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Fröberg's Conjecture through 2d in sufficiently many variables

Primarily AI-generated textHuman understanding: some partsmath.AC — Commutative Algebra

Contributed by Vlado Vojdanovski ↗

Version 1 / Oct 12, 2026 / CC BY 4.0

Abstract

Let k be any infinite field. We prove that, for every fixed d ≥ 1 and all sufficiently large n, the quotient of k[x1, . . . , xn] by any number of general forms of degree d has the Hilbert function predicted by Fröberg’s conjecture through degree 2d. The threshold for n is independent of the field and the number of generators. For generating degrees at least three, a direct construction with two groups of variables gives a nonincreasing recurrence for the two defects at the critical generator counts. Divisibility of these bounded defects and polynomial equidistribution then force their eventual vanishing.

Provenance statement

The work was generated via a guided exploration by Open AI's Astra model.
FormalizationsPalomar

References

  1. M. Boij, E. Dannetun and S. Lundqvist, Independence of generic forms and the Fröberg conjecture , preprint (2026), https://arxiv.org/abs/2605.03872 arXiv:2605.03872 .arXiv
  2. R. Fröberg, An inequality for Hilbert series of graded algebras , Math. Scand. 56 (1985), 117--144. https://doi.org/10.7146/math.scand.a-12092 doi:10.7146/math.scand.a-12092 .DOI
  3. M. Hochster and D. Laksov, The linear syzygies of generic forms , Comm. Algebra 15 (1987), nos. 1--2, 227--239. https://doi.org/10.1080/00927872.1987.10487449 doi:10.1080/00927872.1987.10487449 .DOI
  4. H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins , Math. Ann. 77 (1916), 313--352. https://doi.org/10.1007/BF01475864 doi:10.1007/BF01475864 .DOI

Version history

  1. v1Submitted by Vlado VojdanovskiInitial depositCurrentOct 11, 2026