SYZYGIES and HILBERT FUNCTIONS

Throughout k stands for a field. For simplicity, we assume that k is algebraically closed and has characteristic 0. However, many of the open problems and conjectures make sense without these assumptions. The polynomial ring S = k[x1, . . . , xn] is graded by deg(xi) = 1 for all i. A polynomial f is homogeneous if f ∈ Si for some i, that is, if all monomial terms of f have the same degree. An ideal I is graded (or homogeneous) if it satisfies the following equivalent conditions: ◦ if f ∈ I, then every homogeneous component of f is in I. ◦ if Ii = Si ∩ I, then I = ⊕i∈N Ii. ◦ I has a system of homogeneous generators. Throughout the paper, I stands for a graded ideal in S and R stands for S/I. The quotient R inherits the grading by (S/I)i = Si/Ii for all i.