The Jacobian Conjecture is a long-standing open problem in algebraic geometry. Though the problem is inherently algebraic, it crops up in fields throughout mathematics including perturbation theory, quantum field theory and combinatorics. This thesis is a unified treatment of the combinatorial approaches toward resolving the conjecture, particularly investigating the work done by Wright and Singer. Along with surveying their contributions, we present new proofs of their theorems and motivate their constructions. We also resolve the Symmetric Cubic Linear case, and present new conjectures whose resolution would prove the Jacobian Conjecture to be true.

The Jacobian Conjecture is a long-standing open problem in algebraic geometry. Though the problem is inherently algebraic, it crops up in fields throughout mathematics including perturbation theory, quantum field theory and combinatorics. This thesis is a unified treatment of the combinatorial approaches toward resolving the conjecture, particularly investigating the work done by Wright and Singer. Along with surveying their contributions, we present new proofs of their theorems and motivate their constructions. We also resolve the Symmetric Cubic Linear case, and present new conjectures whose resolution would prove the Jacobian Conjecture to be true.