Semi-invariants of binary forms and symmetrized graph-monomials

TL;DR

This paper introduces a method to construct invariants and semi-invariants of binary forms using symmetrized graph-monomials, with applications in quantum physics and new bounds on polynomial coefficients.

Contribution

It provides a practical condition for nontrivial symmetrization of graph-monomials, enabling the creation of infinite invariant families and semi-invariants, advancing algebraic invariant theory.

Findings

Established a sufficient condition for nontrivial symmetrization.

Constructed infinite families of invariants and semi-invariants.

Derived a new polynomial lower bound on specific coefficients.

Abstract

This article provides a method for constructing invariants and semi-invariants of a binary $N$-ic form over a field $k$ characteristics $0$ or $p > N$. A practical and broadly applicable sufficient condition for ensuring nontriviality of the symmetrization of a graph-monomial is established. This allows construction of infinite families of invariants (especially, skew-invariants) and families of $k$-linearly independent semi-invariants. These constructions are very useful in the quantum physics of Fermions. Additionally, they permit us to establish a new polynomial-type lower bound on the coefficient of $q^{w}$ in $(q - 1) {N + d \choose d}_{q}$ for all sufficiently large integers $d$ and $w \leq N d / 2$.