Eigenminimal Tensors: Constructions, Positive Z-Eigenvectors, and Topology

1. Introduction

The spectral theory of higher-order tensors generalizes fundamental concepts from matrix analysis, yet gives rise to behaviors with no direct matrix counterpart. For a real symmetric tensor $\mathcal{A}\in {Sym}^m\left({\mathbb{R}}^n\right),$ a real Z-eigenpair consists of a scalar $\lambda \in \mathbb{R}$ and a unit vector $x\in {\mathbb{R}}^n$ satisfying

$$\mathcal{A}{x}^{m-1}=\lambda x,\parallel x\parallel =1.$$

The Z-eigenvalue problem was introduced independently, in equivalent variational forms, by Qi and Lim [16,21]. If $f_{\mathcal{A}}\left(x\right)=\mathcal{A}{x}^m,$ then the unit Z-eigenvectors of a symmetric tensor are precisely the critical points of $f_{\mathcal{A}}$ on the unit sphere. This variational interpretation places the problem naturally at the intersection of multilinear algebra, polynomial optimization, and critical-point theory; see, for example, [22].

One of the fundamental questions in tensor spectral theory concerns the existence and uniqueness of distinguished real eigenvectors. For nonnegative tensors, a substantial Perron–Frobenius theory has been developed [3,8,10]. For Z-eigenvectors, however, positivity and irreducibility alone do not in general produce the same uniqueness theory as in the matrix case. Transition probability tensors provide an important setting in which this problem has been studied. Sufficient conditions for uniqueness have been obtained using contraction mappings, monotone operator, and fixed-point methods, while special low-dimensional symmetric cases admit stronger uniqueness results [5,7,15]. Transition probability tensors also arise naturally from higher-order stochastic processes and stationary-distribution problems [2,11,14]. Thus, uniqueness of a positive Z-eigenvector generally requires strong additional structural hypotheses.

The viewpoint of the present paper is different. We begin by imposing the condition that the real Z-eigenconfiguration be made as small as topology permits. Once the total number of real eigendirections is minimized, under additional geometric conditions, the existence of at most one Z-eigenvector in the interior of the positive cone is proven. These results emerge from the structure of the Z-eigenconfiguration rather than from a separate Perron–Frobenius argument.

This approach is motivated by the topology underlying the Z-eigenvalue problem. In earlier work, Chang, Pearson, and Zhang [4] used variational methods and LS category to show that a nonzero real symmetric tensor of even order $2k$ and dimension n possesses at least n real projectively distinct Z-eigendirections. We therefore make the following definition.

Definition 1.

Let $k\ge 1$ and $n\ge 2$. A nonzero tensor $\mathcal{A}\in {Sym}^{2k}\left({\mathbb{R}}^n\right)$ is called eigenminimal if it has exactly n real projective Z-eigendirections.

In terms of the associated even order homogeneous polynomials, eigenminimality yields the minimal number of antipodal pairs of critical points on $S^{n-1}$. For matrices ($k=1$), this is precisely the familiar simple-spectrum situation: a real $n\times n$ symmetric matrix with n distinct eigenvalues has exactly n distinct real eigendirections.

For tensors of higher order, the picture is substantially richer. Cartwright and Sturmfels [6] showed that, for $m>2$, the generic projective complex eigenvector count is

$${\displaystyle \frac{{(m-1)}^n-1}{m-2}},$$

which is typically much larger than n. Using spherical harmonics, Kozhasov [13] constructed generic real symmetric tensors for which all of these projective eigenvectors are real.

A broader goal is to characterize even order real symmetric tensors (not necessarily nonnegative) which possess exactly one positive Z-eigenvector in the strict positive cone. In order to control the geometric location of Z-eigenvectors, a natural step is to study these eigenminimal models.

The main contributions of the paper are highlighted as follows:

1.

We explicitly construct three algebraically distinct eigenminimal families that attain the minimum number of real projective Z-eigendirections, in the stated parameter ranges, through different mechanisms. Every Type I and Type II tensor, and every Type III tensor in the parameter range of Theorem 5, has at most one Z-eigenvector in the strict positive cone. For the first two families, any tensor with no positive Z-eigenvector can be orthogonally transformed to an eigenminimal tensor with a unique Z-eigenvector in the strict positive cone. For the third family, whose Z-eigenvectors have full support, the parameters can be chosen so that the resulting tensor has a unique Z-eigenvector in the strict positive cone.

2.

We prove in Theorem 2 that the nondegenerate eigenminimal locus ${\mathsf{EM}}_{2k,n}^{\mathrm{ND}}$ is a nonempty open subset of the ambient space ${Sym}^{2k}\left({\mathbb{R}}^n\right)$. Consequently, it is a Baire space and is of second category in itself.

3.

We prove in Theorem 6 that every Type II tensor and every Type III tensor in the parameter range of Theorem 4 can be deformed through eigenminimal tensors into the Type I family. In the parameter range of Theorem 5, these deformations can be chosen to retain at most one projective Z-eigendirection meeting ${\mathbb{R}}_{>0}^n$. Consequently, the three families, in the stated parameter ranges, lie in a single path component of ${\mathsf{EM}}_{2k,n}$. However, we show in Proposition 2 that the positive-cone uniqueness shared by these particular families is a consequence of their particular geometric structure, not of the path component of the eigenminimal locus containing them. It is also worth mentioning, we do not know whether the entire eigenminimal locus is path connected.

4.

We prove in Theorem 7 for $k\ge 2$ that ${\mathsf{EM}}_{2k,n}$ is not dense in ${Sym}^{2k}\left({\mathbb{R}}^n\right)$. This gives a stark contrast between eigenminimal (simple-spectrum) real symmetric matrices ($k=1$) and higher order ($k\ge 2$) eigenminimal tensors: simple-spectrum real symmetric matrices form an open dense subset of real symmetric matrices, but ${\mathsf{EM}}_{2k,n}$ has nonempty interior and is not dense in ${Sym}^{2k}\left({\mathbb{R}}^n\right)$.

5.

We also exhibit an eigenminimal binary quartic Example 2 having two distinct projective Z-eigendirections that meet the strict positive cone, showing that positive uniqueness is not a consequence of eigenminimality alone.

Three further connections place the Type I construction in other settings. The radial-lift construction is naturally related to the classical Neumann system, whose anisotropic quadratic potential on the sphere has the same critical-point structure as a Type I tensor. In dimension two optic systems, homogeneous lifts of the Zernike modes of azimuthal order $m>0$ have exactly m real projective Z-eigendirections; hence the astigmatic sector $m=2$ gives precisely the eigenminimal case and is a Type I radial lift. Finally, the parameter space of diagonal Type I tensors has a natural arrangement-theoretic interpretation: eigenminimality fails exactly on the hyperplanes $c_i=c_j,$ so the eigenvalue-collision discriminant is the classical braid arrangement; see, for example, [1]. For general background on hyperplane arrangements and their topology, see [20]. These observations are included at the end as connections to other areas. The central theme throughout, however, is the construction, geometry, and topology of minimal real Z-eigenconfigurations.

In summary, the aim of this paper is not to give a complete classification of tensors with unique Z-eigenvectors or minimal real Z-eigenconfigurations. Rather, it introduces eigenminimality as a framework for studying the global real Z-eigenvector structure and provides several explicit mechanisms by which the topological lower bound can be attained. For Types I and II, positive uniqueness is controlled by orthogonal geometry, while for Type III, in the parameter range of Theorem 5, it follows from the cyclic coupling structure. The binary quartic example 2 shows that neither conclusion follows from eigenminimality alone. Together with perturbative stability, non-density, and the path connectness among the three model families, these results describe several distinct geometric features of the minimal real Z-eigenconfiguration regime.

2. Type I: Simple-Spectrum Radial Lifts

This section introduces a special family of even-order symmetric tensors whose real projective Z-eigendirections attain the lower bound in Definition 1.1. The construction begins with a quadratic form having simple spectrum and raises its degree by multiplication with a radial factor. Since that factor is constant on the unit sphere, the spherical critical-point structure is unchanged.

We begin with a lemma to show that eigenminimality of a tensor is preserved under orthogonal changes of coordinates.

Lemma 1

(Orthogonal equivariance). Let $\mathcal{A}\in {Sym}^m\left({\mathbb{R}}^n\right)$, $f_{\mathcal{A}}\left(x\right)=\mathcal{A}{x}^m$, and $Q\in O\left(n\right)$. We define

$$\tilde{f}\left(x\right):=f_{\mathcal{A}}\left(Q^{\top }x\right),$$

and let $\tilde{\mathcal{A}}\in {Sym}^m\left({\mathbb{R}}^n\right)$ be the symmetric tensor satisfying $\tilde{f}\left(x\right)=\tilde{\mathcal{A}}{x}^m.$ Then $(\lambda ,u)$ is a real unit Z-eigenpair of $\mathcal{A}$ if and only if $(\lambda ,Qu)$ is a real unit Z-eigenpair of $\tilde{\mathcal{A}}$.

Consequently, the $O\left(n\right)$-action preserves the real projective Z-eigenconfiguration and the corresponding Z-eigenvalues. In particular, when m is even, eigenminimality is preserved under orthogonal changes of coordinates.

Proof. 

First, since Q is orthogonal, we have $\parallel Qu\parallel =\parallel u\parallel .$ By the chain rule,

$$\nabla \tilde{f}\left(x\right)={\left(Q^{\top }\right)}^{\top }\nabla f_{\mathcal{A}}\left(Q^{\top }x\right)=Q\nabla f_{\mathcal{A}}\left(Q^{\top }x\right).$$

Since $\nabla f_{\mathcal{A}}\left(x\right)=m\mathcal{A}{x}^{m-1}$ and $\nabla \tilde{f}\left(x\right)=m\tilde{\mathcal{A}}{x}^{m-1},$ we obtain

$$m\tilde{\mathcal{A}}{x}^{m-1}=mQ\mathcal{A}{\left(Q^{\top }x\right)}^{m-1}.$$

Now suppose that $(\lambda ,u)$ is a unit Z-eigenpair of $\mathcal{A}$. Setting $x=Qu$ gives $Q^{\top }x=u$, and hence

$$\tilde{\mathcal{A}}{\left(Qu\right)}^{m-1}=Q\mathcal{A}{u}^{m-1}=Q\left(\lambda u\right)=\lambda Qu.$$

Thus, $(\lambda ,Qu)$ is a unit Z-eigenpair of $\tilde{\mathcal{A}}$.

Conversely, applying the same argument with $Q^{\top }$ shows that every unit Z-eigenpair of $\tilde{\mathcal{A}}$ arises in this way. Therefore, $u⟼Qu$ gives a bijection between the real unit Z-eigenvectors of $\mathcal{A}$ and those of $\tilde{\mathcal{A}}$, preserving their Z-eigenvalues. Passing to projective eigendirections gives the final assertions. □

Definition 2.

Let $C\in {\mathbb{R}}^{n\times n}$ be a symmetric matrix, it then defines the quadratic form

$$q_C\left(x\right)=x^{\top }Cx,$$

and let $m=2k$ with $k\ge 1$. The degree-$2k$ homogeneous polynomial

$$f_C\left(x\right):=q_C\left(x\right){\parallel x\parallel }^{2k-2}=\left(x^{\top }Cx\right){\left(x^{\top }x\right)}^{k-1}$$

(1)

is called the radial lift of $q_C$ to degree $2k$. We denote by ${\mathcal{A}}_C\in {Sym}^{2k}\left({\mathbb{R}}^n\right)$ the unique symmetric tensor satisfying $f_C\left(x\right)={\mathcal{A}}_C{x}^{2k}.$ If C has simple spectrum, that is, C has only simple eigenvalues, then ${\mathcal{A}}_C$ is called a simple-spectrum radial-lift tensor or simply a Type I tensor, shown to be eigenminimal in Theorem 1.

We note that

$$f_C{{|}_{S^{n-1}}=q_C|}_{S^{n-1}}.$$

Thus, although $f_C$ has degree $2k$, its restriction to the sphere has the same critical points as the original quadratic form.

Theorem 1

(Radial lifts are eigenminimal). Let $C\in {\mathbb{R}}^{n\times n}$ be a symmetric matrix with distinct eigenvalues ${\mu }_1,\dots ,{\mu }_n$, and let $u_1,\dots ,u_n$ be a corresponding orthonormal eigenbasis. For any $k\ge 1$, let ${\mathcal{A}}_C$ be the order-$2k$ symmetric tensor associated with (1). Then the real unit Z-eigenvectors of ${\mathcal{A}}_C$ are precisely

$$\{\pm u_1,\dots ,\pm u_n\}.$$

Moreover,

$${\mathcal{A}}_C{u}_i^{2k-1}={\mu }_i{u}_i,1\le i\le n.$$

Consequently, ${\mathcal{A}}_C$ has exactly n real projective Z-eigendirections and is eigenminimal.

Proof. 

For the associated homogeneous polynomial $f_C\left(x\right)={\mathcal{A}}_C{x}^{2k}$, one has

$$\nabla f_C\left(x\right)=2k{\mathcal{A}}_C{x}^{2k-1}.$$

Hence, the unit Z-eigenvectors of ${\mathcal{A}}_C$ are exactly the critical points of $f_C$ restricted to $S^{n-1}$. On the unit sphere,

$$f_C\left(x\right)=x^{\top }Cx,$$

so it suffices to determine the critical points of $f_C\left(x\right)=x^{\top }Cx$ on $S^{n-1}$.

By the method of Lagrange multipliers, a unit vector x is critical if and only if $Cx=\mu x$ for some $\mu \in \mathbb{R}$. Equivalently, the critical points are the unit eigenvectors of C. Since C has simple spectrum, each eigenspace is one-dimensional, and the complete set of critical points is $\{\pm u_1,\dots ,\pm u_n\}$.

It remains to identify the corresponding Z-eigenvalues. For $k\ge 2$, differentiating (1) gives

$$\nabla f_C\left(x\right)={2Cx\parallel x\parallel }^{2k-2}+2(k-1)\left(x^{\top }Cx\right){\parallel x\parallel }^{2k-4}x.$$

For $k=1$, the same conclusion follows directly from $f_C\left(x\right)=x^{\top }Cx$. At a unit eigenvector $u_i$, where $C{u}_i={\mu }_i{u}_i$ and $u_i^{\top }C{u}_i={\mu }_i$, this reduces to

$$\nabla f_C\left(u_i\right)=2k{\mu }_i{u}_i.$$

Therefore,

$${\mathcal{A}}_C{u}_i^{2k-1}={\mu }_i{u}_i.$$

There are exactly n antipodal pairs, so ${\mathcal{A}}_C$ is eigenminimal. □

Corollary 1

(Diagonal radial lifts). Let $c_1,\dots ,c_n$ be distinct real numbers and set $C=diag(c_1,\dots ,c_n)$. Then

$$f_C\left(x\right)=\left({\displaystyle \sum _{i=1}^n}{c}_i{x}_i^2\right){\left({\displaystyle \sum _{i=1}^n}{x}_i^2\right)}^{k-1}.$$

(I)

The associated tensor is eigenminimal, its unit Z-eigenvectors are $\pm e_1,\dots ,\pm e_n$, and the Z-eigenvalue corresponding to $\pm e_i$ is $c_i$.

The crucial assumption is that the eigenvalues of C are distinct. In the diagonal form, this is the condition $c_i\ne c_j$ for $i\ne j$. If an eigenvalue has multiplicity greater than one, then every unit vector in the corresponding eigenspace is a Z-eigenvector, and the tensor has infinitely many real projective Z-eigendirections.

The behavior under orthogonal changes of coordinates distinguishes Z-eigenvectors from H-eigenvectors. The H-eigenvalue equation

$$\mathcal{A}{x}^{m-1}=\lambda x^{[m-1]}$$

involves coordinatewise powers and is therefore not preserved by $O\left(n\right)$. By contrast, Lemma 1 shows that the real Z-eigenconfiguration is preserved under orthogonal changes of coordinates.

Corollary 2

(A unique positive Z-eigenvector). A Type I tensor has at most one projective Z-eigendirection that intersects the strict positive cone ${\mathbb{R}}_{>0}^n$. It has a unique Z-eigenvector in ${\mathbb{R}}_{>0}^n$ if and only if one eigenvector of C can be chosen with all strictly positive coordinates.

Proof. 

The Z-eigendirections are represented by an orthonormal eigenbasis of C. Two nonzero vectors in ${\mathbb{R}}_{>0}^n$ have strictly positive inner product and therefore cannot be orthogonal. Hence, at most one eigendirection can meet the strict positive cone. The characterization of existence is immediate. □

Corollary 3

(Prescribing a unique positive Z-eigenvector). Let $v\in {\mathbb{R}}_{>0}^n$ be a unit vector, and let ${\mu }_1,\dots ,{\mu }_n$ be distinct real numbers. Then, for every $k\ge 1$, there exists a Type I eigenminimal tensor $\mathcal{A}\in {Sym}^{2k}\left({\mathbb{R}}^n\right)$ whose unique unit Z-eigenvector in ${\mathbb{R}}_{>0}^n$ is v. Furthermore, v may be prescribed to have Z-eigenvalue ${\mu }_1$.

Proof. 

Choose $Q\in SO\left(n\right)$ whose first column is v, and set $D=diag({\mu }_1,\dots ,{\mu }_n)$ and $C=QD{Q}^{\top }.$ Then C has simple spectrum, with orthonormal eigenvectors

$$Q{e}_1=v,Q{e}_2,\dots ,Q{e}_n$$

and corresponding eigenvalues ${\mu }_1,\dots ,{\mu }_n$.

Let ${\mathcal{A}}_C$ be the Type I radial-lift tensor associated with

$$f_C\left(x\right)=\left(x^{\top }Cx\right){\parallel x\parallel }^{2k-2}.$$

By Theorem 1, its real unit Z-eigenvectors are precisely $\{\pm Q{e}_1,\dots ,\pm Q{e}_n\},$ with $Q{e}_i$ corresponding to the Z-eigenvalue ${\mu }_i$.

Since $Q{e}_1=v\in {\mathbb{R}}_{>0}^n$, it remains only to prove uniqueness. If $\pm Q{e}_i\in {\mathbb{R}}_{>0}^n$ for some $i\ne 1$, then $\langle v,\pm Q{e}_i\rangle >0,$ because both vectors would have strictly positive coordinates. This contradicts the orthogonality of the columns of Q. Hence, no other unit Z-eigenvector lies in ${\mathbb{R}}_{>0}^n$, and v is the unique one. □

Example 1.

Let $n=2$, let $C=diag(c_1,c_2)$ with $c_1\ne c_2$, and let

$$Q_{\theta }=\left(\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right),0<\theta <{\displaystyle \frac{\pi }{2}}.$$

For $\tilde{C}=Q_{\theta }C{Q}_{\theta }^{\top }$, the eigendirection represented by $Q_{\theta }{e}_1={(cos\theta ,sin\theta )}^{\top }$ lies in ${\mathbb{R}}_{>0}^2$, whereas the other eigendirection does not. Thus the rotated radial lift has exactly one projective Z-eigendirection meeting the strict positive cone.

The preceding uniqueness conclusion is a consequence of the orthogonal geometry of Type I tensors which does not follow from eigenminimality alone.

Example 2

(Eigenminimality does not imply positive uniqueness). Consider the binary quartic

$$f(x,y)=-x^4-x^{3}y+x{y}^3+y^4=(y^4-x^4)+xy(y^2-x^2),$$

with associated tensor $\mathcal{A}\in {Sym}^4\left({\mathbb{R}}^2\right)$.

Passing to polar coordinates on $S^1$, we write $x=cos\theta$ and $y=sin\theta$. Then

$$\begin{array}{cc}\hfill & y^2-x^2={sin}^2\theta -{cos}^2\theta =-cos\left(2\theta \right)\hfill \\ \hfill & y^4-x^4=(y^2-x^2)(y^2+x^2)=-cos\left(2\theta \right)\hfill \\ \hfill & xy=sin\theta cos\theta ={\displaystyle \frac{1}{2}}sin\left(2\theta \right)\hfill \end{array}$$

Substituting these back yields:

$$g\left(\theta \right):=f(cos\theta ,sin\theta )=-cos\left(2\theta \right)-{\displaystyle \frac{1}{4}}sin\left(4\theta \right)$$

It follows that the Z-eigenvectors on $S^1$ correspond to the critical points of $g\left(\theta \right)$.

By solving

$$\begin{array}{cc}\hfill 0& =g^{\prime }\left(\theta \right)=2sin\left(2\theta \right)-cos\left(4\theta \right)\hfill \\ \hfill & =2sin\left(2\theta \right)-\left(1-2{sin}^2\left(2\theta \right)\right),\hfill \end{array}$$

which yields $2{sin}^2\left(2\theta \right)+2sin\left(2\theta \right)-1=0$. Let $w=sin\left(2\theta \right)$, then

$$w={\displaystyle \frac{-2\pm \sqrt{4-4\left(2\right)(-1)}}{2\left(2\right)}}={\displaystyle \frac{-1\pm \sqrt{3}}{2}}$$

Thus, the exact critical condition simplifies to:

$$sin\left(2\theta \right)={\displaystyle \frac{\sqrt{3}-1}{2}}$$

Since $sin\left(2\theta \right)={\displaystyle \frac{\sqrt{3}-1}{2}}>0$, the angle $2\theta$ lies in Quadrants I and II (modulo $2\pi$), this implies

1. 

$2{\theta }_1=arcsin\left({\displaystyle \frac{\sqrt{3}-1}{2}}\right)\in \left(0,{\displaystyle \frac{\pi }{2}}\right)\Rightarrow {\theta }_1\in \left(0,{\displaystyle \frac{\pi }{4}}\right)$

2. 

$2{\theta }_2=\pi -2{\theta }_1\in \left({\displaystyle \frac{\pi }{2}},\pi \right)\Rightarrow {\theta }_2\in \left({\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{2}}\right)$

Hence,

1. 

The tensor has exactly 2 real projective Z-eigendirections represented by ${\theta }_1$ and ${\theta }_2$, confirming that it is eigenminimal ($n=2$).

2. 

Since both ${\theta }_1$ and ${\theta }_2$ lie in $(0,\pi /2)$, both projective Z-eigendirections meet the strict positive cone ${\mathbb{R}}_{>0}^2$.

This confirms that eigenminimality alone does not imply the uniqueness of a positive Z-eigenvector.

Proposition 1

(Nondegeneracy of Type I tensors). Let $C\in {\mathbb{R}}^{n\times n}$ be symmetric with distinct eigenvalues ${\mu }_1,\dots ,{\mu }_n$, and let $u_1,\dots ,u_n$ be a corresponding orthonormal eigenbasis. Then all n antipodal pairs of critical points $\pm u_1,\dots ,\pm u_n$ of $f_C{|}_{S^{n-1}}$, where $f_C\left(x\right)=\left(x^{\top }Cx\right){\parallel x\parallel }^{2k-2},$ are nondegenerate.

Proof. 

Since

$$f_C{{|}_{S^{n-1}}=q_C|}_{S^{n-1}},q_C\left(x\right)=x^{\top }Cx,$$

it suffices to examine the critical points of $q_C$ on $S^{n-1}$.

Fix an eigenvector $u_i$. In an orthonormal eigenbasis of C, the Hessian of $q_C{|}_{S^{n-1}}$ at $u_i$, restricted to the tangent space $T_{u_i}{S}^{n-1}$, has eigenvalues

$$2({\mu }_j-{\mu }_i),j\ne i.$$

Since the eigenvalues of C are distinct, ${\mu }_j-{\mu }_i\ne 0$ for $j\ne i$. Hence, the restricted Hessian is nonsingular, so $u_i$ is a nondegenerate critical point. The same argument applies at $-u_i$. Therefore, all $2n$ critical points $\pm u_1,\dots ,\pm u_n$ are nondegenerate. □

3. Stability of Minimal Z-Eigenconfigurations

The Type I construction shows that the minimum number of real Z-eigendirections can be attained by explicit tensors. We now show that this phenomenon is stable under perturbation in the full space of symmetric tensors. This implies eigenminimality is not confined to the special parameter families used to construct examples.

For fixed $k\ge 1$ and $n\ge 2$, we denote the eigenminimal locus by

$${\mathsf{EM}}_{2k,n}:=\left\{\mathcal{A}\in {Sym}^{2k}\left({\mathbb{R}}^n\right)\setminus \left\{0\right\}:\mathcal{A}\mathrm{is}\mathrm{eigenminimal}\right\}.$$

Equivalently, $\mathcal{A}\in {\mathsf{EM}}_{2k,n}$ if the restriction $h_{\mathcal{A}}:=f_{\mathcal{A}}{|}_{S^{n-1}}$ for $f_{\mathcal{A}}\left(x\right)=\mathcal{A}{x}^{2k},$ has exactly $2n$ critical points.

We call an eigenminimal tensor $\mathcal{A}$ nondegenerate if every critical point of $h_{\mathcal{A}}$ is nondegenerate, i.e. the Hessian of $h_{\mathcal{A}}$ restricted to the corresponding tangent space of $S^{n-1}$ is nonsingular. Let ${\mathsf{EM}}_{2k,n}^{\mathrm{ND}}\subseteq {\mathsf{EM}}_{2k,n}$ denote the set of nondegenerate eigenminimal tensors.

Theorem 2

(Stability Principle). The nondegenerate eigenminimal locus ${\mathsf{EM}}_{2k,n}^{\mathrm{ND}}$ is a nonempty open subset of ${Sym}^{2k}\left({\mathbb{R}}^n\right)$ with respect to the Euclidean topology on the tensor coefficients. Consequently, it is itself a Baire space of the second category.

Proof. 

By the construction of Type I tensors, ${\mathsf{EM}}_{2k,n}^{\mathrm{ND}}\ne \varnothing$. Let ${\mathcal{A}}_0\in {\mathsf{EM}}_{2k,n}^{\mathrm{ND}}$ and set $h_0:=h_{{\mathcal{A}}_0}.$ Since ${\mathcal{A}}_0$ is eigenminimal, $h_0$ has exactly $2n$ nondegenerate critical points, which we write as $\pm u_1,\dots ,\pm u_n.$

Choose pairwise disjoint neighborhoods

$$U_1^{+},\dots ,U_n^{+},U_1^{-},\dots ,U_n^{-}$$

of these critical points. By nondegeneracy and the Implicit Function Theorem, if $\mathcal{A}$ is sufficiently close to ${\mathcal{A}}_0$, then $h_{\mathcal{A}}$ has a unique nondegenerate critical point in each $U_r^{\pm }$.

It remains to exclude additional critical points. Set

$$\Omega :=S^{n-1}\setminus {\displaystyle \bigcup _{r=1}^n}\left(U_r^{+}\cup U_r^{-}\right).$$

The set $\Omega$ is compact and contains no critical point of $h_0$. Hence,

$$\delta :=\underset{x\in \Omega }{min}\parallel {\nabla }_{S^{n-1}}{h}_0\left(x\right)\parallel >0.$$

The map $\mathcal{A}\mapsto h_{\mathcal{A}}$ depends continuously on the tensor coefficients in the $C^2$ topology on $S^{n-1}$. Therefore, after shrinking the neighborhood of ${\mathcal{A}}_0$ if necessary,

$$\underset{x\in S^{n-1}}{sup}∥{\nabla }_{S^{n-1}}{h}_{\mathcal{A}}\left(x\right)-{\nabla }_{S^{n-1}}{h}_0\left(x\right)∥<{\displaystyle \frac{\delta }{2}}.$$

It follows that, for every $x\in \Omega$,

$$\parallel {\nabla }_{S^{n-1}}{h}_{\mathcal{A}}\left(x\right)\parallel \ge \parallel {\nabla }_{S^{n-1}}{h}_0\left(x\right)\parallel -∥{\nabla }_{S^{n-1}}{h}_{\mathcal{A}}\left(x\right)-{\nabla }_{S^{n-1}}{h}_0\left(x\right)∥>{\displaystyle \frac{\delta }{2}}.$$

Thus, $h_{\mathcal{A}}$ has no critical points in $\Omega$. Consequently, $h_{\mathcal{A}}$ has exactly the $2n$ nondegenerate critical points lying in the neighborhoods $U_r^{\pm }$. Since the tensor order is even, $h_{\mathcal{A}}(-x)=h_{\mathcal{A}}\left(x\right),$ so these critical points occur in antipodal pairs. Hence, $\mathcal{A}$ has exactly n real projective Z-eigendirections and remains nondegenerate. Thus, $\mathcal{A}\in {\mathsf{EM}}_{2k,n}^{\mathrm{ND}},$ and the assertion follows. □

Corollary 4

(Eigenminimal tensors have nonempty interior). For every $k\ge 1$ and $n\ge 2$, the set ${\mathsf{EM}}_{2k,n}$ has nonempty interior in ${Sym}^{2k}\left({\mathbb{R}}^n\right)$.

Proof. 

By Theorem 1 and Proposition 1, every simple-spectrum Type I radial lift is a nondegenerate eigenminimal tensor. Theorem 2 therefore gives an open neighborhood of each such tensor contained in ${\mathsf{EM}}_{2k,n}$. □

The preceding theorem is particularly useful for perturbative constructions.

Corollary 5

(Stability along parameter families). Let $t\mapsto {\mathcal{A}}_t\in {Sym}^{2k}\left({\mathbb{R}}^n\right)$ be a continuous family defined for t in an interval containing 0, and suppose that ${\mathcal{A}}_0\in {\mathsf{EM}}_{2k,n}^{\mathrm{ND}}$. Then there exists $\epsilon >0$ such that

$$\left|t\right|<\epsilon ⟹{\mathcal{A}}_t\in {\mathsf{EM}}_{2k,n}^{\mathrm{ND}}.$$

In particular, the number of real projective Z-eigendirections remains equal to n for all sufficiently small t.

Proof. 

By Theorem 2, ${\mathsf{EM}}_{2k,n}^{\mathrm{ND}}$ is an open neighborhood of ${\mathcal{A}}_0$. The conclusion follows immediately from the continuity of $t\mapsto {\mathcal{A}}_t$. □

In other words, sufficiently small perturbations of a nondegenerate eigenminimal tensor may move the eigendirections and their Z-eigenvalues, but they cannot create or destroy real projective Z-eigendirections. This stability principle will be used later in constructing the Type III cyclic family.

4. Type II: Hub-Coupled Radial Lifts

We next introduce a second eigenminimal family in which one coordinate plays a distinguished role. We regard $x_n$ as a hub coordinate and $x_1,\dots ,x_{n-1}$ as peripheral coordinates. The construction combines a radial lift of a diagonal quadratic form on the peripheral variables with additional terms coupling each peripheral variable to the hub. The ordering and sign assumptions below rule out critical points having mixed support.

Definition 3.

Let $n\ge 2$, let $m=2k$ with $k\ge 2$, and choose real parameters satisfying

$$0<a_1<a_2<\cdots <a_{n-1}\le b,c_1<c_2<\cdots <c_{n-1}<0.$$

(2)

Define the degree-$2k$ homogeneous polynomial

$$f_H\left(x\right):={\displaystyle \sum _{i=1}^{n-1}}{a}_i{x}_i^2{x}_n^{2k-2}+b{x}_n^{2k}+\left({\displaystyle \sum _{i=1}^{n-1}}{c}_i{x}_i^2\right){\left({\displaystyle \sum _{j=1}^n}{x}_j^2\right)}^{k-1}.$$

(II)

The symmetric tensor ${\mathcal{A}}_H\in {Sym}^{2k}\left({\mathbb{R}}^n\right)$ determined by $f_H\left(x\right)={\mathcal{A}}_H{x}^{2k}$ is called a hub-coupled radial-lift tensor or simply a Type II tensor. Theorem 3 shows that every Type II tensor is eigenminimal.

Theorem 3

(Hub-coupled radial lifts are eigenminimal). Let ${\mathcal{A}}_H$ be the tensor in Definition 3. Its real unit Z-eigenvectors are precisely $\{\pm e_1,\dots ,\pm e_n\}.$ The corresponding Z-eigenvalues are

$${\lambda }_i=c_i(1\le i\le n-1),{\lambda }_n=b.$$

Consequently, ${\mathcal{A}}_H$ has exactly n real projective Z-eigendirections and is eigenminimal.

Proof. 

On $S^{n-1}$, the radial factor in the final term of (II) equals one. Hence,

$$h\left(x\right):=f_H{|}_{S^{n-1}}\left(x\right)={\displaystyle \sum _{i=1}^{n-1}}{a}_i{x}_i^2{x}_n^{2k-2}+b{x}_n^{2k}+{\displaystyle \sum _{i=1}^{n-1}}{c}_i{x}_i^2.$$

(3)

The unit Z-eigenvectors of ${\mathcal{A}}_H$ are exactly the critical points of h on $S^{n-1}$. Writing the Lagrange equations in the form ${\displaystyle \frac{1}{2}}\nabla h\left(x\right)=\mu x$ gives

$$\begin{array}{cccc}\hfill x_i\left(a_i{x}_n^{2k-2}+c_i-\mu \right)& =0,\hfill & \hfill & 1\le i\le n-1,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill x_n\left((k-1)x_n^{2k-4}{\displaystyle \sum _{j=1}^{n-1}}{a}_j{x}_j^2+kb{x}_n^{2k-2}-\mu \right)& =0.\hfill \end{array}$$

(5)

Let

$$I\left(x\right):=\{i\in \{1,\dots ,n-1\}:x_i\ne 0\}$$

be the peripheral support of x.

Suppose first that $x_n=0$. Since $x\in S^{n-1}$, the set $I\left(x\right)$ is nonempty. For each $i\in I\left(x\right)$, equation (4) gives $\mu =c_i$. The numbers $c_1,\dots ,c_{n-1}$ are distinct, so $I\left(x\right)$ contains exactly one index. The sphere constraint then yields $x=\pm e_i$ for some $i<n$.

Now suppose that $x_n\ne 0$, and put $t=x_n^2>0$. Dividing (5) by $x_n$ gives

$$\mu =(k-1)t^{k-2}{\displaystyle \sum _{j=1}^{n-1}}{a}_j{x}_j^2+kb{t}^{k-1}.$$

(6)

We first show that $\left|I\right(x\left)\right|\le 1$. Indeed, if $i<j$ both belong to $I\left(x\right)$, then equation (4) implies

$$a_i{t}^{k-1}+c_i=a_j{t}^{k-1}+c_j,$$

and therefore

$$(a_i-a_j)t^{k-1}=c_j-c_i.$$

The left-hand side is negative, whereas the right-hand side is positive, a contradiction.

It remains to exclude $\left|I\right(x\left)\right|=1$. Suppose $I\left(x\right)=\left\{i\right\}$. Then $x_i^2=1-t$, and equations (4) and (6) give, respectively,

$$\mu =a_i{t}^{k-1}+c_i$$

and

$$\mu =(k-1)a_i(1-t)t^{k-2}+kb{t}^{k-1}.$$

Equating these expressions and simplifying yields

$$0=c_i-(k-1)a_i{t}^{k-2}+k(a_i-b)t^{k-1}.$$

(7)

Every term on the right is nonpositive, and the first two are strictly negative: $c_i<0$, $a_i>0$, $t>0$, and $a_i\le b$, which means (7) is impossible. Hence, $I\left(x\right)=\varnothing$, and the sphere constraint gives $x=\pm e_n$.

We have therefore found exactly the $2n$ unit Z-eigenvectors $\pm e_1,\dots ,\pm e_n$. For a unit Z-eigenvector x, its Z-eigenvalue is $\lambda =f_H\left(x\right)$. Evaluating (II) at the coordinate vectors gives $f_H\left(e_i\right)=c_i$ for $i<n$ and $f_H\left(e_n\right)=b$. This proves the stated eigenvalue formulas and the eigenminimality. □

We note the polynomial in (II) is not the radial lift of a single quadratic form.

4.1. Orthogonal Images and Positive Z-Eigenvectors

Lemma 1 immediately produces orthogonal images of the Type II family.

Corollary 6

(Orthogonal images of Type II tensors). Let $Q\in O\left(n\right)$ and define

$${\tilde{f}}_Q\left(x\right):=f_H\left(Q^{\top }x\right).$$

If ${\tilde{\mathcal{A}}}_Q$ is the symmetric tensor associated with ${\tilde{f}}_Q$, then its real unit Z-eigenvectors are precisely $\{\pm Q{e}_1,\dots ,\pm Q{e}_n\}.$ The corresponding Z-eigenvalues are $c_1,\dots ,c_{n-1},b,$ respectively. In particular, ${\tilde{\mathcal{A}}}_Q$ is eigenminimal.

Proof. 

This follows immediately from Theorem 3 and Lemma 1. □

Corollary 7

(A unique positive Z-eigenvector). In the setting of Corollary 6, suppose that $Q{e}_r\in {\mathbb{R}}_{>0}^n$ for some $1\le r\le n$. Then $Q{e}_r$ is the unique unit Z-eigenvector of ${\tilde{\mathcal{A}}}_Q$ in the strict positive cone.

Proof. 

By Corollary 6, every unit Z-eigenvector is of the form $\pm Q{e}_i$. If another such vector were strictly positive, then its inner product with $Q{e}_r$ would be positive. This contradicts the orthogonality of the columns of Q. □

5. Type III: Full-Support Cyclic Perturbations

The first two constructions produce eigenminimal tensors whose Z-eigenvectors can be identified exactly. We now introduce a third family with odd-parity terms interactions. The interactions are arranged cyclically, so a zero in any coordinate propagates around the entire cycle. Consequently, every real Z-eigenvector has full support in the displayed cyclic coordinate system. A simple-spectrum radial-lift core provides a nondegenerate eigenminimal configuration, and sufficiently small cyclic coupling preserves the minimal number of real projective Z-eigendirections.

The small-coupling assumption allows us to apply the stability principle of Section 3 to the Type I tensor obtained at $\epsilon =0$. This gives a rigorous perturbative guarantee that the minimal real Z-eigenconfiguration persists for sufficiently small $\left|\epsilon \right|$. We make no claim here that eigenminimality persists for arbitrary coupling strength.

5.1. Type III Eigenminimal Family

Definition 4.

Let $n\ge 2$, let $m=2k$ with $k\ge 2$, and choose pairwise distinct real numbers $c_1,\dots ,c_n$ together with positive coupling constants $b_1,\dots ,b_n$. We now adopt cyclic indexing, so that $x_{n+1}=x_1$, and set

$$q_c\left(x\right):={\displaystyle \sum _{i=1}^n}{c}_i{x}_i^2\mathrm{and}{p}_b\left(x\right):={\displaystyle \sum _{i=1}^n}{b}_i{x}_i{x}_{i+1}^{2k-1}.$$

For $\epsilon \in \mathbb{R}$, we define

$$\begin{array}{cc}\hfill f_{\epsilon }\left(x\right)& :=q_c\left(x\right){\parallel x\parallel }^{2k-2}+\epsilon p_b\left(x\right)\hfill \\ \hfill & =\left({\displaystyle \sum _{i=1}^n}{c}_i{x}_i^2\right){\left({\displaystyle \sum _{j=1}^n}{x}_j^2\right)}^{k-1}+\epsilon {\displaystyle \sum _{i=1}^n}{b}_i{x}_i{x}_{i+1}^{2k-1}.\hfill \end{array}$$

(III)

Let ${\mathcal{A}}_{\epsilon }\in {Sym}^{2k}\left({\mathbb{R}}^n\right)$ be the symmetric tensor determined by $f_{\epsilon }\left(x\right)={\mathcal{A}}_{\epsilon }{x}^{2k}$. For $\epsilon \ne 0$, we call ${\mathcal{A}}_{\epsilon }$ a full-support cyclic perturbation or simply a Type III cyclic tensor.

We show in the following Theorem 4, for sufficiently small nonzero cyclic perturbations, Type III cyclic tensors are also eigenminimal.

Although each coupling monomial contains odd powers of individual coordinates, its total degree is even. Thus $f_{\epsilon }(-x)=f_{\epsilon }\left(x\right)$, and the real unit Z-eigenvectors occur in antipodal pairs with the same Z-eigenvalue.

Lemma 2

(Full support). Let $\epsilon \ne 0$. Every real Z-eigenvector of ${\mathcal{A}}_{\epsilon }$ has full support. Equivalently, if $x\in S^{n-1}$ is a real Z-eigenvector, then

$$x_i\ne 0,1\le i\le n.$$

Proof. 

Use cyclic conventions for both the coordinates and the coupling constants. A direct differentiation gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial f_{\epsilon }}{\partial x_i}}\left(x\right)& =2x_i\left(c_i{\parallel x\parallel }^{2k-2}+(k-1)q_c\left(x\right){\parallel x\parallel }^{2k-4}\right)\hfill \\ \hfill & +\epsilon \left(b_i{x}_{i+1}^{2k-1}+(2k-1)b_{i-1}{x}_{i-1}{x}_i^{2k-2}\right).\hfill \end{array}$$

(8)

Suppose that $x_i=0$ for some i. Since x is a unit Z-eigenvector,

$${\displaystyle \frac{\partial f_{\epsilon }}{\partial x_i}}\left(x\right)=2k\lambda x_i=0.$$

Because $k\ge 2$, substituting $x_i=0$ into (8) leaves $\epsilon b_i{x}_{i+1}^{2k-1}=0.$ Since $\epsilon \ne 0$ and $b_i>0$, it follows that $x_{i+1}=0$. Repeating this argument around the cycle forces every coordinate of x to vanish, contradicting $x\in S^{n-1}$. Hence, every coordinate is nonzero. □

Theorem 4

(Small cyclic perturbations are eigenminimal). Let ${\mathcal{A}}_{\epsilon }$ be the tensor in Definition 4. There exists ${\epsilon }_0>0$ such that, whenever $0<\left|\epsilon \right|<{\epsilon }_0,$ the real unit Z-eigenvectors of ${\mathcal{A}}_{\epsilon }$ consist of exactly n antipodal pairs $\{\pm u_1\left(\epsilon \right),\dots ,\pm u_n\left(\epsilon \right)\}.$ The branches may be chosen smoothly for $\left|\epsilon \right|<{\epsilon }_0$ starting from $u_r\left(0\right)=e_r$ for $1\le r\le n.$ For every nonzero ε in this range,

$$supp{u}_r\left(\epsilon \right)=\{1,\dots ,n\}.$$

All of these critical points are nondegenerate after ${\epsilon }_0$ is chosen small enough. Moreover, the corresponding Z-eigenvalues ${\lambda }_r\left(\epsilon \right)=f_{\epsilon }\left(u_r\left(\epsilon \right)\right)$ depend smoothly on ε and satisfy ${\lambda }_r\left(0\right)=c_r$. Consequently, ${\mathcal{A}}_{\epsilon }$ has exactly n real projective Z-eigendirections and is eigenminimal.

Proof. 

At $\epsilon =0$,

$$f_0\left(x\right)=\left({\displaystyle \sum _{i=1}^n}{c}_i{x}_i^2\right){\parallel x\parallel }^{2k-2}$$

is a simple-spectrum Type I radial lift. By Theorem 1 and Proposition 1, ${\mathcal{A}}_0$ is nondegenerate and eigenminimal. Since $\epsilon \mapsto {\mathcal{A}}_{\epsilon }$ is continuous, Corollary 5 gives ${\epsilon }_0>0$ such that ${\mathcal{A}}_{\epsilon }$ remains nondegenerate and eigenminimal whenever $\left|\epsilon \right|<{\epsilon }_0$. The Implicit Function Theorem therefore gives smooth critical-point branches

$$\pm u_1\left(\epsilon \right),\dots ,\pm u_n\left(\epsilon \right),u_r\left(0\right)=e_r,$$

which account for all real unit Z-eigenvectors after adjusting ${\epsilon }_0$ if necessary. For $\epsilon \ne 0$, Lemma 2 gives

$$supp{u}_r\left(\epsilon \right)=\{1,\dots ,n\}.$$

Lastly, ${\lambda }_r\left(\epsilon \right)=f_{\epsilon }\left(u_r\left(\epsilon \right)\right)$ depends smoothly on $\epsilon$ and satisfies ${\lambda }_r\left(0\right)=c_r$. □

Lemma 3

(Boundary exclusion on the spherical orthant). Let $K=S^{n-1}\cap {\mathbb{R}}_{\ge 0}^n,$ and let h be a $C^1$ function on a neighborhood of K. We denote $\partial K$ the boundary of K relative to $S^{n-1}$. If for every $x\in \partial K$, there is an index i such that

$$x_i=0\mathit{and}{\displaystyle \frac{\partial h}{\partial x_i}}\left(x\right)>0.$$

then h cannot attain its maximum on $\partial K$. Similarly, if for every $x\in \partial K$, there is an index i such that

$$x_i=0\mathit{and}{\displaystyle \frac{\partial h}{\partial x_i}}\left(x\right)<0,$$

then h cannot attain its minimum on $\partial K$.

Proof. 

Let $x\in \partial K$ and suppose that $x_i=0$. Consider the inward spherical curve

$$\gamma \left(s\right)={\displaystyle \frac{x+s{e}_i}{\sqrt{1+s^2}}},s\ge 0.$$

Since $\parallel x\parallel =1$ and $x_i=0$, we have $\parallel x+s{e}_i{\parallel }^2=1+s^2,$ so $\gamma \left(s\right)\in K$. Moreover,

$$\gamma \left(0\right)=x\mathrm{and}{\gamma }^{\prime }\left(0\right)=e_i.$$

Therefore,

$${\left.{\displaystyle \frac{d}{ds}}h\left(\gamma \left(s\right)\right)\right|}_{s=0}={\displaystyle \frac{\partial h}{\partial x_i}}\left(x\right).$$

If this derivative is positive, then $h\left(\gamma \right(s\left)\right)>h\left(x\right)$ for all sufficiently small $s>0$, so x cannot be a maximum of h on K. The same argument applies to a negative derivative. □

Theorem 5

(A unique positive Z-eigenvector for Type III). Let ${\mathcal{A}}_{\epsilon }$ be the Type III tensor of Definition 4. Let $r_{+}$ and $r_{-}$ be the unique indices satisfying

$$c_{r_{+}}=\underset{1\le i\le n}{max}{c}_i\mathrm{and}{c}_{r_{-}}=\underset{1\le i\le n}{min}{c}_i.$$

After decreasing the constant ${\epsilon }_0$ in Theorem 4 if necessary, the following hold:

1. 

If $0<\epsilon <{\epsilon }_0,$ then ${\mathcal{A}}_{\epsilon }$ has exactly one unit Z-eigenvector in ${\mathbb{R}}_{>0}^n$. It belongs to the branch $u_{r_{+}}\left(\epsilon \right)$ issuing from $e_{r_{+}}$.

2. 

If $-{\epsilon }_0<\epsilon <0,$ then ${\mathcal{A}}_{\epsilon }$ has exactly one unit Z-eigenvector in ${\mathbb{R}}_{>0}^n$. It belongs to the branch $u_{r_{-}}\left(\epsilon \right)$ issuing from $e_{r_{-}}$.

Proof. 

On $S^{n-1}$, the Type III polynomial restricts to

$$h_{\epsilon }\left(x\right)=q_c\left(x\right)+\epsilon p_b\left(x\right)={\displaystyle \sum _{i=1}^n}{c}_i{x}_i^2+\epsilon {\displaystyle \sum _{i=1}^n}{b}_i{x}_i{x}_{i+1}^{2k-1}.$$

Its critical points on $S^{n-1}$ are precisely the unit Z-eigenvectors of ${\mathcal{A}}_{\epsilon }$.

Consider the restriction of $h_{\epsilon }$ to the compact spherical orthant $K=S^{n-1}\cap {\mathbb{R}}_{\ge 0}^n.$

By the Extreme Value Theorem, $h_{\epsilon }\left(x\right)$ attains both a global maximum and minimum on K. Suppose $x\in \partial K$. Since $x\ne 0$ and the coordinates are cyclically indexed, there is an index i such that

$$x_i=0\mathrm{and}{x}_{i+1}>0.$$

At such a point,

$${\displaystyle \frac{\partial h_{\epsilon }}{\partial x_i}}\left(x\right)=\epsilon b_i{x}_{i+1}^{2k-1}.$$

Since $b_i>0$ and $x_{i+1}>0$, this derivative is positive when $\epsilon >0$ and negative when $\epsilon <0$. According to Lemma 3, when $\epsilon >0$, a global maximum of $h_{\epsilon }$ on K lies in ${\mathbb{R}}_{>0}^n$, while, when $\epsilon <0$, a global minimum lies in ${\mathbb{R}}_{>0}^n$.

We now prove uniqueness. For any critical point $x\in K$ with Lagrange multiplier $\mu$, the critical-point equations ${\displaystyle \frac{\partial h_{\epsilon }}{\partial x_i}}=2\mu x_i$ simplify to:

$$2(\mu -c_i)x_i=\epsilon \left(b_i{x}_{i+1}^{2k-1}+(2k-1)b_{i-1}{x}_{i-1}{x}_i^{2k-2}\right)$$

Since $x\in {\mathbb{R}}_{>0}^n$ and $b_j>0$, the term inside the parenthesis is strictly positive. Dividing by $2x_i>0$ shows that $\mu -c_i$ has the same sign as $\epsilon$ for every i:

1.

If $\epsilon >0$: $\mu >c_i$ for all $i\Rightarrow \mu >{max}_i{c}_i=c_{r_{+}}$.

2.

If $\epsilon <0$: $\mu <c_i$ for all $i\Rightarrow \mu <{min}_i{c}_i=c_{r_{-}}$.

By Theorem 4, for sufficiently small $\left|\epsilon \right|<{\epsilon }_0$, all real unit Z-eigenvectors lie on the n antipodal branch pairs

$$\pm u_r\left(\epsilon \right),1\le r\le n,$$

with $u_r\left(0\right)=e_r$. Their corresponding Lagrange multipliers satisfy

$$\underset{\epsilon \to 0}{lim}{\mu }_r\left(\epsilon \right)=c_r.$$

Since $c_1,\dots ,c_n$ are distinct, by continuity, we have that

1.

For $\epsilon >0$. After shrinking ${\epsilon }_0>0$ if necessary, we have ${\mu }_r\left(\epsilon \right)<c_{r_{+}}$ ($r\ne r_{+}$) for $0<\epsilon <{\epsilon }_0$. But every positive critical point satisfies $\mu >c_{r_{+}}$. Hence, no branch with $r\ne r_{+}$ can meet the strict positive cone. Since a positive critical point has already been shown to exist, it must belong to the branch $u_{r_{+}}\left(\epsilon \right)$.

2.

Similar argument applies to $\epsilon <0$. This completes the proof.

□

To consider the relation to the previous two eigenminimal families, we note that when $\epsilon =0$, equation (III) is the diagonal simple-spectrum radial lift from Section 2. Turning on the cyclic interaction breaks the coordinatewise sign symmetries and moves all n eigendirections away from the coordinate axes. Unlike the hub-coupled family of Section 4, no single coordinate is structurally designated as a hub: every coordinate is involved in one incoming and one outgoing cyclic interaction. In summary, the three constructions use different mechanisms: spectral simplicity, hub exclusion, and cyclic full-support propagation.

Full support in the Type III construction is a property of the displayed cyclic coordinate system, whereas eigenminimality itself is preserved under arbitrary orthogonal changes of coordinates. An arbitrary orthogonal image remains eigenminimal, but an orthogonal transformation need not preserve the positive cone, so the positive-eigenvector conclusion does not automatically pass to arbitrary orthogonal images.

Corollary 8

(Orthogonal images of Type III tensors). Assume $0<\left|\epsilon \right|<{\epsilon }_0,$ let $Q\in O\left(n\right)$, and define

$${\tilde{f}}_{\epsilon ,Q}\left(x\right):=f_{\epsilon }\left(Q^{\top }x\right).$$

If ${\tilde{\mathcal{A}}}_{\epsilon ,Q}$ is the associated symmetric tensor, then its real unit Z-eigenvectors are precisely

$$\{\pm Q{u}_1\left(\epsilon \right),\dots ,\pm Q{u}_n\left(\epsilon \right)\}.$$

The corresponding Z-eigenvalues are

$${\lambda }_1\left(\epsilon \right),\dots ,{\lambda }_n\left(\epsilon \right),$$

and ${\tilde{\mathcal{A}}}_{\epsilon ,Q}$ is eigenminimal.

Proof. 

This follows immediately from Theorem 4 and Lemma 1. □

5.2. A Common Path Component

Although the three preceding families attain the minimal real Z-eigenconfiguration by different mechanisms, they are not separated topologically inside the eigenminimal locus ${\mathsf{EM}}_{2k,n}$. In fact, each of the three families can be connected through eigenminimal tensors to the Type I family.

Definition 5.

For $n\ge 2$ and $k\ge 1$, we define

$${\mathsf{EM}}_{2k,n}^{\le 1}:=\left\{\mathcal{A}\in {\mathsf{EM}}_{2k,n}:\mathcal{A}\mathit{has}\mathit{at}\mathit{most}\mathit{one}\mathit{real}\mathit{projective}Z\text{-}\mathit{eigendirection}\mathit{in}{\mathbb{R}}_{>0}^n\right\}.$$

Theorem 6

(Common path component of the three model families). Let $n\ge 2$ and $k\ge 2$. We have that

1. 

Every Type I, Type II, and Type III tensor in the parameter range of Theorem 4 lie in a single path component of ${\mathsf{EM}}_{2k,n}.$

2. 

Every Type I, Type II, and Type III tensor in the smaller parameter range of Theorem 5 lie in a single path component of ${\mathsf{EM}}_{2k,n}^{\le 1}.$

Proof. 

We first show that the Type I locus is path connected. We write

$$C=Qdiag({\mu }_1,\dots ,{\mu }_n)Q^{\top },{\mu }_1<\cdots <{\mu }_n.$$

After changing the sign of one eigenvector if necessary, we may assume $Q\in SO\left(n\right)$. Choose a path $Q_t$ in $SO\left(n\right)$ from Q to $I_n$, the identity matrix. Then

$$C_t=Q_{t}diag({\mu }_1,\dots ,{\mu }_n)Q_t^{\top }$$

has simple spectrum for every t. Thus, the associated radial lifts give a Type I path from ${\mathcal{A}}_C$ to the diagonal tensor associated with $diag({\mu }_1,\dots ,{\mu }_n)$. If $d_1<\cdots <d_n$ are fixed, then

$${\mu }_i\left(t\right)=(1-t){\mu }_i+t{d}_i$$

remain strictly ordered, so every diagonal Type I tensor is connected to the fixed radial lift associated with $D=diag(d_1,\dots ,d_n)$. Hence, the Type I locus is path connected. By Corollary 2, this path lies in ${\mathsf{EM}}_{2k,n}^{\le 1}$.

For a Type II tensor, we consider the path

$$f_{H,t}:=t\left({\displaystyle \sum _{i=1}^{n-1}}{a}_i{x}_i^2{x}_n^{2k-2}+b{x}_n^{2k}\right)+\left({\displaystyle \sum _{i=1}^{n-1}}{c}_i{x}_i^2\right){\parallel x\parallel }^{2k-2},0\le t\le 1.$$

For $t>0$, this is again Type II, since

$$0<t{a}_1<\cdots <t{a}_{n-1}\le tb.$$

At $t=0$, it is the Type I radial lift associated with $diag(c_1,\dots ,c_{n-1},0),$ which has simple spectrum. Moreover, for $t>0$, its eigendirections are the coordinate axes, so the entire path lies in ${\mathsf{EM}}_{2k,n}^{\le 1}$.

For a Type III tensor, we use the path

$$f_{t\epsilon }:=q_c\left(x\right){\parallel x\parallel }^{2k-2}+t\epsilon p_b\left(x\right),0\le t\le 1.$$

If $0<\left|\epsilon \right|<{\epsilon }_0$ is in the range of Theorem 4, then $0<\left|t\epsilon \right|<{\epsilon }_0$ for every $t>0$, so the path is eigenminimal. At $t=0$, it is the Type I radial lift associated with $diag(c_1,\dots ,c_n)$. If $\epsilon$ lies in the smaller range of Theorem 5, then every $t>0$ point on the path has exactly one projective Z-eigendirection meeting ${\mathbb{R}}_{>0}^n$, while the Type I endpoint has at most one. Hence, this path lies in ${\mathsf{EM}}_{2k,n}^{\le 1}$. Since the Type I locus is path connected, the two assertions follow. □

The above theorem does not assert that ${\mathsf{EM}}_{2k,n}^{\le 1}$ itself is path connected.

Proposition 2

(The positive-cone count is not a path-component invariant). The number of real projective Z-eigendirections inside ${\mathbb{R}}_{>0}^n$ is not constant on path components of the eigenminimal locus. In particular, an eigenminimal tensor lying in the same path component as the Type I family need not have at most one projective Z-eigendirection meeting the strict positive cone.

Proof. 

Consider the one-parameter family of binary quartics ($k=2$ and $n=2$):

$$f_t(x,y)=y^4-x^4+txy(y^2-x^2),0\le t\le 1.$$

At $t=0$,

$$f_0(x,y)=(y^2-x^2)(x^2+y^2),$$

which is the Type I radial lift associated with

$$C=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right).$$

Thus $f_0$ is eigenminimal.

We show that the entire path remains eigenminimal. On $S^1$, we write $x=cos\theta$ and $y=sin\theta$. Then

$$g_t\left(\theta \right):=f_t(cos\theta ,sin\theta )=-cos\left(2\theta \right)-{\displaystyle \frac{t}{4}}sin\left(4\theta \right),$$

so

$$g_t^{\prime }\left(\theta \right)=2sin\left(2\theta \right)-tcos\left(4\theta \right).$$

For $t>0$, set $w=sin\left(2\theta \right).$ Using $cos\left(4\theta \right)=1-2w^2,$ the critical-point equation becomes

$$2t{w}^2+2w-t=0.$$

Its roots are

$$w_{\pm }\left(t\right)={\displaystyle \frac{-1\pm \sqrt{1+2t^2}}{2t}}.$$

For $0<t\le 1$,

$$0<w_{+}\left(t\right)<1,w_{-}\left(t\right)<-1.$$

Hence, only $w_{+}\left(t\right)$ can occur as $sin\left(2\theta \right)$, and the critical-point equation reduces to

$$sin\left(2\theta \right)=w_{+}\left(t\right).$$

Modulo $\pi$ in $\theta$, this equation has exactly two solutions. Consequently, $f_t$ has exactly two real projective Z-eigendirections for every $0<t\le 1$. Since the same is true at $t=0$, the entire path $\{f_t:0\le t\le 1\}$ lies in ${\mathsf{EM}}_{4,2}$.

Moreover, because $w_{+}\left(t\right)>0$, for every $t>0$ the two projective eigendirections have representatives with angles

$$0<{\theta }_1\left(t\right)<{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{4}}<{\theta }_2\left(t\right)<{\displaystyle \frac{\pi }{2}}.$$

Thus, both projective Z-eigendirections meet ${\mathbb{R}}_{>0}^2$. At $t=1$ we recover the tensor of Example 2,

$$f_1(x,y)=-x^4-x^{3}y+x{y}^3+y^4.$$

Therefore, this tensor lies in the same path component of the eigenminimal locus as a Type I tensor, even though both of its projective Z-eigendirections meet the strict positive cone. □

This shows that the positive-cone uniqueness shared by the Type I and Type II families is a consequence of their particular geometric structure, not of the path component of the eigenminimal locus containing them.

6. Global Size of the Eigenminimal Locus

Section 3 shows that the eigenminimal locus ${\mathsf{EM}}_{2k,n}$ has nonempty interior in the full space ${Sym}^{2k}\left({\mathbb{R}}^n\right)$. We now show that, beginning in order four, eigenminimality is nevertheless not dense. This implies both minimal and nonminimal real Z-eigenconfigurations persist on open subsets of ${Sym}^{2k}\left({\mathbb{R}}^n\right)$

Theorem 7

(Eigenminimal tensors are not dense for higher even order). Let $n\ge 2$ and $k\ge 2$. Then ${\mathsf{EM}}_{2k,n}$ is not dense in ${Sym}^{2k}\left({\mathbb{R}}^n\right)$. More precisely, its complement also contains a nonempty open subset.

Proof. 

Choose $a_1,\dots ,a_n>0$ and consider the homogeneous polynomial

$$g\left(x\right)={\displaystyle \sum _{i=1}^n}{a}_i{x}_i^{2k},$$

with associated symmetric tensor $\mathcal{B}$. The critical points of ${g|}_{S^{n-1}}$ satisfy

$$2k{a}_i{x}_i^{2k-1}=2\mu x_i\mathrm{for}1\le i\le n.$$

Hence, whenever $x_i\ne 0$, $k{a}_i{\left|x_i\right|}^{2k-2}=\mu .$ In particular, $\mu >0$. Fix a nonempty support $S\subseteq \{1,\dots ,n\}.$ For $i\in S$, the critical-point equations yield

$$|x_i{|}^2=a_i^{-1/(k-1)}{\left({\displaystyle \frac{\mu }{k}}\right)}^{1/(k-1)}.$$

Summing over all indices in S and setting ${\parallel x\parallel }^2=1$ gives:

$${\left({\displaystyle \frac{\mu }{k}}\right)}^{1/(k-1)}\sum _{j\in S}{a}_j^{-1/(k-1)}=1⟹{\left({\displaystyle \frac{\mu }{k}}\right)}^{1/(2k-2)}={\displaystyle \frac{1}{{\left({\sum }_{j\in S}{a}_j^{-1/(k-1)}\right)}^{1/2}}},$$

whence,

$$|x_i|={\displaystyle \frac{a_i^{-1/(2k-2)}}{{\left({\displaystyle \sum _{j\in S}{a}_j^{-1/(k-1)}}\right)}^{1/2}}}.$$

Thus, if $\left|S\right|=s$, there are $2^s$ choices of signs, giving $2^{s-1}$ projectively distinct Z-eigendirections. Summing over all nonempty supports gives

$$\#\left\{\mathrm{real}\mathrm{projective}\text{Z-eigendirections}\mathrm{of}\mathcal{B}\right\}={\displaystyle \sum _{s=1}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{s}}\right)2^{s-1}={\displaystyle \frac{3^n-1}{2}}.$$

Clearly for $n\ge 2$, ${\displaystyle \frac{3^n-1}{2}}>n$, the tensor $\mathcal{B}$ is therefore not eigenminimal.

We next show that all of these critical points are nondegenerate. Let x be a critical point with support S, and consider the Lagrangian

$$L\left(x\right)=g\left(x\right)-\mu (\parallel x{\parallel }^2-1).$$

Its Hessian is diagonal. Using the fact

$$k{a}_i{\left|x_i\right|}^{2k-2}=\mu \mathrm{for}i\in S,$$

the diagonal entries are

$$4(k-1)\mu (i\in S)\mathrm{and}-2\mu (i\notin S).$$

Since x is supported on S, the tangent space decomposes as

$$T_x{S}^{n-1}=\left(T_x{S}^{n-1}\cap {\mathbb{R}}^S\right)\oplus {\mathbb{R}}^{S^c}.$$

The constrained Hessian therefore acts by the nonzero scalar $4(k-1)\mu$ on the first summand and by $-2\mu$ on the second. Thus, every critical point of ${g|}_{S^{n-1}}$ is nondegenerate.

By the same Implicit Function Theorem and compactness argument used in Theorem 2, all of these critical points persist under sufficiently small perturbations of $\mathcal{B}$, and no additional critical points appear. Hence there exists an open neighborhood $\mathcal{U}$ of $\mathcal{B}$ such that every $\mathcal{A}\in \mathcal{U}$ has exactly ${\displaystyle \frac{3^n-1}{2}}$ real projective Z-eigendirections. Since this number is strictly greater than n for $n\ge 2$,

$$\mathcal{U}\cap {\mathsf{EM}}_{2k,n}=\varnothing .$$

Therefore, the complement of ${\mathsf{EM}}_{2k,n}$ contains a nonempty open set, and ${\mathsf{EM}}_{2k,n}$ is not dense. □

Combining this theorem with the stability principle from Section 3, we arrive at

Corollary 9.

Let $n\ge 2$ and $k\ge 2$. Then both ${\mathsf{EM}}_{2k,n}$ and ${Sym}^{2k}\left({\mathbb{R}}^n\right)\setminus {\mathsf{EM}}_{2k,n}$ have nonempty interior.

Proof. 

By Corollary 4, ${\mathsf{EM}}_{2k,n}$ has nonempty interior. By Theorem 7, its complement also contains a nonempty open subset. □

Remark 1.

The matrix case provides a useful contrast. When $k=1$, eigenminimality for a real symmetric matrix is equivalent to simple spectrum. Since simple-spectrum symmetric matrices form an open dense subset of ${Sym}^2\left({\mathbb{R}}^n\right)$, eigenminimality is open and dense in order two.

For $k\ge 2$, the situation is fundamentally different. The eigenminimal locus contains full-dimensional open subsets, but so does its complement. Thus, minimal real Z-eigenconfigurations are neither lower-dimensional exceptions nor generic dense phenomena. Instead, tensor space contains open regions supporting distinct, locally stable real Z-eigenconfiguration types.

7. Further Connections

We now present three connections that place the Type I construction in broader mathematical and applied contexts.

7.1. The Neumann System and Type I Eigenminimal Tensors

The classical Neumann system describes a particle constrained to $S^{n-1}$ and subject to an anisotropic quadratic potential

$$V\left(x\right)={\displaystyle \sum _{i=1}^n}{c}_i{x}_i^2,\mathrm{where}{c}_1,\dots ,c_n\in \mathbb{R}\mathrm{are}\mathrm{distinct}.$$

It is a classical completely integrable Hamiltonian system and is closely related to geodesic motion on quadrics and to spectral theory [9,12,18]. Passing to the homogeneous polynomial $f\left(x\right)=V\left(x\right){\left({\sum }_{i=1}^n{x}_i^2\right)}^{k-1}$ does not change the restriction to the unit sphere. Consequently, the equilibrium directions of the Neumann potential are exactly the Z-eigendirections of the associated Type I radial-lift tensor.

7.2. Zernike Modes and Type I Eigenminimal Tensors

There is a simple connection between Type I eigenminimal tensors in dimension two and the angular part of the real Zernike basis. Several indexing and normalization schemes are used in the optics literature, including Noll’s single-index ordering, the traditional double-index notation, and conventions used for reporting ocular wavefront aberrations [17,19,23,24]. In the usual double-index notation, a real Zernike mode has the form

$$Z_N^m(\rho ,\theta )=R_N^m\left(\rho \right)cos\left(m\theta \right)$$

or

$$Z_N^{-m}(\rho ,\theta )=R_N^m\left(\rho \right)sin\left(m\theta \right),$$

with $R_N^m\left(1\right)=1$. Hence its restriction to the unit circle is $cos\left(m\theta \right)$ or $sin\left(m\theta \right)$.

For even $N=2k$, define the homogeneous representatives

$$H_{N,m}^c(x,y)={(x^2+y^2)}^{(N-m)/2}Re\left({(x+iy)}^m\right),$$

and

$$H_{N,m}^s(x,y)={(x^2+y^2)}^{(N-m)/2}Im\left({(x+iy)}^m\right).$$

On $S^1$ these restrict to $cos\left(m\theta \right)$ and $sin\left(m\theta \right)$, respectively.

Proposition 3

(The eigenminimal Zernike sector). Let $N=2k$ and $m>0$ be an admissible Zernike azimuthal order. The tensors associated with $H_{N,m}^c$ and $H_{N,m}^s$ have exactly m real projective Z-eigendirections. Consequently, they are eigenminimal if and only if $m=2.$ In this case they belong to the Type I eigenminimal family.

Proof. 

On $S^1$, the two homogeneous representatives are

$$cos\left(m\theta \right)\mathrm{and}sin\left(m\theta \right).$$

Each has $2m$ critical points on the unit circle, hence m antipodal pairs and therefore m real projective Z-eigendirections. Since the minimum in dimension two is 2, eigenminimality occurs exactly when $m=2$:

$$H_{2k,2}^c(x,y)=(x^2-y^2){(x^2+y^2)}^{k-1},$$

and

$$H_{2k,2}^s(x,y)=2xy{(x^2+y^2)}^{k-1}.$$

These have the Type I form $\left(x^{\top }Cx\right){\parallel x\parallel }^{2k-2},$ with, respectively,

$$C=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),C=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$$

Both matrices have simple spectrum, so both representatives are Type I eigenminimal tensors. □

The case $m=2$ is the astigmatic Zernike sector. Thus, after homogeneous lifting, the astigmatic modes are precisely the nonradial even Zernike modes that attain the eigenminimal minimum in dimension two. For $m\ge 4$ there are $m>2$ projective Z-eigendirections, while the radial mode $m=0$ is constant on $S^1$ and has infinitely many.

7.3. An Arrangement Viewpoint on the Type I Parameter Space

The Type I family has a natural interpretation in terms of the braid arrangement. Consider the diagonal radial lifts

$$f_c\left(x\right)=\left({\displaystyle \sum _{i=1}^n}{c}_i{x}_i^2\right){\left({\displaystyle \sum _{i=1}^n}{x}_i^2\right)}^{k-1},c=(c_1,\dots ,c_n)\in {\mathbb{R}}^n.$$

By Theorem 1, the associated tensor is eigenminimal precisely when the parameters $c_1,\dots ,c_n$ are pairwise distinct.

For $1\le i<j\le n$, consider the hyperplane

$$H_{ij}=\{c_i=c_j\},$$

and let

$${\mathcal{B}}_n=\{H_{ij}:1\le i<j\le n\}$$

be the braid arrangement of type $A_{n-1}$.

Proposition 4.

The real parameter space of diagonal Type I eigenminimal tensors is

$${\mathcal{P}}_{\mathrm{I}}^{\mathbb{R}}={\mathbb{R}}^n\setminus \bigcup _{i<j}{H}_{ij}.$$

Hence, the failure locus for Type I eigenminimality is exactly the braid arrangement. The complement has $n!$ chambers, corresponding to the possible strict orderings of $c_1,\dots ,c_n$.

Proof. 

For $C=diag(c_1,\dots ,c_n)$, simple spectrum is equivalent to $c_i\ne c_j$ for $i\ne j$. The result therefore follows directly from Theorem 1. If $c_i=c_j$, the corresponding eigenspace has dimension at least two, so the radial lift has infinitely many projective Z-eigendirections. □

Remark 2.

The same braid arrangement also appears naturally in the Type III construction. Indeed, the Type III family is obtained by perturbing a Type I radial lift with pairwise distinct parameters $c_1,\dots ,c_n$. Thus, the hyperplanes $c_i=c_j$ again describe degeneracy of the radial core. The Type III results are perturbative, however, and do not give a description of its full eigenminimal parameter space as an arrangement complement.

8. Conclusions

We have introduced eigenminimal tensors as even-order symmetric tensors whose real projective Z-eigenconfiguration attains the minimum allowed by Lusternik–Schnirelmann theory. The resulting viewpoint is global: rather than isolating a distinguished Z-eigenvector from the outset, one first controls the complete real eigenconfiguration and then asks what additional geometry can be extracted from that minimal configuration.

Three explicit families realize this minimum by different mechanisms. Type I tensors arise from simple-spectrum quadratic forms through radial lifting and inherit the orthogonal eigendirections of the underlying matrix problem. Type II tensors use a hub-coupled structure together with ordering and sign conditions to exclude mixed-support critical points. Type III tensors are obtained by small cyclic perturbations of a nondegenerate radial-lift core; their cyclic interactions force every real Z-eigenvector to have full support. For Types I and II, orthogonality implies that at most one projective eigendirection can meet the strict positive cone. For sufficiently small nonzero Type III perturbations, we prove directly that there is exactly one unit Z-eigenvector in the strict positive cone. On the other hand, an explicit eigenminimal binary quartic has two projective Z-eigendirections meeting the strict positive cone. Thus, positive uniqueness is not a consequence of eigenminimality alone, but can arise from additional geometric structure within particular eigenminimal families.

Eigenminimality also exhibits substantial stability. Nondegenerate eigenminimal tensors persist under sufficiently small perturbations, so ${\mathsf{EM}}_{2k,n}$ has nonempty interior in ${Sym}^{2k}\left({\mathbb{R}}^n\right)$. For $k\ge 2$, however, the eigenminimal locus is not dense: its complement also contains nonempty open subsets. Hence, higher-order symmetric tensor space contains distinct locally stable regions supporting minimal and nonminimal real Z-eigenconfigurations. Moreover, although the three model families attain eigenminimality through different mechanisms, they all lie in a single path component of ${\mathsf{EM}}_{2k,n}$. This does not assert that the full eigenminimal locus is path connected, and the global topology of ${\mathsf{EM}}_{2k,n}$ remains to be understood.

The connections with the Neumann system, Zernike modes, and the braid arrangement further show that minimal real Z-eigenconfigurations interact naturally with classical structures from dynamical systems, optics, and arrangement theory. These examples suggest several directions for further study, including the construction and classification of additional eigenminimal families, the determination of the path components and other topological features of the eigenminimal locus, and the identification of structural conditions under which eigenminimality yields existence or uniqueness of Z-eigenvectors in the strict positive cone or in other geometrically distinguished regions. More broadly, the results suggest that questions of distinguished real Z-eigenvectors can profitably be studied through the geometry of the complete real Z-eigenconfiguration.