Generic sequences of polynomials
Let \(k\) be an infinite field, \(S = k[x_1,\ldots,x_n]\) the polynomial ring with \(n\) variables and let \(I=(f_1,\dots,f_r) \subset S\) be a homogeneous ideal defined by a generic sequence of \(r\) homogeneous polynomials of degree \(d_1,\ldots,d_r\). Given a power series \(\sum_{d=0}^{\infty} a_d\, t^d\) with integer coefficients, let define the power series \(\left| \sum_{d=0}^{\infty} a_d\, t^d \right| = \sum_{d=0}^{\infty} b_d\, t^d\), where \(b_d = a_d\) if \(a_i > 0,\;0 \leq i \leq d\) and \(b_d = 0\) otherwise. \textit{R. Fröberg} [Math. Scand. 56, 117--144 (1985; Zbl 0582.13007)] conjectured that the Hilbert series of \(S/I\) is \[ H_{S/I}(t)=\left|\frac{\prod_{i=1}^{r}(1-t^{d_i})}{(1-t)^n}\right|. \] Under the same hypothesis \textit{G. Moreno-Socías} [Autour de la fonction de Hilbert-Samuel (escaliers d'idéaux polynomiaux) (École Polytechnique), (1991)] conjectured that the initial ideal \(J = \mathbf{in}(I)\) with respect to the degree reverse lexicographic order is a weak reverse lexicographic ideal, namely if \(x^\mu\) is a minimal generator of \(J\), then any monomial of the same degree which precedes \(x^\mu\) belongs to \(J\). The main result of this paper is the proof that Moreno-Socías' conjecture implies Fröberg's conjecture. The author achieves this result by introducing some conjectures all equivalent to Fröberg's one and by exploiting the properties of the degree reverse lexicographic order. In the last part of the paper, the author discusses the condition for an Hilbert series to be realized by a weak reverse lexicographic ideal and proves that a Hilbert series of the type \(\left|\frac{\prod_{i=1}^{r}(1-t^{d_i})}{(1-t)^n}\right|\) admits such an ideal.
scientific article; zbMATH DE number 5787063
scientific article