All cyclic p-roots of index 3, found by symmetry-preserving calculations

Abstract

When using a Groebner basis to solve the highly symmetric system of algebraic equations defining the cyclic p-roots, one has the feeling that much of the advantage of computerized symbolic algebra over hand calculation is lost through the fact that the symmetry is immediately ``thrown out'' by the calculations. In this paper, the problem of finding (for all relevant primes p) all cyclic p-roots of index 3 is treated with the symmetry preserved through the calculations. Once we had found the relevant formulas, using MAPLE and MATHEMATICA, the calculations could even be made by hand. On the other hand, with respect to a straightforward attack with Groebner basis, it is not even clear how this could be organized for a general p. In other terminologies, our results involve listings of all bi-unimodular sequences constant on the cosets of the group G_0 of cubic residues, or equivalently all circulant complex Hadamard matrices related to G_0. The corresponding problem for bi-unimodular sequences of index 2 was solved by the first named author in 1989 and shortly after solved independently by de la Harpe and Jones in the case p = 1 (mod 4) and by Munemasa and Watatani in the case p = 3 (mod 4).