Fröberg–Pardue Hilbert series conjecture for generic sequences

Determine whether, for a generic sequence F = {f1, …, fm} of homogeneous polynomials in k[x1, …, xn] with degrees deg(fi) = di, the Hilbert series of the quotient ring R/⟨F⟩ equals the positive truncation of the rational function ∏i=1^m (1 − z^{di})/(1 − z)^n, that is, establish H_{R/⟨F⟩}(z) = [∏i=1^m (1 − z^{di})/(1 − z)^n].

Background

Beyond the weakly reverse-lexicographic structure, a precise prediction for the Hilbert series of R/⟨F⟩ for generic sequences has been conjectured by Fröberg and later discussed by Pardue. This conjecture provides a closed-form expression (with positive-part truncation) for the Hilbert series in terms of the degrees of the generators.

The paper leverages this conjectured formula to drive its Hilbert-based construction of leading monomials: by matching the Hilbert function of a candidate monomial ideal with the conjectured Hilbert series, the algorithm can infer how many leading monomials must appear in each degree.

The authors assume this conjectural formula (or deduce it via the Moreno–Socías conjecture using Pardue’s theorem) to justify and validate the LGB procedure, underscoring its importance in predicting Gröbner basis structure for generic inputs.

References

Conjecture [] Let F = {f_1, \dots, f_m} \subset R be a generic sequence of homogeneous polynomials with degrees \deg(f_i) = d_i . Then the Hilbert series of R/\langle F \rangle satisfies

H_{R/\langle F \rangle}(z) = \left[ \frac{\prod_{i=1}m (1 - z{d_i})}{(1 - z)n} \right].

— An Algorithm for Computing the Leading Monomials of a Minimal Groebner Basis of Generic Sequences (2505.10246 - Sakata et al., 15 May 2025) in Subsection “Generic Sequences” (Section 2)