Monogenic period equations are cyclotomic polynomials

We study monogeneity in period equations, psi(e)(x), the auxiliary equations introduced by Gauss to solve cyclotomic polynomials by radicals. All monogenic psi(e)(x) of degrees 4 = 4, we conjecture all monogenic period equations to be cyclotomic polynomials. Totally real period equations are of interest in applications of quadratic discrete-time dynamical systems

On common index divisors and non-monogenity of number fields defined by x5+ax3+bx^5+ax^3+b

Let $K$ be a number field generated by a complex root $\theta$ of a monic irreducible trinomial $F(x)= x^5+ax^3+b \in \mathbb{Z}[x]$ where $ab \neq 0$. In this paper, we provide some explicit conditions only on $a$ and $b$ for which $2$ divides the index of $K$. We give necessary and sufficient condition for which $3$ divides the index of $K$. As an application of our results, we provide some infinite parametric families of number fields which are non-monogenic.Comment: submitted. arXiv admin note: substantial text overlap with arXiv:2203.10413, arXiv:2203.07625, arXiv:2206.1434

Polynomial Roots and Calabi-Yau Geometries

The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer polynomials arising naturally in the context of Calabi-Yau spaces, notably Poincare and Newton polynomials, and observe various salient features and geometrical patterns.Comment: 22 pages, 13 Figure