On Sign Pattern Matrices that Allow or Require Algebraic Positivity

A square matrix $M$ with real entries is said to be algebraically positive (AP) if there exists a real polynomial $p$ such that all entries of the matrix $p(M)>0$. A square sign pattern matrix $S$ is said to allow algebraic positivity if there is an algebraically positive matrix $M$ whose sign pattern class is $S$. On the other hand, $S$ is said to require algebraic positivity if any matrix $M$, having sign pattern class $S$, is algebraically positive. Motivated by open problems raised in the work of Kirkland, Qiao and Zhan (2016) on AP matrices, we list down all nonequivalent irreducible $3\times 3$ sign pattern matrices and classify each of them into three groups (i) those that require AP, (ii) those that allow but not require AP, or (iii) those that do not allow AP. We also give a necessary condition for an irreducible $n\times n$ sign pattern to allow algebraic positivity.