In the ﬁrst two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic ﬁelds could be explicitly parametrized. In particular, a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state.
In the ﬁrst three parts of this series, we considered quadratic, cubic and quartic rings (i.e., rings free of ranks 2, 3, and 4 over Z) respectively, and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space. These orbit results are summarized in Table 1.
Steiner symmetrization, one of the simplest and most powerful symmetrization
processes ever introduced in analysis, is a classical and very well-known
device, which has seen a number of remarkable applications to problems of
geometric and functional nature. Its importance stems from the fact that,
besides preserving Lebesgue measure, it acts monotonically on several geometric
and analytic quantities associated with subsets of Rn. Among these,
perimeter certainly holds a prominent position.