The modular counterparts of Cayley's hyperdeterminants

Abstract

LetHbe a hypersurface of degreeminPG(n, q),q=ph,pprime.(1) Ifm<n+ 1,Hhas 1 (modp) points.(2) Ifm=n+ 1,Hhas 1 (modp) points ⇔Hp−1has no termWe show some applications, including the generalised Hasse invariant for hypersurfaces of degreen+ 1 inPG(n, F), various porperties of finite projective spaces, and in particular ap-modular invariant detpof any (n+ 1)r+2= (n+ 1)×…×(n+ 1) array on hypercubeAover a field characteristicp. This invariant is multiplicative in that detp(AB) = detp(B), whenever the product (or convolution of the two arraysAandBis defined, and both arrays are not 1-dimensional vectors. (IfAis (n+ 1)r+2andBis (n+ 1)s+2, thenABis (n+ 1)r+s+2.) The geometrical meaning of the invariant is that over finite fields of characteristicpthe number of projections ofAfromr+ 1 points in any givenr+ 1 directions of the array to a non-zero point in the final direction is 0 (modp). Equivalently, the number of projections ofAfromrpoints in any givenrdirections to a non-singular (n+ 1)2matrix is 0 (modp). Historical aspects of invariant theory and connections with Cayley's hyperdeterminant Det for characteristic 0 fields are mentioned.