1. The interpolation problem

We give an overview of the exciting class of problems in algebraic geometry known as interpolation problems: basically, when points (or more generally zero-dimensional schemes) in projective space may fail to impose independent conditions on polynomials of a given degree, and by how much.

We work over an arbitrary field K. Our starting point is this elementary theorem:

Theorem 1.1. Given any, there is a unique of degree at most d such that

More generally:

Theorem 1.2. Given any, natural numbers with, and

there is a unique of degree at most d such that

The problem we'll address here is simple: What can we say along the same lines for polynomials in several variables?

First, introduce some language/notation. The “starting point” statement Theorem 1.1 says that the evaluation map

is surjective; or, equivalently,

for any distinct points whenever. More generally, Theorem 1.2 says that

when. To generalize this, let be an subscheme of dimension 0 and degree n. We say that F imposes independent conditions on hypersurfaces of degree d if the evaluation map

is surjective, that is, if

we'll say it imposes maximal conditions if p has maximal rank—that is, is either injective or surjective, or equivalently if.