A characterization of the sign patterns that allow algebraic positivity

A real square matrix is algebraically positive if some real polynomial in the matrix has only positive entries. A sign pattern, which is a matrix with entries $+$, $-$ and $0$, allows algebraic positivity if some real matrix with that pattern of signs is algebraically positive. Kirkland, Qiao and Zhan asked for a characterization of the sign patterns that allow algebraic positivity. We prove that an $n\times n$ sign pattern $S$ with $n\ge2$ allows algebraic positivity if and only if $S$ is irreducible and some real matrix with sign pattern $S$ or $-S$ has positive row sums that equal its column sums. Studying these patterns, Das called $S$ AP-irreducible if $S$ is irreducible, every row and every column of $S$ contains a $+$, and $B_S$ is irreducible, where $B_S$ is the nonnegative pattern with an entry $+$ exactly where $S$ has a $+$ or the transpose of $S$ has a $-$. He conjectured that $S$ allows algebraic positivity if and only if $S$ or $-S$ is AP-irreducible. This conjecture holds for the orders 2, 3 and 4 by the characterization, whereas a family of patterns with zero diagonal refutes it for every order $n\ge5$. These patterns are minimally AP-irreducible, which means that they lose AP-irreducibility when any one nonzero entry is replaced by $0$. Because they have $2n$ nonzero entries, they also refute his conjecture that a minimally AP-irreducible pattern of order $n$ with zero diagonal has at most $2n-2$ nonzero entries.