Quantify additional syzygies arising from interactions between similarity classes

Quantify the number and structure of independent syzygies that arise from interactions between distinct #1(x)-similarity classes of #1(x)-hypergeometric solutions of a linear Mahler operator L(x,M). Specifically, given bases (y_i) within each similarity class and the combined vector (y_1, …, y_s, M y_1, …, M y_s), derive general bounds or explicit formulas for the rank contribution beyond the known 2s−1 syzygies per individual class, thereby characterizing the cross-class interaction terms in the syzygy module over #1[x].

Background

In the Hermite–Padé approach, the authors paper the module of syzygies of the tuple (z_1, …, z_N, M z_1, …, M z_N) associated with a basis of power series solutions for a linear Mahler equation Ly=0. They show that each #1(x)-similarity class of #1(x)-hypergeometric solutions contributes 2s−1 independent syzygies, where s is the dimension of the class.

However, they observe that when several similarity classes are present, additional syzygies may be created by interactions between the classes. These cross-class interaction syzygies can increase the rank of the overall syzygy module beyond the sum of the within-class contributions, but the authors do not currently have a general quantitative description of this phenomenon.