On the Expressive Power of Read-Once Determinants

Abstract

We introduce and study the notion of read-k projections of the determinant: a polynomial $f \in \mathbb {F}[x_1, \ldots , x_n]$ is called a 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 $Perm_n$ 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 $mon(Perm_n)$ 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.