Regularity and non-emptyness of linear systems in ${\mathbb{P}}^n$

The main goal of this paper is to present a new algorithm bounding the regularity and ``alpha’’ (the lowest degree of existing hypersurface) of a linear system of hypersurfaces (in ${\mathbb{P}}^n$) passing through multiple points in general position. To do the above we formulate and prove new theorem, which allows to show non-specialty of linear system by splitting it into non-special (and simpler) systems. As a result we give new bounds for multiple point Seshadri constants on ${\text{PP}}^2$.