New effective bounds on the dimension of a linear system in P2

The main goal of this paper is to present an algorithm bounding the dimension of a linear system of plane curves of given degree (or monomial basis) with multiple points in general position. As a result we prove the Harbourne–Hirschowitz conjecture when the multiplicities of base points are bounded by 11. This gives a partial answer to the question of when bivariate polynomial interpolation is possible.