Cubic Equations Through the Looking Glass of Sylvester

Summary

One can hardly believe that there is still something to be said about cubic equations. To dodge this doubt, we will instead try and say something about Sylvester. He doubtless found a way of solving cubic equations. As mentioned by Rota, it was the only method in this vein that he could remember. We realize that in the generic case Sylvester’s magnificent approach aimed at reduced cubic equations boils down to an easy identity expressing a cubic polynomial as a sum of two third powers of linear forms. This leads to Cardano’s formula for cubic equations involving the third roots of unity.

Acknowledgments

The author is grateful to Joseph P.S. Kung, Peter Paule, Catherine H.F. Yan, Doron Zeilberger and the referees for their insightful comments and suggestions. This work was supported by the National Science Foundation of China.