General interpolation problem for fat points

Determine the dimension of the linear system of degree-d forms in projective space that vanish to prescribed multiplicities at general points, for arbitrary numbers of points and multiplicities.

Background

The paper relates the interpolation problem in algebraic geometry to Hilbert functions of ideals generated by powers of general linear forms. Given general points in projective space and prescribed multiplicities, the problem asks for the dimension of the corresponding linear system of forms vanishing at those points. The AlexanderHirschowitz theorem resolves the double-point case, and results of Laface, Postinghel, and Santana Sánchez resolve configurations lying on a rational normal curve, but the general problem remains unresolved.