Manifold submetries are maps that generalize Riemannian submersions, isometric group actions, and isoparametric maps. Locally around a point, manifold submetries are determined by a manifold submetry from a round sphere. In this talk, we will prove a surprising equivalence between the class of manifold submetries from round spheres and a special class of polynomial algebras. This has applications, for example, to Invariant Theory. This talk is based on a joint work with Ricardo Mendes.