Skip to main content Accesibility Help
×
×
Home

The modular counterparts of Cayley's hyperdeterminants

  • David G. Glynn (a1)
Abstract

Let H be a hypersurface of degree m in PG(n, q), q = ph, p prime.

(1) If m < n + 1, H has 1 (mod p) points.

(2) If m = n + 1, H has 1 (mod p) points ⇔ Hp−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 detp 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 detp(AB) = detp(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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The modular counterparts of Cayley's hyperdeterminants
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      The modular counterparts of Cayley's hyperdeterminants
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      The modular counterparts of Cayley's hyperdeterminants
      Available formats
      ×
Copyright
References
Hide All
[1]Assmus, E. and Key, J.D., Designs and their Codes (Cambridge University Press, Cambridge, 1993).
[2]Cayley, A., ‘On the theory of linear transformations’, Cambr. Math. J. 4 (1845), 193209.
[3]Cayley, A., ‘On linear transformations’, Cambr. and Dublin Math. J. 1 (1846), 104122.
[4]Cayley, A., ‘Note sur un système de certaines formules’, J. Reine Angew. Math. 39 (1850), 1415.
[5]Cayley, A., Collected Mathematical Papers 1 (Cambridge University Press, Cambridge, 1889).
[6]Coolidge, J.L., A history of geometrical methods (Oxford University Press, Oxford, 1940).
[7]Dickson, L.E., History of the theory of numbers, III, Chapter 19 (Chelsea Publishing Co., New York, 1966). (See also Vol I, pp.231–233, with a report on A. Hurwitz’ work).
[8]Gauß, C.F., Disquisitiones arithmeticae, 1801.
[9]Glynn, D.G., ‘On cubic curves in projective planes of characteristic two’, Australas. J. Combin. 17 (1998), 120.
[10]Glynn, D.G. and Hirschfeld, J.W.P., ‘On the classification of geometric codes by polynomial functions’, Des. Codes Cryptogr. 6 (1995), 189204.
[11]Gelfand, I.M., Kapranov, M.M. and Zelevinsky, A.V., Discriminants, resultants and multidimensional determinants (Birkhäuser, Boston, Basel, Berlin, 1994).
[12]Hartshorne, R., Algebraic geometry (Springer, Berlin, Heidelberg, New York, 1983), pp. 332340.
[13]Hirschfeld, J.W.P., Projective geometry over a finite field (Oxford University Press, Oxford, 1979).
[14]Schläfli, L., ‘Über die Resultante eines Systemes mehrerer algebraischen Gleichungen’, Denkschr. der Kaiserlicher Akad. der Wiss., math-naturwiss. Klasse 4 (1852).
[15]Schläfli, L., Gesammelte Abhandlungen 2 (Birkhäuser Verlag, Basel, 1953).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed