MacMahon Partition Analysis and the Poincar\'{e} series of the algebras of invariants of ternary, quaternary and quinary forms

TL;DR

This paper applies MacMahon partition analysis to compute the Poincaré series of invariant algebras for small-order ternary, quaternary, and quinary forms, advancing understanding of their algebraic structures.

Contribution

It introduces a novel application of MacMahon partition analysis to explicitly calculate Poincaré series for specific invariant algebras.

Findings

Successfully computed Poincaré series for selected forms

Demonstrated effectiveness of MacMahon partition analysis in invariant theory

Provided explicit algebraic invariants for small forms

Abstract

By using MacMahon partition analysis technique, the Poincar\'e series for the algebras of invariants of the ternary, quaternary and quinary forms of small orders are calculated.