On the Hilbert series of ideals generated by generic forms

Abstract

There is a longstanding conjecture by Fröberg about the Hilbert series of the ring R/I, where R is a polynomial ring, and I an ideal generated by generic forms. We prove this conjecture true in the case when I is generated by a large number of forms, all of the same degree. We also conjecture that an ideal generated by m'th powers of forms of degree d gives the same Hilbert series as an ideal generated by generic forms of degree md. We verify this in several cases. This also gives a proof of the first conjecture in some new cases.

Questions and conjectures on extremal Hilbert series

“…, d r ), then the conjecture is proved for these values. There are also some further special results when all d i are equal, [33]; (n, 2, 3), n ≤ 11 and (n, 3, r), n ≤ 8, by Fröberg and Hollman [13]. It is also proved by Hochster and Laksov that the formula is correct in the first nontrivial degree min{d i } + 1, [19].…”

The generic dimension of spaces of $\mathbf{A}$-harmonic polynomials

“…This generalizes Theorem 3.2, but now genericity is no longer expressed in terms of regular sequences. The Fröberg Conjecture is known to be true if r = N + 1, or if N ≤ 3 and in some other cases; see [6] or [19] for details and references and a recent new result in [20]. A direct (not based on (5.4)) but not fully conclusive treatment of r > N = 2 can be found in [28] when K = R. However, the method requires finding some "admissible" sequence of integers for which no general algorithm is provided.…”

Decomposability of Tensors

“…Here, we explain how the binary case can be treated; see [155] (14); see [156]. In the case of binary forms, by semicontinuity, we may specialize the G i 's to be powers of linear forms.…”