The permanent conjecture ★★

Author(s): Kahn

\begin{conjecture} If $A$ is an invertible $n \times n$ matrix, then there is an $n \times n$ submatrix $B$ of $[A A]$ so that $perm(B)$ is nonzero. \end{conjecture}

Keywords: invertible; matrix; permanent

A nowhere-zero point in a linear mapping ★★★

Author(s): Jaeger

\begin{conjecture} If ${\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb F}$, then there are column vectors $x,y \in {\mathbb F}^n$ which have no coordinates equal to zero such that $Ax=y$. \end{conjecture}

Keywords: invertible; nowhere-zero flow

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