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Fröberg’s Conjecture on the Hilbert–Poincaré series of generic ideals

Determine whether, for an infinite field K, a generic sequence of homogeneous polynomials f1, …, fm in R = K[x1, …, xn] of degrees d1, …, dm generates an ideal I whose Hilbert–Poincaré series HS_{R/I}(z) equals the truncation after the last consecutive positive coefficient of the rational function ∏_{i=1}^{m}(1 − z^{d_i})/(1 − z)^{n}; equivalently, show that the sequence (f1, …, fm) is cryptographic semi-regular.

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Background

The authors connect Fröberg’s conjecture to the notion of cryptographic semi-regularity and the explicit form of the Hilbert–Poincaré series expected for generic sequences. They note that Fröberg’s conjecture is equivalent to Pardue’s Conjecture B, thereby linking the existence of the specific Hilbert series to the semi-regularity of generic sequences.

Confirming Fröberg’s conjecture would provide a precise Hilbert–Poincaré series for generic systems, enabling sharp complexity estimates for Gröbner basis computations and reinforcing the semi-regularity framework used in cryptographic applications and algebraic geometry.

References

In 1985, Fröberg already conjectured in that, when $K$ is an infinite field, a generic sequence of homogeneous polynomials $f_1,\ldots,f_m \in R$ of degrees $d_1,\ldots , d_m$ generates an ideal $I$ with the Hilbert-Poincar {e} series of the form eq:semiregHil2, namely $(f_1,\ldots , f_m)$ is cryptographic semi-regular.

eq:semiregHil2:

HSR/I(z)=[∏j=1m(1−zdj)(1−z)n].{\rm HS}_{R/I}(z) = \left[ \frac{\prod_{j=1}^{m}(1-z^{d_j})}{(1- z)^n} \right].

— On Hilbert-Poincaré series of affine semi-regular polynomial sequences and related Gröbner bases (2401.07768 - Kudo et al., 15 Jan 2024) in Subsection 2.3 (Cryptographic semi-regular sequences)