Quartic equations and algorithms for riemann tensor classification

Abstract

The Petrov classification of the Weyl conformal curvature and the Plebanski or Segre classification of the Ricci tensor of spacetimes in general relativity both depend on multiplicities of the roots of quartic equations. The coefficients in these quartic equations may be complicated functions of the space-time coordinates. We review briefly the general theory of quartic equations and then consider practical algorithms for determination of the multiplicities of their roots and hence for the classification of Riemann tensors. Preliminary results of tests of computer implementations of these algorithms, using the computer algebra system SHEEP, are reported.