Abstract :

Let R = k[x1; : : : ; xn] be a polynomial ring over a eld k and I be a homogeneous ideal of height n so that A = R=I = Ls t=0 At has dimension zero. Then h(t) = dimkAt is called the Hilbert Function of the Graded Artin Algebra A. We will discuss some results and questions concerning when this Hilbert fucntion is unimodal. Most of the known cases depend on the fact that the embedding dimension, n is small or the socle degree s is small enough or that I is extra special with a lot of structure.