Polarizations of powers of graded maximal ideals

Abstract

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal (x1,x2,…,xm)n of a polynomial ring in m variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case m=3 and also in the power two case n=2 the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We give conjectures relating to topological properties and to algebraic geometry, in particular that any polarization of an Artinian monomial ideal defines a simplicial ball.