Relating Different Definitions of Linear Series on Tropical Curves

Abstract: We compare tropical linear series as defined by Farkas, Jensen, and Payne with combinatorial limit linear series as defined by Amini and Gierczak. We show that tropical linear series of rank $r$ are combinatorial limit linear series of rank $r$, provide a counterexample of the converse, and provide a slight simplification for the definition of a combinatorial limit linear series. We also discuss realizability of permutation arrays as local arrays of linear series on tropical curves.