# An algebraic independence result related to a conjecture of Dixmier on binary form invariants

## TL;DR

This paper proves algebraic independence of a subfamily of invariants related to Dixmier's conjecture on binary form invariants, using elementary combinatorial methods and explicit computations for binary octavics.

## Contribution

It generalizes Dixmier's proposed invariants and establishes their algebraic independence using a simple combinatorial proof approach.

## Key findings

- A subfamily of invariants is algebraically independent.
- Elementary proof using Chu-Vandermonde Theorem for Dixon's Summation.
- Explicit invariants computed for binary octavic forms.

## Abstract

In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier's collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of binomial coefficients are nonzero. Along the way we provide a very elementary proof \`a la Racah, namely, only using the Chu-Vandermonde Theorem, for Dixon's Summation Theorem. We also provide explicit computations of invariants, for the binary octavic, which can serve as ideal introductory examples to Gordan's 1868 method in classical invariant theory.