7 problems

Let $A$ be a totally positive element of $GL_n$, and let $R$ be a ratio of products of minors as above. In a weighted planar-network parametrization, call $R=p/q$ subtraction free…

Hall–Johnson conjecture. The supremum of $R(A)$ over all $4\times4$ positive definite matrices $A$ is $27/16$.

Let $A$ be a totally positive element of $GL_n$, and let … where $I_k,I'_k,J_k,J'_k\subseteq\{1,2,\ldots,n\}$ satisfy $|I_k|=|I'_k|$ and $|J_k|=|J'_k|$. A ratio is subtraction free…

Let $A$ be a totally positive element of $GL_n$, and let … where $I_k,I'_k,J_k,J'_k\subseteq\{1,2,\ldots,n\}$ satisfy $|I_k|=|I'_k|$ and $|J_k|=|J'_k|$. A ratio is bounded if it is…

Determinantal threshold conjecture. For $k=1$ or $2$ and $1\leq m\leq14$, one has

Dong–Ji's conjecture. For the sequence $a_n=q(n)$, one has $A>0$ for $n\geq230$, $B>0$ for $n\geq272$, and $I>0$ for $n\geq267$. The source records that the first two thresholds we…

Let $A$ be a totally positive $n\times n$ matrix. For index sequences $\alpha$ and $\beta$ defining products of minors, write $\alpha/\beta$ for the corresponding ratio. A ratio is…