On the expressive power of read-once determinants

Abstract

We introduce and study the notion of read-$k$ projections of the determinant: a polynomial $f\in \mathbb{F}[x_1,\dots ,x_n]$ is called a {\it read-$k$ projection of determinant} if $f=det(M)$, where entries of matrix $M$ are either field elements or variables such that each variable appears at most $k$ times in $M$. A monomial set $S$ is said to be expressible as read-$k$ projection of determinant if there is a read-$k$ projection of determinant $f$ such that the monomial set of $f$ is equal to $S$. We obtain basic results relating read-$k$ determinantal projections to the well-studied notion of determinantal complexity. We show that for sufficiently large $n$, the $n\times n$ permanent polynomial $Per{m}_n$ and the elementary symmetric polynomials of degree $d$ on $n$ variables $S_n^d$ for $2\le d\le n-2$ are not expressible as read-once projection of determinant, whereas $mon(Per{m}_n)$ and $mon(S_n^d)$ are expressible as read-once projections of determinant. We also give examples of monomial sets which are not expressible as read-once projections of determinant.