Permanent formulae from the Veronesean

The two formulae for the permanent of a d × d matrix given by Ryser (1963) and Glynn (2010) fit into a similar pattern that allows generalization because both are related to polarization identities for symmetric tensors, and to the classical theorem of P. Serret in algebraic geometry. The difference between any two formulae of this type corresponds to a set of dependent points on the “Veronese variety” (or “Veronesean”) v _d (d ? 1]), where v _d (n]) is the image of the Veronese map v _d acting on n], the n-dimensional projective space over a suitable field. To understand this we construct dependent sets on the Veronesean and show how to construct small independent sets of size nd + 2 on v _d (n]). For d = 2 such sets of 2n + 2 points in n] have been called “associated” and we observe that they correspond to self-dual codes of length 2n + 2.