The special symplectic structure of binary cubics

TL;DR

This paper explores the symplectic structure of binary cubic polynomials, analyzing their geometric and algebraic properties through moment maps, orbit classifications, and classical formulas, revealing deep symplectic insights related to Lie algebra G2.

Contribution

It provides a comprehensive symplectic analysis of binary cubics, including orbit classification, a symplectic derivation of Cardano's formulas, and a generalization of Eisenstein syzygy, connecting classical algebra with symplectic geometry.

Findings

Complete orbit classification under SL(2,k) and GL(2,k)

Symplectic derivation of Cardano's formulas

Generalization of Eisenstein syzygy

Abstract

Let $k$ be a field of characteristic not 2 or 3. Let $V$ be the $k$-space of binary cubic polynomials. The natural symplectic structure on $k^2$ promotes to a symplectic structure $\omega$ on $V$ and from the natural symplectic action of $\textrm{Sl}(2,k)$ one obtains the symplectic module $(V,\omega)$. We give a complete analysis of this symplectic module from the point of view of the associated moment map, its norm square $Q$ (essentially the classical discriminant) and the symplectic gradient of $Q$. Among the results are a symplectic derivation of the Cardano-Tartaglia formulas for the roots of a cubic, detailed parameters for all $\textrm{Sl}(2,k)$ and $\textrm{Gl}(2,k)$-orbits, in particular identifying a group structure on the set of $\textrm{Sl}(2,k)$-orbits of fixed nonzero discriminant, and a purely symplectic generalization of the classical Eisenstein syzygy for the covariants of a binary cubic. Such fine symplectic analysis is due to the special symplectic nature inherited from the ambient exceptional Lie algebra $\mathfrak G_2$.