10 problems

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.

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…

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…

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…