The balancing index over the alternating group

TL;DR

This paper investigates the balancing index of polynomials over the alternating group, focusing on the minimal positive sum of coefficients needed to produce symmetric polynomials using only even permutations.

Contribution

It introduces a restricted version of the balancing index problem limited to even permutations within the alternating group.

Findings

Characterization of the balancing index over the alternating group

Identification of minimal positive sums for specific classes of polynomials

Extension of previous results from the symmetric group to the alternating group

Abstract

The balancing index of a polynomial $f \in \mathbb{Z}[x_1,\dots,x_n]$ is the least positive sum of coefficients in an integer linear combination of permuted copies of $f$ which produces a symmetric polynomial. Here we consider the restricted problem in which only even permutations are used.