Analogue of Sylvester-Cayley formula for invariants of $n$-ary form

TL;DR

This paper calculates the number of linearly independent homogeneous invariants of a given degree for n-ary forms of degree d, extending classical invariant theory results.

Contribution

It provides a formula for counting invariants of n-ary forms, generalizing the Sylvester-Cayley formula to higher arity.

Findings

Derived a formula for the number of invariants of n-ary forms

Extended classical invariant theory to n-ary cases

Provides explicit calculations for specific n and d

Abstract

The number $\nu_{n,d}(k)$ of linearly independed homogeneous invariants of degree $k$ for the $n$-ary form of degree $d$ is calculated.