Invariants of Binary Forms

TL;DR

This paper provides a simplified proof for the invariants of binary forms over complex numbers and extends the results to fields of characteristic greater than 5, including pairs of binary cubics.

Contribution

A straightforward proof for invariants of binary forms in characteristic p > 5, building on classical invariant theory and extending to pairs of binary cubics.

Findings

Simplified proof applicable in characteristic p > 5

Extension of invariants to pairs of binary cubics

Connection to classical and modern invariant theory

Abstract

Basic invariants of binary forms over $\mathbb C$ up to degree 6 (and lower degrees) were constructed by Clebsch and Bolza in the 19-th century using complicated symbolic calculations. Igusa extended this to algebraically closed fields of any characteristic using difficult techniques of algebraic geometry. In this paper a simple proof is supplied that works in characteristic $p > 5$ and uses some concepts of invariant theory developed by Hilbert (in characteristic 0) and Mumford, Haboush et al. in positive characteristic. Further the analogue for pairs of binary cubics is also treated.