2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126108In this article we introduce the flow polynomial of a digraph and use it to study nowhere-zero flows from a commutative algebraic perspective. Using Hilbert's 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.CombinatoricsDiscrete MathematicsCommutative Algebra05C, 05E, 13C, 13P, 68R, 68W, 90C, 94Ctext