to V$, $\phi_{0}:\mathbb{R}\oplus\mathbb{R}^{n}\to V$, and $\phi^{\prime}:\mathbb{R}^{n}\to\mathbb{R}\oplus\mathbb{R}^{n}$, respectively. We will describe $\pi(\mathcal{B})$ in terms of the set

[TABLE]

Let $b_{0},b_{1},\dots,b_{n}$ denote the standard basis of $\mathbb{R}^{n+1}=\mathbb{R}\oplus\mathbb{R}^{n}$.

It is easy to see that the set $C$ is a closed convex cone containing $b_{0}$ as an interior point. In addition, $C$ can be seen as a linear section of the convex cone $\mathcal{K}$ through the interior point $e$. To be more precise, let $v_{1},v_{2},\ldots,v_{n}\in V$ and let $\pi_{v_{1},\ldots,v_{n}}:V^{\ast}\to(\mathbb{R}^{n})^{\ast}$ be defined by

[TABLE]

If $\pi=\pi_{v_{1},\ldots,v_{n}}$, then

[TABLE]

holds for all $(x_{0},x)^{T}\in\mathbb{R}\oplus\mathbb{R}^{n}$ where $x=(x_{1},\ldots,x_{n})^{T}$. The image of $\phi_{0}$ is the subspace $\operatorname{im}(\phi_{0})=\operatorname{span}(e,v_{1},\ldots,v_{n})$, which intersects the convex cone $\mathcal{K}$ in the interior point $e$. If $e,v_{1},v_{2},\ldots,v_{n}$ are linearly independent, then the monomorphism $\phi_{0}$ identifies $C$ with a linear section of $\mathcal{K}$ through $e$.

3.3 shows that the set $\operatorname{h}_{b_{0}}(C^{\vee})=\{(y_{0},y)\in C^{\vee}\colon y_{0}=1\}$ is a (compact, convex) base of the dual convex cone $C^{\vee}$, as $b_{0}$ is an interior point of $C$. Obviously, the set $\operatorname{h}_{b_{0}}(C^{\vee})$ is isometric under $\pi_{2}$ to the set

[TABLE]

We show this set is a linear image of the base $\mathcal{B}$ of $\mathcal{K}^{\vee}$.

Proposition 5.2**.**

The image of the convex cone $\mathcal{K}^{\vee}$ under the linear map $\pi_{0}$ is the closed convex cone $\pi_{0}(\mathcal{K}^{\vee})=C^{\vee}$. The image of the set $\mathcal{B}$ is the compact base $\pi_{0}(\mathcal{B})=\operatorname{h}_{b_{0}}(C^{\vee})$ of $C^{\vee}$. In addition we have $\pi(\mathcal{B})=\pi_{2}\circ\operatorname{h}_{b_{0}}(C^{\vee})$.

Proof.

To prove the equation $C^{\vee}=\pi_{0}(\mathcal{K}^{\vee})$, first let $y=\pi_{0}(\ell)\in\pi_{0}(\mathcal{K}^{\vee})$ for some $\ell\in\mathcal{K}^{\vee}$. For all $x\in C$ we have $\phi_{0}(x)\in\mathcal{K}$ and hence

[TABLE]

This shows that $y\in C^{\vee}$ and therefore $\pi_{0}(\mathcal{K}^{\vee})\subset C^{\vee}$. We prove a partial converse by duality, i.e. we show $\left(\pi_{0}(\mathcal{K}^{\vee})\right)^{\vee}\subset C$. Let $x\in\left(\pi_{0}(\mathcal{K}^{\vee})\right)^{\vee}$. Then

[TABLE]

which implies $\phi_{0}(x)\in\mathcal{K}$ or in other words $x\in\phi_{0}^{-1}(\mathcal{K})=C$.

We finish proving $C^{\vee}=\pi_{0}(\mathcal{K}^{\vee})$ by showing that the convex cone $\pi_{0}(\mathcal{K}^{\vee})$ is closed. By 3.1 it suffices to prove that $\pi_{0}(\mathcal{B})$ is a compact base of $\pi_{0}(\mathcal{K}^{\vee})$. The set $\pi_{0}(\mathcal{B})$ is compact as $\mathcal{B}$ is compact. The set $\mathcal{K}^{\vee}\setminus\{0\}$ is the cone generated by $\mathcal{B}$, hence the set $\pi_{0}(\mathcal{K}^{\vee})\setminus\{0\}$ is the cone generated by $\pi_{0}(\mathcal{B})$. It remains to show that the origin does not lie in the affine hull of $\pi_{0}(\mathcal{B})$. Since $\ell(e)=1$ holds for all $\ell\in\mathcal{B}$, the affine hull of $\mathcal{B}$ does not intersect the annihilator $e^{\perp}=\{\ell\in V^{\ast}:\ell(e)=0\}$. As $e\in\operatorname{im}(\phi_{0})$ and $\ker(\pi_{0})=\operatorname{im}(\phi_{0})^{\perp}$, we have $\ker\pi_{0}\subset e^{\perp}$. This shows $\operatorname{aff}(\mathcal{B})\cap\ker\pi_{0}=\emptyset$, which proves that the origin does not lie in $\pi_{0}(\operatorname{aff}(\mathcal{B}))=\operatorname{aff}(\pi_{0}(\mathcal{B}))$.

Using the equation $\pi_{0}(\mathcal{K}^{\vee})=C^{\vee}$ and the projection $\pi_{1}:(\mathbb{R}\oplus\mathbb{R}^{n})^{\ast}\to\mathbb{R}^{\ast}$, $(y_{0},y)\mapsto y_{0}$ onto the first summand, we get

[TABLE]

Finally, $\pi(\mathcal{B})=\pi_{2}\circ\pi_{0}(\mathcal{B})=\pi_{2}\circ\operatorname{h}_{b_{0}}(C^{\vee})$ completes the proof. ∎

5.2 induces a duality of convex sets. Let $b_{0}^{\ast},b_{1}^{\ast},\dots,b_{n}^{\ast}$ denote the standard basis of $(\mathbb{R}^{n+1})^{\ast}=(\mathbb{R}\oplus\mathbb{R}^{n})^{\ast}$. Let $\widetilde{\pi_{2}}\colon\mathbb{R}\oplus\mathbb{R}^{n}\to\mathbb{R}^{n}$, $(x_{0},x)^{T}\mapsto x$ denote the projection onto the second summand. Clearly, the affine slice $\operatorname{h}_{b_{0}^{\ast}}(C)=\{(x_{0},x)^{T}\in C\colon x_{0}=1\}$ of the convex cone $C$ is isometric under $\widetilde{\pi_{2}}$ to the set

[TABLE]

We show this set is the dual convex set to a linear image of the set $\mathcal{B}$.

Corollary 5.3**.**

The dual convex set to $\pi(\mathcal{B})$ is \widetilde{\pi_{2}}\circ\operatorname{h}_{b_{0}^{\ast}}(C)=\big{(}\pi(\mathcal{B})\big{)}^{\circ}. If the set $\pi(\mathcal{B})$ contains the origin, then \pi(\mathcal{B})=\big{(}\widetilde{\pi_{2}}\circ\operatorname{h}_{b_{0}^{\ast}}(C)\big{)}^{\circ}.

Proof.

Let $x\in\mathbb{R}^{n}$. Then

[TABLE]

The first equivalence follows from 5.2. Regarding the fourth equivalence, note that $\operatorname{h}_{b_{0}}(C^{\vee})$ is a base of the convex cone $C^{\vee}$. The remaining equivalences follow immediately from the definitions. As $\pi(\mathcal{B})$ is a closed convex set, the second statement follows from the equation $S=(S^{\circ})^{\circ}$, which holds for all closed convex subsets $S\subset(\mathbb{R}^{n})^{\ast}$ containing the origin [49]. ∎

The convex duality of 5.3 has been used earlier [14, 32, 30] in the context of the joint numerical range. The conic duality of 5.2 has advantages in our situation.

Remark 5.4**.**

If the convex set $\pi(\mathcal{B})$ contains the origin, we could use the convex duality

[TABLE]

of 5.3 to describe the set $\pi(\mathcal{B})$. 5.5 and Figure 4 present an example where Equation 11 fails as $0\in\pi(\mathcal{B})$ fails. A remedy would be to translate $\pi(\mathcal{B})$ along a vector $(\lambda_{1},\ldots,\lambda_{n})\in(\mathbb{R}^{n})^{\ast}$ and apply Equation 11 to the translated set. Indeed, if $v_{1},\ldots,v_{n}\in V$, then the map in Equation 10 yields

[TABLE]

The translation is unnecessary if we use the conic duality

[TABLE]

of 5.2, which is valid even if $0\in\pi(\mathcal{B})$ fails.

The main reason why we employ the conic duality of Equation 12 is that it conveys the algebraic geometry of the dual hyperbolicity cone $C^{\vee}$ described in Section 4 to the convex set $\pi(\mathcal{B})$ directly. The algebraic geometry of the hyperbolicity cone $C$ is hidden behind a convex duality if we use the Equation 11 to describe the set $\pi(\mathcal{B})$.

Example 5.5**.**

Let $\mathcal{K}=\{(\xi_{1},\xi_{2})^{T}\in\mathbb{R}^{2}:\xi_{1},\xi_{2}\geq 0\}$ be the nonnegative quadrant of the plane $V=\mathbb{R}^{2}$ and let $e=(1,1)^{T}\in V$. The segment

[TABLE]

is a base of the dual convex cone $\mathcal{K}^{\vee}=\{z^{T}:z\in\mathcal{K}\}$. Let $v=(2,\tfrac{1}{3})^{T}\in V$. The image of $\mathcal{B}$ under the map $\pi:V^{\ast}\to\mathbb{R}^{\ast}$, $z\mapsto z(v)$ is the interval $[\tfrac{1}{3},2]$, which does not contain the origin, $0\not\in\pi(\mathcal{B})$. We visualize 5.3, Equation 12, and the failure of Equation 11 in Figure 4.

We formulate the results from Section 4 in terms of bases of cones, taking the convex cone $\mathcal{K}\subset V$ in 5.1 equal to a hyperbolicity cone.

Theorem 5.6**.**

Let $p$ be a hyperbolic polynomial on $V$ with hyperbolicity direction $e\in V$ and hyperbolicity cone $C_{e}(p)\subset V$. The image of the compact convex base $\operatorname{h}_{e}(C_{e}(p)^{\vee})$ of the dual convex cone $C_{e}(p)^{\vee}$ under the linear map $\pi$ is

[TABLE]

Here, $p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{r}^{m_{r}}$ is a factorization of the pullback $p\circ\phi_{0}\in\mathbb{R}[x_{0},x_{1},\dots,x_{n}]$ into irreducible polynomials, and

[TABLE]

Proof.

The pullback $p\circ\phi_{0}$ is a hyperbolic polynomial on $\mathbb{R}\oplus\mathbb{R}^{n}$ with hyperbolicity direction $b_{0}$. As $\phi_{0}((x_{0},x)^{T})=x_{0}e+\phi(x)$ holds for all points $(x_{0},x)^{T}\in\mathbb{R}\oplus\mathbb{R}^{n}$, we have $p\circ\phi_{0}(b_{0})=p(e)>0$. The equation

[TABLE]

shows that the polynomial $p\circ\phi_{0}(tb_{0}-a)$ in one variable $t$ has only real roots for every point $a\in\mathbb{R}\oplus\mathbb{R}^{n}$ so that $p\circ\phi_{0}$ is indeed hyperbolic with respect to the point $b_{0}$. Equation 13 also shows

[TABLE]

Using the hyperbolicity cone $\mathcal{K}=C_{e}(p)$ in 5.2, we get

[TABLE]

If $p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{r}^{m_{r}}=p\circ\phi_{0}$ is a factorization into irreducible polynomials, then

[TABLE]

holds. 4.5 proves $C_{b_{0}}(p_{i})^{\vee}=\operatorname{clos}(\operatorname{cc}(C_{i}))$, $i=1,\dots,r$, where

[TABLE]

Now Equation 7 in 3.5 yields

[TABLE]

The claim follows since the map $\pi_{2}:(\mathbb{R}\oplus\mathbb{R}^{n})^{\ast}\to(\mathbb{R}^{n})^{\ast}$ restricts to an affine isomorphism from $\operatorname{h}_{b_{0}}((\mathbb{R}\oplus\mathbb{R}^{n})^{\ast})$ onto $(\mathbb{R}^{n})^{\ast}$, and since $\pi_{2}\circ\operatorname{h}_{b_{0}}(C_{i})=T_{i}$ holds for $i=1,\dots,r$. ∎

Like in 4.14, some of the factors of the pullback $p\circ\phi_{0}=p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{r}^{m_{r}}$ are redundant in 5.6. Let $I\subset\{1,2,\dots,r\}$ be the subset such that $i\in I$ if and only if the hypersurface $\mathcal{V}(p_{i})$ belongs to the algebraic boundary $\partial_{a}C_{b_{0}}(p\circ\phi_{0})$ of the hyperbolicity cone $C_{b_{0}}(p\circ\phi_{0})$ for $i=1,\dots,r$. Then \pi(\mathcal{B})=\operatorname{conv}\Big{(}\operatorname{clos}\big{(}\bigcup_{i\in I}T_{i}\big{)}\Big{)}.

6. The Cone of Positive-Semidefinite Matrices

Here we prove the theorems stated in the introduction by applying our results to the hyperbolicity cone of positive semi-definite matrices. We discuss block diagonalization versus factorization of the determinant. We present examples, among them the promised example by Chien and Nakazato.

Let $V=H_{d}$ be the space of Hermitian $d\times d$ matrices. The determinant is a hyperbolic polynomial on $V$ with respect to any positive definite matrix. The hyperbolicity cone is the set of positive semi-definite matrices

[TABLE]

This convex cone is a self-dual convex cone, i.e. $\mathcal{K}^{\vee}=\mathcal{K}$, as we identify the dual space $V^{\ast}$ with $V$ using the scalar product $\langle A,B\rangle=\operatorname{tr}(AB)$, $A,B\in H_{d}$. In the notation of 5.1, the base $\operatorname{h}_{\mathds{1}}(\mathcal{K}^{\vee})$ of $\mathcal{K}^{\vee}$ is the set of density matrices introduced in Equation 1,

[TABLE]

Let $A_{1},\dots,A_{n}\in H_{d}$ be Hermitian matrices and let $\pi:H_{d}\to(\mathbb{R}^{n})^{\ast}$ be the linear map defined by $\pi(A)=\langle A,A_{i}\rangle_{i=1}^{n}$ for all $A\in H_{d}$. The image of the set $\mathcal{B}$ under $\pi$ is the joint numerical range defined in Equation 2,

[TABLE]

The dual map $\phi=\pi^{\ast}$ has the values $\phi(x)=x_{1}A_{1}+\cdots+x_{n}A_{n}$ and the map $\phi_{0}:\mathbb{R}\oplus\mathbb{R}^{n}\to V$, see 5.1, has the values $\phi_{0}[(x_{0},x)^{T}]=x_{0}\mathds{1}+\phi(x)$ for all $x_{0}\in\mathbb{R}$ and $x=(x_{1},\dots,x_{n})^{T}\in\mathbb{R}^{n}$. Hence, the pullback $\det\circ\phi_{0}$ takes the form

[TABLE]

With this preparation, 1.2 follows from 5.6 immediately. To obtain Kippenhahn’s original result, as stated in 1.1, we use 4.10 instead of 4.7 in the proof of 5.6.

Remark 6.1**.**

In the matrix setting of this section, the convex cone $\phi_{0}^{-1}(\mathcal{K})$ is the spectrahedral cone [46]

[TABLE]

which is the hyperbolicity cone $C_{b_{0}}(p)$ of the polynomial in Equation 14, as we observed more generally in the proof of 5.6 above.

Researchers have studied the relationships between the possibility to block diagonalize the Hermitian matrices $A_{1},\ldots,A_{n}\in H_{d}$ simultaneously and to factor the determinant $p$ in Equation 14. The matrices $A_{1},\ldots,A_{n}$ are called unitarily reducible if there is a $d$-by-$d$ unitary $U$ and an integer $0<j<d$ such that $UA_{i}U^{\ast}=A_{i,1}\oplus A_{i,2}$ is a block diagonal matrix, where $A_{i,1}\in H_{j}$ and $A_{i,2}\in H_{d-j}$ for all $i=1,\dots,n$. Otherwise the matrices $A_{1},\ldots,A_{n}$ are unitarily irreducible.

Clearly, the polynomial $p\in\mathbb{R}[x_{0},x_{1},x_{2}]$ is the product of the polynomials corresponding to the two diagonal blocks for every pair of unitarily reducible matrices $A_{1},A_{2}\in H_{d}$. Interestingly, $p$ may even factor when the matrices $A_{1},A_{2}$ are unitarily irreducible.

Remark 6.2** (Kippenhahn’s conjecture).**

Kippenhahn [38, Prop. 28a] conjectured that if $p$ has a repeated factor, then the matrices $A_{1},A_{2}$ are unitarily reducible. Laffey [39] found a first counterexample to the conjecture by presenting a pair of unitarily irreducible hermitian $8$-by-$8$ matrices $A_{1},A_{2}\in H_{8}$, where $p$ is the square of a polynomial of degree four. The paper [9] summarizes more recent work on the topic and shows that every pair of hermitian $6$-by-$6$ matrices $A_{1},A_{2}\in H_{6}$ is unitarily reducible if $p$ is the square of a polynomial of degree three that defines a smooth cubic.

Similarly, one may ask whether the polynomial $p\in\mathbb{R}[x_{0},x_{1},x_{2}]$ can be a product of several irreducible factors of multiplicity one if the hermitian matrices $A_{1},A_{2}\in H_{d}$ are unitarily irreducible. Such questions have been studied in detail by Kerner and Vinnikov ([36, 37]), resulting in a number of necessary and sufficient conditions. For example, the union of three lines in the projective plane admits a unitarily irreducible determinantal representation (see [36, Remark 3.6]).

Here is a simple example demonstrating that the dual varieties to the irreducible components of the hypersurface $\mathcal{V}(p)$ may have different dimensions.

Example 6.3**.**

Let

[TABLE]

The irreducible components of $\mathcal{V}(p)$ are $X_{1}=\{x\in\mathbb{P}^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=x_{0}^{2}\}$ and $X_{2}=\{x\in\mathbb{P}^{3}:x_{0}+2x_{1}=0\}$. The dual varieties are $X_{1}^{\ast}=\{y\in(\mathbb{P}^{3})^{\ast}:y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=y_{0}^{2}\}$, by 3.8, and $X_{2}^{\ast}=\{(1:2:0:0)\}$. Both $X_{1}^{\ast}$ and $X_{2}^{\ast}$ are smooth, so $T_{1}=\{y\in(\mathbb{R}^{3})^{\ast}:y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1\}$ and $T_{2}=\{(2,0,0)\}$. The joint numerical range $W=\operatorname{conv}(T_{1}\cup T_{2})$ is the convex hull of a sphere and a point outside the sphere.

In our last example, we discuss the original example given by Chien and Nakazato from the point of view of real algebraic geometry in detail. The example explains why singular points of the dual varieties $\mathcal{V}(p_{i})^{\ast}$ have to be excluded from the statement of 5.6.

Example 6.4** (Chien and Nakazato [14]).**

Let

[TABLE]

The hyperbolic cubic form

[TABLE]

is irreducible. The dual variety to $X=\{x\in\mathbb{P}^{3}:p(x)=0\}$ is the hypersurface $X^{\ast}=\{y\in(\mathbb{P}^{3})^{\ast}:q(y)=0\}$ defined by the homogeneous quartic form

[TABLE]

It is easy to show that the singular locus $X^{\ast}\setminus(X^{\ast})_{\mathrm{reg}}$ of $X^{*}$ is the line $\{(y_{0},y_{1},y_{2},y_{3})\in(\mathbb{P}^{3})^{\ast}:y_{2}=y_{3}=0\}$.

The surface $T=\{(y_{1},y_{2},y_{3})\in(\mathbb{R}^{3})^{\ast}\mid(1:y_{1}:y_{2}:y_{3})\in X^{\ast}\}$, depicted in Figure 5, is the zero-set of the polynomial $Q(y_{1},y_{2},y_{3})=q(1,y_{1},y_{2},y_{3})$. All points $(y_{1},0,0)\in T$ on the line of singular points $y_{2}=y_{3}=0$ in the interval $|y_{1}|\leq 1$ are central points of $X^{\ast}$. This can be seen from a parametrization of $X$ described in [52]. In addition, the points $(y_{1},0,0)\in T$ with $|y_{1}|>1$ are not central. This can be seen from the roots of the quadratic polynomial $R_{y_{1},y_{3}}\in\mathbb{R}[y_{2}]$ satisfying $R_{y_{1},y_{3}}(y_{2}^{2})=Q(y_{1},y_{2},y_{3})$. In fact, if $(y_{1},y_{3})\in(\mathbb{R}^{2})^{\ast}$, then the line $\{(y_{1},\lambda,y_{3})\colon\lambda\in\mathbb{R}\}$ intersects $T$ if and only if $R_{y_{1},y_{3}}$ has a non-negative root $y_{2}$. It is not hard to see (for example using Sturm’s theorem [7]) that, under the assumption $y_{3}\neq 0$, the polynomial $R_{y_{1},y_{3}}$ has a non-negative root if and only if $(y_{1},y_{3})$ lies in the convex hull of the union of the singleton $\{(1,0,0)\}$ with the ellipse given by $y_{1}^{2}+5y_{3}^{2}-2y_{1}y_{3}+2y_{1}-6y_{3}+1=y_{2}=0$, the yellow curve in Figure 5 b). Therefore, the points $(y_{1},0,0)$ with $|y_{1}|>1$ are not central. These are the points which are removed from $X^{\ast}(\mathbb{R})$ in the statement of 1.2. The convex hull of the remainder of $T$ is the joint numerical range $W$.

For a general, regular point $(y_{0}:y_{1}:y_{2}:y_{3})$ on $X^{*}$, the dual hyperplane in $\mathbb{P}^{3}$ defined by $x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}=0$ intersects $X$ in an irreducible but singular cubic. The hyperplane sections of the hypersurface $X$ corresponding to points of the line of singular points of $X^{\ast}$ are special in the following way: The sections are reducible curves in $\mathbb{P}^{2}$ that factor into a conic and a line, which is tangent to the conic at a real point. Not all of the points on this line are central points of $X^{*}$. The central points of $X^{*}$ on this line are exactly those that we can perturb such that the dual hyperplane section of $X$ deforms to a real cubic with a real singularity. The other points on this line can only be perturbed on $X^{*}$ to complex points, which means that the dual hyperplane section of $X$ only deforms to complex singular hyperplane sections that are complex cubics with complex singularities.

To see this explicitly, we compute the restrictions of $p$ to hyperplanes in $\mathbb{P}^{3}$ defined by $x_{0}y_{0}+x_{1}y_{1}=0$. Let us assume for simplicity for now that $y_{0}\neq 0$ so that $x_{0}=ax_{1}$ for some $a\in\mathbb{R}$. Then $p$ factors

[TABLE]

So in the plane defined by $x_{0}y_{0}+x_{1}y_{1}=0$, we see the conic

[TABLE]

and the line defined by $x_{0}=x_{1}=0$, which is tangent to the conic at the point $(0:0:0:1)$, which is a singular point of $X$. (The computation for the case $y_{0}=0$ is similar.) For every real value of the parameter $a$, the conic is real and indefinite of full rank except for $a=-1$ and $a=1$. These two values of $a$ bound the interval of central points on the line of singular points on $X^{\ast}$; see Figure 5.

Section 6 of [56] presents further examples of surfaces analogous to the surface $T$ in 6.4 that contain straight lines. These lines are identified in the paper arXiv:1603.06569v3 [math.FA] on the preprint server arXiv.

So why do naive generalizations of Kippenhahn’s Theorem fail in higher dimensions? The reason is that hyperplanes and lines are only the same in $\mathbb{P}^{2}$. A central point of the argument in the proof of Kippenhahn’s Theorem is 4.2, which argues that there are no lines through the interior of the hyperbolicity cone that are tangent to the hyperbolic hypersurface (4.3). The hyperplane corresponding to a real point of the dual variety may well meet an interior point of the hyperbolicity cone as long as the hyperplane contains no tangent line to the hyperbolic hypersurface incident with this interior point. Our generalization holds essentially because this can only happen for hyperplanes that are not central points of the dual variety.

Acknowledgements

We thank H. Nakazato for helpful comments. SW thanks K. Życzkowski for discussions of numerical ranges in quantum mechanics. We would like to thank the anonymous referees for their suggestions to improve the paper.