18 problems

Let $R$ be a commutative ring, let $n$ be a positive integer, and let $A$ be an $n\times n$ symmetric matrix over $R$. For each $S\subseteq [n]$, write $A_S$ for the principal mino…

Let the principal minor map send a complex symmetric matrix to the collection of all its principal minors. Let the group $\operatorname{SL}_2(\mathbb{C})^n\rtimes S_n$ act on the t…

Let $A$ be an $n\times n$ matrix over a field, and for $S\subseteq[n]$ let $A_S$ be the determinant of the principal submatrix indexed by $S$. The principal minor map sends $A$ to…

Let $P_n$ be the prime ideal of the algebraic variety that is the image of the principal-minor map for complex symmetric $n\times n$-matrices. Let the hyperdeterminantal module be…

Let $K$ be a valued field with an automorphic involution preserving the valuation. Let $A=B+C$, where $B\in K^{n\times n}$ is skew-Hermitian and $C\in K^{n\times n}$ is Hermitian o…

Square-free Fourier principal-minor conjecture. If $N$ is square-free, then all principal minors of ${\mathcal{F}}_N$ are nonzero.

Column-permutation conjecture. For every natural number $N$, there exists a permutation $\sigma$ such that the Fourier matrix with permuted columns has no zero principal minors.

Caragea–Lee conjecture. If $N$ is square-free, then every principal minor of $\mathcal F_N$ is nonzero.

Let $\Sigma \in \operatorname{PR}_5$ be a real symmetric matrix whose principal-minor sign pattern $\operatorname{signs}_\Sigma$ agrees with $s_*$ on all subsets of sizes $1$ and…

Cai's conjecture. Under certain assumptions on the moments of the non-Gaussian variables $x_{ij}$, the asymptotic behavior of $T_{m,n,p}$ is similar to that of $T_{m,n,p}$ in the G…

Robbins-number amalgamation conjecture. The principal minor sequence of this amalgamation is the sequence of Robbins numbers.

Let $n\geq 3$ and let $A\in\{0,1\}^{n\times n}$ be a nonsingular symmetric 0–1 matrix. The enhanced principal rank characteristic sequence of a symmetric matrix records, for each o…

Let $A$ and $B$ be $n\times n$ skew-symmetric real matrices. Two such matrices have equal corresponding principal minors of all orders if, for every subset $X\subseteq [n]$, the pr…

Let $M=(x_{ij})$ be an $n\times n$ matrix, with diagonal variables $x_i:=x_{ii}$. Let $M'$ be obtained from $M$ by setting $c=o(n)$ variables to constants. For constant $k$, partit…

Degree-12 equations conjecture. The variety $\mathcal{V}_n$ is cut out by equations of degree $12$.

Let $Z_n$ be the variety of principal minors of symmetric $n\times n$ matrices, and let $HD$ denote the hyperdeterminantal module, namely the span of the…

Finite asymptotic-nullity characterization conjecture. Given any $n$, there exists a finite list of polynomial matrices $P_1,\ldots,P_\ell$ such that a homogeneous ratio…

Let $\frac{\beta}{\alpha}$ be a ratio of products of principal minors, and let $\log\left(\frac{\alpha}{\beta}\right)$ denote its formal logarithm. Write $\mathcal B_n$ for the sem…