# matrix

## The Alon-Tarsi basis conjecture ★★

Author(s): Alon; Linial; Meshulam

\begin{conjecture} If $B_1,B_2,\ldots B_p$ are invertible $n \times n$ matrices with entries in ${\mathbb Z}_p$ for a prime $p$, then there is a $n \times (p-1)n$ submatrix $A$ of $[B_1 B_2 \ldots B_p]$ so that $A$ is an AT-base. \end{conjecture}

## 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

## The additive basis conjecture ★★★

Author(s): Jaeger; Linial; Payan; Tarsi

\begin{conjecture} For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$) so that the union (as multisets) of any $c(p)$ bases of the vector space $({\mathbb Z}_p)^n$ contains an additive basis. \end{conjecture}