We develop techniques for determining the dimension of linear systems of divisors based at a zero-dimensional in P^n by partitioning the monomial basis for H^0(O(d)). The methods 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 P^2 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.