New methods for determining speciality of linear systems based at fat points in Pn

In this paper we develop techniques for determining the dimension of linear systems of divisors based at a collection of general fat points in Pn by partitioning the monomial basis for H0(OPn(d)). The methods that we develop can be viewed as extensions of those developed by Dumnicki. We apply these techniques to produce new lower bounds on multi-point Seshadri constants of P2 and to provide a new proof of a known result confirming the perfect-power cases of Iarrobino's analogue to Nagata's Conjecture in higher dimension.