Toric ideal Birch's Theorem 4 Chemical reaction network 4 Complex balancing 4 Cycles 4 Deficiency zero 4 Detailed balancing 4 Following problem 4 Graph labeling 4 Graph labelings 4 Incidence matrices 4 Incidence matrix 4 Kernel 4 Labeling 4 Labelings 4 Matrix-tree theorem 4 Moduli space 4 Multiplicative version 4 Polyhedron 4 Polynomial algorithm 4 Positive integers 4 Possible solutions 4 Toric ideals 4 Trajectory 4