Covariant algebra of the binary nonic and the binary decimic

TL;DR

This paper establishes a minimal generating set for the algebra of SL(2,C)-covariant polynomials on binary forms of degrees 9 and 10, confirming longstanding conjectures through advanced computational methods.

Contribution

It provides the first rigorous proof of the minimal generators for these covariant algebras, utilizing improved Gordan's algorithm techniques.

Findings

Minimal 476 generators for degree 9 forms

Minimal 510 generators for degree 10 forms

Results confirm previous conjectures

Abstract

We give a minimal system of 476 generators (resp. 510 generators) for the algebra of SL(2,C)-covariant polynomials on binary forms of degree 9 (resp. degree 10). These results were only known as conjectures so far. The computations rely on Gordan's algorithm, and some new improvements.