Fröberg’s Hilbert-series conjecture for equal-degree forms in four variables
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Fröberg’s conjecture predicts the generic Hilbert series of ideals generated by forms of specified degrees. Here the setting is four variables over a characteristic-zero field, with all forms of degree 5 or all of degree 7 and any generator count r≥1.
Why it matters and what counts as progress
Why it matters: The quintic and septic cases test a general Hilbert-series prediction in four variables beyond previously resolved degree ranges.
What counts as progress: Prove the predicted series in the unresolved middle counts and propagate it to all r, or extend the result to further degrees.
How the problem developed
- 1985 · Conjecture [1]
Ralf Fröberg Proposed the predicted Hilbert series for ideals generated by general forms; the conjecture remains open in full generality.
- 2014 · Partial result [2]
Alessandro Oneto Proved the conjectured series for generic ideals generated by eight equal-degree forms in four variables, a single generator count rather than a full degree slice.
- 25 Aug 2026 · Reported theorem [3]
Qihang Wang and Dongming Zhang Claimed the predicted series for every generator count in four variables for equal-degree quintics and septics over characteristic-zero fields.
AI contribution
Fröberg's Conjecture for Quintics and Septics in Four Variables
Over any characteristic-zero field, Wang and Zhang prove Fröberg’s predicted generic Hilbert series for equal-degree forms in four variables when the degree is 5 or 7 and the generator count is any r≥1. Exact rank certificates cover the previously open middle ranges r=6–11 and r=6–21, respectively; propagation covers the remaining counts.
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