2023/11/15 by Lenny Jones, Jones, Lenny
Mathematics · #Algebraic and Geometric Analysis #Advanced Differential Equations and Dynamical Systems #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2311.08875
Let f(x)∈ \mathbb Z[x] be monic of degree N≥ 2. Suppose that f(x) is monogenic, and that f(x) is the characteristic polynomial of the Nth order linear recurrence sequence Υf:=(Un)n≥ 0 with initial conditions U0=U1=⋯ =UN-2=0 and UN-1=1. Let p be a prime such that f(x) is irreducible over \mathbb Fp and f(xp) is irreducible over \mathbb Q. We prove that f(xp) is monogenic if and only if π(p2)≠ π(p), where π(m) denotes the period of Υf modulo m. These results extend previous work of the author, and provide a new and simple test for the monogenicity of f(xp). We also provide some infinite families of such polynomials. This article extends previous work of the author.