We introduce and study the notion of read-k projections of the determinant:
a polynomial fβF[x1β,β¦,xnβ] 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 nΓn permanent polynomial Permnβ and the elementary symmetric
polynomials of degree d on n variables Sndβ for 2β€dβ€nβ2 are
not expressible as read-once projection of determinant, whereas mon(Permnβ)
and mon(Sndβ) 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