16 problems

Let $n,d$ be positive integers, let $T,\mathbf{t}\in\mathbb{C}^n\otimes((\mathbb{C}^n)^*)^{\otimes d}$, and assume that $\mathbf{t}$ is non-singular. For $\lambda_0\in\mathbb{C}$,…

Let $n,d$ be positive integers, let $T,\mathbf{t}\in\mathbb{C}^n\otimes((\mathbb{C}^n)^*)^{\otimes d}$, and assume that $\mathbf{t}$ is non-singular. For $\lambda_0\in\mathbb{C}$,…

Let $H$ be a connected $k$-uniform hypergraph, and let $\rho$ be the spectral radius of its adjacency tensor $\mathcal{A}(H)$. Let $\mathcal{L}(H)$ denote its Laplacian tensor. The…

Let $\mathcal{A}$ be a symmetric biquadratic M-tensor: a symmetric biquadratic tensor that can be written as $\mathcal{A}=\alpha\mathcal{I}-\mathcal{B}$, where $\mathcal{I}$ is the…

Concave multilinearization inequality. Then

Let $H$ be a connected $k$-uniform hypergraph, let $\mathcal{L}(H)$ be its Laplacian tensor, let $\mathcal{A}(H)$ be its adjacency tensor, and let $rho$ be the spectral radius of…

Let $H$ be a connected $k$-uniform hypergraph, let $\mathcal{A}(H)$ be its adjacency tensor, let $rho$ be its spectral radius, and let $\mathbb{V}_{rho}(\mathcal{A}(H))$ be the pro…

Let $\mathcal{A}$ be a $k$-th order, $n$-dimensional tensor, let $lambda$ be an eigenvalue, and let … be its eigenvariety. Suppose that this variety has irreducible components…

Let $\mathbf{w} \in \mathbb R^{n \circ l}$ be an even-order symmetric real tensor, and let $\lambda_k^v(\mathbf{w})$ denote its $k$th variational eigenvalue, defined by … where $S$…

Finite-fiber conjecture for symmetric tensors. All fibers of $\Phi_{\mathrm{sym}}$ are finite.

Characteristic polynomial fiber-dimension conjecture. The generic fiber of $\Phi$ is equidimensional of dimension $n$.

For a tensor, let $\textrm{am}(\lambda)$ be the algebraic multiplicity of an eigenvalue $\lambda$, meaning its multiplicity as a root of the characteristic polynomial, and let…

Let $(\mathbf{x}(s),\lambda(s))$ be the solution curve considered in Theorem 2.8, with $s\in[0,s_{max})$, and suppose the generic condition in part $(iv)$ of that theorem is not im…

Let a lower-order Hankel tensor and its associated higher-order Hankel tensor have the same generating vector, with the higher order a multiple of the lower order. A Hankel tensor…

Let $G=(V,E)$ be an ordinary graph, let $k=2r$ be even, and let $G^k=(V^k,E^k)$ be the $k$-power hypergraph of $G$. Let $\mathcal L^k$ and $\mathcal Q^k$ be the Laplacian and signl…

Let $\boldsymbol{\mathscr{A}}$ be a tensor and let $\{\boldsymbol{x}_k\}$ be the sequence generated by the shifted power method. Suppose the method's convergence proof applies when…