On the minimal polynomials of the arguments of dilogarithm ladders

TL;DR

This paper investigates the minimal polynomials of algebraic numbers involved in dilogarithm ladders, constructing an infinite family of such ladders with complex minimal polynomial structures using advanced geometric identities.

Contribution

It introduces a new method to construct infinitely many dilogarithm ladders with complex minimal polynomials, expanding understanding of algebraic relations in polylogarithm ladders.

Findings

Constructed an infinite family of dilogarithm ladders with arbitrarily many nonzero polynomial coefficients.

Derived a dilogarithm identity via Seifert volumes and Dehn surgery techniques.

Extended classical relations to more complex algebraic structures in dilogarithm ladders.

Abstract

Letting $L_{n}(N, u)$ denote a polylogarithm ladder of weight $n$ and index $N$ with $u$ as an algebraic number, there is a rich history surrounding how mathematical objects of this form can be constructed for a given weight or index. This raises questions as to what minimal polynomials for $u$ are permissible in such constructions. Classical relations for the dilogarithm $\text{Li}_{2}$ provide families of weight-2 ladders in such a way so that the base equations for $u$ consist of a fixed number of terms, and subsequent constructions for dilogarithm ladders rely on sporadic cases whereby $u$ is defined via a cyclotomic equation, as in the supernumary ladders due to Abouzahra and Lewin. This motivates our construction of an infinite family of dilogarithm ladders so as to obtain arbitrarily many terms with nonzero coefficients for the minimal polynomials for $u$. Our construction relies on a derivation of a dilogarithm identity introduced by Khoi in 2014 via the Seifert volumes of manifolds obtained from the use of Dehn surgery.