The hyperdeterminant of a polynomial

The coefficients of a degree $d$ homogeneous polynomial may be arranged in a $d$-dimensional matrix. Analogous to the determinant of a matrix, Cayley introduced the notion of the hyperdeterminant of a multi-dimensional matrix, and we consider this hyperdeterminant applied to a polynomial. In this talk we will describe some of the beautiful symmetry, geometry and combinatorics of this symmetrized hyperdeterminant. In particular we will give a geometric description via dual varieties, and we will use this to interpret the combinatorial formula for the degree of the hyperdeterminant.