Nowhere-zero flow polynomials

In this article, we introduce certain flow polynomials associated with digraphs and use them to study nowhere-zero flows from a commutative algebraic perspective. Using Hilbertʹs Nullstellensatz, we establish a relation between nowhere-zero flows and dual flows. For planar graphs this gives a relation between nowhere-zero flows and flows of their planar duals. It also yields an appealing proof that every bridgeless triangulated graph has a nowhere-zero four-flow.