Conditions under which Halburd’s degree relations fail to determine evolution

Determine the precise conditions on singularity patterns and equations under which the degree relations in Halburd’s method do not completely determine the time evolution of the counting sequences, particularly in cases with short singularity patterns.

Background

When applying Halburd’s method, the authors identify a structural issue: the number of linear degree relations may be insufficient to determine the time evolution of auxiliary counting sequences derived from singularity patterns. Empirically, this occurs for short patterns, but the exact criteria are unknown.