Abstract
There is polynomial function X q in the entries of an m × m(q − 1) matrix over a field of prime characteristic p, where q = p h is a power of p, that has very similar properties to the determinant of a square matrix. It is invariant under multiplication on the left by a non-singular matrix, and under permutations of the columns. This gives a way to extend the invariant theory of sets of points in projective spaces of prime characteristic, to make visible hidden structure. There are connections with coding theory, permanents, and additive bases of vector spaces.
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References
Alon N., Linial N., Meshulam R.: Additive bases of vector spaces over finite fields. J. Comb. Theory A 57, 203–210 (1991)
Brualdi R.A., Ryser H.J.: Combinatorial matrix theory. In: Encyclopedia of Mathematics and its Applications, vol. 39. Cambridge University Press, Cambridge (1991).
Cayley A.: On the theory of determinants. Camb. Phil. Trans. VIII, 1–16 (1849). (The paper was actually given at a meeting of the society in 1843. See Sect. 2 for the hyperdeterminant).
Dickson L.E.: History of the theory of numbers. In: Divisibility and Primality, vol. I. Chelsea Publishing Company, New York (1966).
Glynn D.G.: The modular counterparts of Cayley’s hyperdeterminants. Bull. Aust. Math. Soc. 57, 479–492 (1998)
Glynn D.G.: The permanent of a square matrix. Europ. J. Comb. (2010). doi:10.1016/j.ejc.2010.01.010.
Glynn D.G., Hirschfeld J.W.P.: On the classification of geometric codes by polynomial functions. Des. Codes Cryptogr. 6, 189–204 (1995)
Goodman R., Wallach N.R.: Representations and invariants of the classical groups. In: Encyclopedia of Mathematics and its Applications, vol. 68. Cambridge University Press, Cambridge (1968).
Hartshorne R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol 52, Springer, New York (1977).
Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998)
Hirschfeld J.W.P., Thas J.A.: General Galois Geometries. Oxford University Press, Oxford (1991)
Hodge W.V.D., Pedoe D.: Methods of Algebraic Geometry, 2 vols. Cambridge University Press, Cambridge (1968)
Pascal E.: Die Determinanten, translated by H. Leitzmann (1900) (The German edition, proof-read among others by the younger H. Grassmann, was preceded by an Italian edition, “I Determinanti”, Hoepli, Milan, 1897).
Pless V.S., Huffman W.C. (eds), Brualdi R.A. (Assoc. ed): Handbook of Coding Theory, 2 vols. North-Holland (1998).
van Lint J.H., Wilson R.M.: A Course in Combinatorics. Cambridge University Press, Cambridge (1991)
Weyl H.: The Classical Groups, their Invariants and Representations. Princeton University Press, Princeton N.J. (1939)
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Communicated by W. H. Haemers.
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Glynn, D.G. An invariant for matrices and sets of points in prime characteristic. Des. Codes Cryptogr. 58, 155–172 (2011). https://doi.org/10.1007/s10623-010-9392-x
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DOI: https://doi.org/10.1007/s10623-010-9392-x
Keywords
- Invariant
- Covariant
- P-modular
- Determinant
- Permanent
- Prime characteristic
- Projective geometry
- Code
- Complete weight enumerator
- Additive basis