Geometric Combinatorics of Polynomials I: the case of a single polynomial

There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce new object encodes of the existing have previously appeared in literature. Concretely, for every p d roots degree at least 2, produce canonical compact planar 2-complex is metric version tiled phase diagram. It has locally Euclidean away from finite set interior points indexed by critical , each its 2-cells rectangle. From rectangular use graphs known as cacti banyans read off several pieces data: chain partition lattice, cyclic factorization -cycle, real noncrossing (also primitive -major), monodromy permutations polynomial. This first series.