Balanced $0,1$-words and the Galois group of $(x+1)^n-\lambda x^p$

TL;DR

This paper investigates the structure of the Galois group of certain polynomials through the combinatorial analysis of binary words with fixed zero fractions, revealing new algebraic insights.

Contribution

It introduces a novel connection between combinatorial properties of binary words and the Galois group of specific polynomials, extending previous algebraic understanding.

Findings

Characterization of the Galois group structure

Link between binary word patterns and polynomial monodromy

New algebraic insights into polynomial symmetries

Abstract

We study the number of $0,1$-words where the fraction of 0 is "almost" fixed for any initial subword. It turns out that this study use and reveal the structure of the Galois group (the monodromy group) of the polynomials $(x+1)^n-\lambda x^p$. ($p$ is not necessary a prime here.)