On the structure of Multiple q-Zeta Values
Benjamin Brindle
TL;DR
The paper develops a unified box-product approach to the algebraic structure of formal $q$-analogues of Multiple Zeta Values (qMZVs) and investigates Bachmann’s conjecture on the spanning set for the SZ-qMZV algebra. It defines the box product, links it to the stuffle product via duality, and shows how to extract the maximal-zero component to control filtrations by zeros, depth, and weight. It proves refined Bachmann conjecture results for several regimes, notably for all $(z,d,w)$ with $z+d\le 6$ and for $1\le d\le 4$ (with explicit cases $ (2,3,w),(2,4,w),(3,4,w)$), using a framework that circumvents direct dependence on classical MZV relations and relies instead on $\,\mathbb{Q}$-linear relations of a specific shape. The work connects box and stuffle structures, provides conjectural bases and kernel descriptions for box maps, and offers a computationally supported roadmap toward a general proof of Bachmann’s conjecture, while also outlining a path to handle the $z<d$ case. Overall, the results advance the understanding of the algebraic structure of SZ-qMZVs and lay groundwork for a complete resolution of Bachmann-type spanning questions via duality and the box-product formalism.
Abstract
In 2015, Bachmann \cite{Ba3} conjectured that the~$\Q$-vector space~$\Zq$ of (formal)~$q$-analogues of Multiple Zeta Values (\qmzv s) is spanned by a very particular set compared to known spanning sets. In this work, we prove that this conjecture is true for a subspace of~$\Zq$ spanned by words satisfying some condition on their number of zeros and depth. According to this partial result, we give an explicit approach to the whole conjecture, based on particular~$\Q$-linear relations among formal Multiple~$q$-Zeta Values which are implied by duality.