The modular counterparts of Cayley's hyperdeterminants

Let H be a hypersurface of degree m in PG(n, q), q = p^h, p prime. (1) If m < n + 1, H has 1 (mod p) points. (2) If m = n + 1, H has 1 (mod p) points ⇔ H^p-1 has no term x^p-1₀ . . . x^p-1_n. We show some applications, including the generalised Hasse invariant for hypersurfaces of degree n+1 in PG(n, F), various properties of finite projective spaces, and in particular a p-modular invariant det_p of any (n + 1)^r+2 = (n + 1) x ⋯ x (n + 1) array or hypercube A over a field of prime characteristic p. This invariant is multiplicative in that det_p (AB) = det_p (A)det_p (B), whenever the product (or convolution) of the two arrays A and B is defined, and both arrays are not 1-dimensional vectors. (If A is (n + 1)^r+2 and B is (n + 1)^s+2, then AB is (n + 1)^r+s+2 .) The geometrical meaning of the invariant is that over finite fields of characteristic p the number of projections of A from r + 1 points in any given r + 1 directions of the array to a non-zero point in the final direction is 0 (mod p). Equivalently, the number of projections of A from r points in any given r directions to a non- singular (n + 1)² matrix is 0 (mod p). Historical aspects of invariant theory and connections with Cayley's hyperdeterminant Det for characteristic 0 fields are mentioned.