Hilbert Functions of Standard Graded Algebras over a Field

In this talk, we introduce Hilbert functions of a graded algebras over a field, and one of the long standing conjectures concerning them, Fröberg conjecture. Then we study relations between Fröberg conjecture on Hilbert series and Moreno-Socias Conjecture. Consequently, we show that Fröberg conjecture holds for special cases as an example. 1. Almost Reverse Lexicographic Monomial Ideal Let R = k[x1, . . . , xn] be the polynomial ring over a field k. Throughout this talk, we assume that k is a field of characteristic 0. If I is an ideal in R, by a definition, a Hilbert function of R/I is a numerical function from Z≥0 into Z≥0 defined to be H(R/I, d) = dimk Rd − dimk Id, for each d. We use only the reverse lexicographic order as a multiplicative term order. A monomial ideal I in R is said to be almost reverse lexicographic if I contains every monomial M which is bigger than a minimal generator of I having the same degree with M . One of conjectures associated with the almost reverse lexicographic ideal is Moreno-Socias conjecture [13]. Conjecture 1.1 (Moreno-Socias). If I is a homogeneous ideal generated by generic forms in R, then the generic initial ideal gin(I) of I is almost reverse lexicographic. Another longstanding conjecture on generic algebras is Fröberg conjecture [7]. Conjecture 1.2 (Fröberg). If I is a homogeneous ideal generated by generic forms F1, . . . , Fr in R of degrees deg Fi = di, then the Hilbert series SR/I(z) of R/I is 2000 Mathematics Subject Classification. 13A02, 13C05, 13D40, 13E10, 13P10.