Fröberg’s conjecture on Hilbert series of ideals generated by generic forms

Establish that for a sequence (f1, ..., fm) of homogeneous polynomials in the polynomial ring k[x1, ..., xn] with algebraically independent coefficients and degrees d1, ..., dm, the Hilbert series of the quotient ring k[x1, ..., xn]/⟨f1, ..., fm⟩ equals the truncated power-series expansion of (∏_{i=1}^{m}(1 − t^{di}))/(1 − t)^{n}, denoted by [ (∏_{i=1}^{m}(1 − t^{di}))/(1 − t)^{n} ]_+.

Background

The paper’s complexity analysis requires Hilbert functions of certain determinantal ideals and column-generated ideals J_k(A). To obtain these Hilbert functions, the authors assume Fröberg’s conjecture, which predicts the Hilbert series of quotients by ideals generated by generic homogeneous forms.

Invoking Fröberg’s conjecture allows the authors to derive explicit formulas for Hilbert functions (via truncation of a rational function), which are then used to count rows and ranks in Macaulay matrices and to bound the arithmetic complexity of their enhanced F5-type algorithm for critical point computations.

References

We believe that $b'_{g(n,d)+1} > 0$ for the remaining values of $n$ and $d$.

— On the Hilbert series of ideals generated by general linear forms  (2608.22823 - Boij et al., 24 Aug 2026) in Remark following Theorem 2.4

Although we have proven that the Iarrobino conjecture does not hold for $n$ large enough, it is open for small values of $n$, in particular for $n=3$ and $m\geq 10$, where it coincides with the SHGH conjecture.

— On the Hilbert series of ideals generated by general linear forms  (2608.22823 - Boij et al., 24 Aug 2026) in Final Remark of Section 8, “The conjecture by Harbourne, Schenck, and Seceleanu”