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In this paper we develop techniques for determining the dimension of linearsystems of divisors based at a collection of general fat points in P^n bypartitioning the monomial basis for the vector space of global sections ofO(d). The methods we develop can be viewed as extensions of those developed byDumnicki. We apply these techniques to produce new lower bounds on multi-pointSeshadri constants of P^2 and to provide a new proof of a known resultconfirming the perfect-power cases of Iarrobino's analogue to Nagata'sConjecture in higher dimension.

New methods for determining speciality of linear systems based at fat points in P^n