1998/06/01 by David G. Glynn · 2 citations
Engineering · Mathematics · Computer Science · #graph theory and CDMA systems #Finite Group Theory Research #Coding theory and cryptography #Mathematics #Combinatorics #Finite field #Invariant (physics) #Degree (music) #Multiplicative function #Discrete mathematics #Mathematical analysis
paper · pdf · doi:10.1017/s0004972700031890
openalex publication_date 1998/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 We show some applications, including the generalised Hasse invariant for hypersurfaces of degree n + 1 in PG ( n, F ), various porperties of finite projective spaces, and in particular a p -modular invariant det p of any ( n + 1) r+2 = ( n + 1)×…×( n + 1) array on hypercube A over a field characteristic p . This invariant is multiplicative in that det p ( AB ) = 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.