The Waring Rank of the 3 x 3 Permanent

Abstract

Let f be a homogeneous polynomial of degree d with coefficients in a field F satisfying char F = 0 or char F > d. The Waring rank of f is the smallest integer r such that f is a linear combination of r powers of F-linear forms. We show that the Waring rank of the polynomial x1 y2 z3 + x1 y3 z2 + x2 y1 z3 + x2 y3 z1 + x3 y1 z2 + x3 y2 z1 is at least 16, which matches the known upper bound.