Hilbert series for ideals generated by generic forms
A homogeneous ideal \(I = (f_ 1, \dots, f_ g)\) in the polynomial ring \(A = \mathbb{C} [x_ 1, \dots, x_ n]\) is said to be of type \((n,d_ 1, \dots, d_ g)\) if \(\deg f_ i = d_ i\) for each \(i\). The aim is to see how large an ideal \(I\) of given type can be, as measured by the Hilbert series \(\text{Hilb}_{A/I} (t) = \sum \dim_ \mathbb{C} (A/I)_ it^ i\). -- \textit{R. Fröberg} [Math. Scand. 56, 117-144 (1985; Zbl 0582.13007)] found a lower bound for \(\text{Hilb}_{A/I} (t)\) in the lexicographic order for series in \(t\). For suitably ``generic'' ideals \(I\), it is conjectured that \(\text{Hilb}_{A/I} (t)\) is equal to this lower bound. Since generic ideals give the smallest Hilbert series, the production of one ideal, of a given type, which has the conjectured Hilbert series will prove the conjecture for that type. Using an adaptation of Bayer and Stillman's (1990) computer algebra system MACAULAY, the conjecture is proved for the ideal types given by \((n \leq 11\), \(d_ i = 2)\) and \((n \leq 8\), \(d_ i = 3)\). The authors also investigate the extent to which ideals generated by powers of generic linear forms satisfy the conjectured equality.
scientific article; zbMATH DE number 638322