ProofAtlas

Commutative algebra · Hilbert series · generic homogeneous ideals

Fröberg’s Conjecture

Collaboration beta

Does the Hilbert series of an ideal generated by generic homogeneous forms always equal the positive truncation of the expected product formula?

HR/(f1,…,fs)(t)=⌈∏i=1s(1-tdi)(1-t)m⌉
Known results and sources
An ivory coefficient staircase rises from a black-green graded monomial lattice and ends at a sharp gold truncation boundary, evoking the predicted Hilbert series of a generic homogeneous ideal without asserting that Fröberg’s conjecture is proved.
A graded coefficient staircase meets the positive-truncation boundary at the heart of Fröberg’s conjecture.

Research problem

Exact mathematical statement

Let R=k[x1,…,xm]R=k[x_1,…,x_m], with kk of characteristic zero, and let f1,…,fsf_1,…,f_s be generic homogeneous forms of degrees d1,…,dsd_1,…,d_s. Fröberg’s conjecture predicts

HR/(f1,…,fs)(t)=⌈∏i=1s(1-tdi)(1-t)m⌉,H_{R/(f_1,…,f_s)}(t)=\left\lceil\frac{\prod_{i=1}^{s}(1-t^{d_i})}{(1-t)^m}\right\rceil,

where positive truncation stops immediately before the first nonpositive coefficient.

Problem infographic

Problem at a glance

A deep forest-green scientific plate explains Fröberg’s conjecture for generic homogeneous forms f₁ through fₛ in R=k[x₁,…,xₘ] over characteristic zero. The expected series E(t)=∏ᵢ₌₁ˢ(1−tᵈᶦ)/(1−t)ᵐ appears above a coefficient landscape. Ivory and green bars retain the positive coefficient prefix through degree N−1; a vermilion marker places the first nonpositive coefficient at degree N on the zero axis, and the discarded cobalt tail lies below it. The plate asks whether the quotient Hilbert series always equals this positive truncation for generic forms of every degree list and states that the general case remains open.
Fröberg’s conjecture predicts the Hilbert series of a quotient by generic homogeneous forms from the expected product series. Positive truncation keeps only the coefficient prefix before the first nonpositive term; the plate distinguishes that retained prefix from the discarded raw tail. Whether this formula holds for generic forms of every degree list remains open in general.

Current mathematical picture

Where work on Fröberg’s Conjecture stands

Open conjecture

The current research map records the current work’s almost-reverse-lexicographic reduction, normalized primitive-map induction, solved early and small-column ranges, monic factor exchange, finite residual strip, exact one-tail repair mechanism, selected exact obstructions and repairs, and the present conormal frontier. The full mixed-degree conjecture remains open. Project theorems and computations remain provisional pending the current work’s stated literature, proof, and reproducibility audits; in particular, the 295-block scan is reported evidence with missing archived payloads, not a proof or a complete independently reproduced inventory.

Strongest supported footholdOne tail repairs arbitrary corank

An invertible conormal coefficient matrix is shown to make the first nonzero Schur-complement term repair the complete corank-k defect with one monomial tail.

Evidence posture · Reported result
Leading routeSorted residual single-tail conormal repair

Primary current route: combine degreewise openness, monic factor exchange, the residual strip, and one-tail repair; close the remaining blocks via Weak Conormal Escape, the small conormal-kernel bound, or the smallest-equation determinant.

Route status · Active route
Useful failureUniversal strong-Lefschetz power

Refuted as a universal primitive witness by the exact central projected-Veronese conic collapse; strong Lefschetz remains useful for ordinary ranks and complement dimensions.

Route status · Refuted route
Main reductionMonic factor exchange established

Exchanging retained and appended monic equations solves every block at or below the largest retained monic degree.

Evidence posture · Reported reduction
Completed special casePascal–Lefschetz families proved

Long-factor, higher-jet, and single-tangent constructions supply explicit determinant mechanisms and infinite solved families.

Evidence posture · Reported special case
Priority open bridgeProve smallest-equation full repair

At a suitable maximum-rank pure-power specialization in every residual block, prove that one first-equation coefficient matrix is invertible.

Task status · Work already reported in progress
Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Fröberg’s Conjecture in numbers

4.4kretained lines of mathematical investigation4,398 in the current working snapshot
Argument development
3,623 · 82%
Explored or eliminated routes
159 · 4%
Computational analysis
156 · 4%
Open obligations
241 · 5%
Definitions and setup
219 · 5%
26selected mapped statements11routes investigated11reported milestones7open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byRalf FröbergProposed1985Status checkedAug 2, 2026
Source recordedAug 2, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

27 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

27 selected steps

Scroll horizontally to explore the route

Working route overview for Fröberg’s ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Fröberg’s conjecture — Depends on missing premiseFröberg’s conjectureAlmost-revlex witness reduction — ChallengedAlmost-revlex witnessreductionDual primitive criterion — Depends on missing premiseDual primitive criterionFinite residual frontier — Depends on missing premiseFinite residual frontierFirst two socle-deficit strips — Depends on missing premiseFirst two socle-deficitstripsNormalized primitive-map induction — Depends on missing premiseNormalized primitive-mapinductionSorted residual-strip lemma — Depends on missing premiseSorted residual-strip lemmaStationary level–Jacobian contradiction — Depends on missing premiseStationary level–JacobiancontradictionBinary primitive theorem — Depends on missing premiseBinary primitive theoremCentral xyz repair — Depends on missing premiseCentral xyz repairCross-string conormal escape — Depends on missing premiseCross-string conormal escapeExact Hall obstruction at the rigid monomial prefix — Depends on missing premiseExact Hall obstruction atthe rigid monomial prefixSorted residual single-tail conormal repair — activeSorted residual single-tailconormal repairProjected-Veronese jets and Pascal–Lefschetz blocks — activeProjected-Veronese jets andPascal–Lefschetz blocksStationary level–Jacobian contradiction — activeStationary level–JacobiancontradictionAppended-quadratic special target — activeAppended-quadratic specialtargetUse a single strong-Lefschetz power g=L^d as the universal primitive witness. — stoppedUse a singlestrong-Lefschetz power g=L^das…Restrict the universal multiplier to one product g=ℓ₁⋯ℓ_d of linear forms. — stoppedRestrict the universalmultiplier to one productg=ℓ₁⋯ℓ_d…Restrict multipliers to k[L,z] and rely on Pascal total positivity within individual Lefschetz strings. — stoppedRestrict multipliers tok[L,z] and rely on Pascaltotal…Use only pure degree-transfer deformations x_i^{e_i}→x_i^{e_i}−t x_j^{e_i}. — stoppedUse only puredegree-transfer deformationsx_i^{e_i}→x_i^{e_i}−t…Audit the literature reduction — OpenAudit the literaturereductionProve Weak Conormal Escape — Work reported in progressProve Weak Conormal EscapeProve smallest-equation full repair — Work reported in progressProve smallest-equation fullrepairBound the conormal tensor kernel — OpenBound the conormal tensorkernelClose the δ=3 and δ=4 strips — OpenClose the δ=3 and δ=4 stripsProve cross-string coupling — OpenProve cross-string couplingRegenerate computational evidence — OpenRegenerate computationalevidence
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeSorted residual single-tail conormal repair

Primary current route: combine degreewise openness, monic factor exchange, the residual strip, and one-tail repair; close the remaining blocks via Weak Conormal Escape, the small conormal-kernel bound, or the smallest-equation determinant.

Route status · Active route
Active routeProjected-Veronese jets and Pascal–Lefschetz blocks

Active secondary route with solved one-/two-column, binary, long-factor, and single-tangent families; the missing theorem is general prescribed multivariate jet nondefect with cross-string coupling.

Route status · Active route
Active routeStationary level–Jacobian contradiction

Geometric alternative: assume every tangent vanishes, construct the level quotient A/Ann(V), and contradict its residual degree inequalities and conormal/Jacobian containments.

Route status · Active route
Active routeAppended-quadratic special target

Promising special target supported by universal support, broad exact scans, and simpler conormal degrees; repeated-label noncancellation or a smallest-equation repair theorem is still required.

Route status · Active route

Explored alternatives

Other routes

7 recorded
Route held in reserveReverse Bott / fusion / deformed Clebsch–Gordan

Structured backup retaining a solved local projector, fusion reference basis, deterministic carries, and local Hermite/Toeplitz blocks; the global quotient-corrected multiplicity theorem remains open.

Route status · Route held in reserve
Route held in reserveCurvilinear generic Toeplitz determinant

Viable backup with an explicit curvilinear complete intersection and one determinant per type, but no uniform generic carry/block noncancellation theorem.

Route status · Route held in reserve
Refuted routeUniversal strong-Lefschetz power

Refuted as a universal primitive witness by the exact central projected-Veronese conic collapse; strong Lefschetz remains useful for ordinary ranks and complement dimensions.

Route status · Refuted route
Browse 4 more explored routes
Narrowed routeOne split multiplier

Narrowed away from universal use by compelling but unreproduced (5⁴), d=2, q=8 rank evidence; higher secants and special split certificates remain viable.

Route status · Narrowed route
Refuted routeMultipliers confined to k[L,z]

Refuted as a universal route by an exact rank-5/6 block; future Pascal arguments must add off-diagonal cross-string coupling.

Route status · Refuted route
Useful but insufficientPure degree-transfer tails

Too rigid as a universal repair family in the tested equal-degree block; mixed monomial tails and the general socle-complement construction remain viable.

Route status · Useful but insufficient
Refuted routeAlmost-revlex monomial prefix witness

Refuted as a universal residual witness by an exact Hall cut, while the same block succeeds at a generic-coordinate pure-power specialization.

Route status · Refuted route

Route statements and reductions

Statements the next route can inspect and build on

Route statementOne- and two-primitive-column blocks

Every primitive block with c_q=1 or c_q=2 has a nonempty primitive locus in arbitrary embedding dimension.

Source-reported route statement · dependencies incomplete
Route statementBinary primitive theorem

For A=k[x,y]/(x^a,y^b), every appended degree and every positive primitive degree admit coordinates and a monomial multiplier giving the required direct sum.

Source-reported route statement · dependencies incomplete
Route statementLong-factor Pascal–Lefschetz families

The current work proves explicit long-factor, higher-jet, and single-tangent primitive blocks by Pascal–Lefschetz determinant factorizations; in the sharp single-tangent form m≥D and g=z^{m−D+r}(z+L) solve the degree-m block.

Source-reported route statement · dependencies incomplete
Route statementFinite residual frontier

Every unresolved normalized block has n≥3, q=d+s with 1≤s<(D−d)/2, H_A(s)≥2, and primitive count c_q≥3; its dual determinant is a square H_A(s)-dimensional restricted multiplication problem.

Source-reported route statement · dependencies incomplete
Route statementSingle-monomial full-repair lemma

At a corank-k pure-power primitive specialization, if one conormal coefficient matrix C_{i,α}=([x^α]v_aQ_i(r_b)) is invertible, deforming only the i-th equation by the socle-complement monomial makes the primitive matrix generically invertible.

Source-reported route statement · dependencies incomplete
Route statementWeak Conormal Escape

At a generic point of every maximum deficient-rank residual component, there exist v in the dual cokernel, r in the kernel, and an equation index i such that vQ_i(r)≠0.

Source-reported route statement · dependencies incomplete
Route statementSmallest-equation single-tail lemma

Every residual block admits a suitable maximum-rank pure-power specialization for which one coefficient matrix ([x^α]v_aQ₁(r_b)) is invertible, so one monomial tail in the smallest defining equation repairs the full defect.

Source-reported route statement · dependencies incomplete
Route statementStationary level–Jacobian contradiction

If total conormal stationarity occurs, the quotient B=A/Ann_A(V) is level with tightly constrained kernel, conormal, square, and derivative containments; ruling out such a quotient throughout the residual strip is an equivalent closing route.

Source-reported route statement · dependencies incomplete
Route statementCross-string conormal escape

Residual kernel-to-cokernel matrices can be made block triangular by primitive Lefschetz degree with nonzero Pascal/Toeplitz diagonal blocks and enough off-diagonal entries to couple every remaining string.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks
01
Regenerate computational evidence

Rerun every publication-relevant modular witness and the 917-, 1,053-, and 295-block scans under the canonical ordering, retaining complete payloads, type lists, code hashes, and nullspace normalization data.

Suggested move: Implement the canonical ascending exponent-tuple order and rerun rather than reconstructing missing historic payloads.
Ready to work on
02
Audit the literature reduction

Verify the exact Pardue/Moreno–Socías formulation, hypotheses, and citations connecting almost-revlex complete-intersection witnesses to the full mixed-degree Fröberg conjecture.

Suggested move: Compare the current work’s reduction statement against the original sources and record the exact characteristic, genericity, initial-ideal, and degree-type hypotheses.
Ready to work on
03
Close the δ=3 and δ=4 strips

Prove the general n×n restricted multiplication determinants in the first two socle-deficit strips.

Suggested move: Exploit s=1 and the closed primitive-count formulas, beginning with a generalization of the solved first-above-generator family.
Ready to work on
04
Prove cross-string coupling

Extend the exact Pascal–Lefschetz mechanisms from individual strings to a uniform residual kernel-to-cokernel coupling theorem.

Suggested move: Use the residual labeled-conormal graph to seek peelable label signatures or smallest-equation matchings rather than returning to the full primitive support graph.
Prerequisites still open
05
Bound the conormal tensor kernel

Prove dim ker Ψ*≤k−1 in every residual corank-k block, or establish another condition forcing an invertible matrix in the normal-direction space.

Suggested move: Analyze whether nonzero tensor relations are rank one and whether all such tensors share one factor forming a proper ruling.
Prerequisites still open
06
Prove smallest-equation full repair

At a suitable maximum-rank pure-power specialization in every residual block, prove that one first-equation coefficient matrix is invertible.

Suggested move: Seek an extremal noncancelling coefficient or triangular basis for the matrices ([x^α]v_aQ₁(r_b)), accounting explicitly for repeated labels.
Work already reported in progress
07
Prove Weak Conormal Escape

Show that every generic maximum deficient-rank residual component has some nonzero product vQ_i(r).

Suggested move: Start with corank one and use the inverse-system Gorenstein quotient, the low source degree s, and the annihilator containments to force a contradiction.
Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026
Current statusOpen conjecture

The full mixed-degree commutative conjecture remains open. A May 2026 paper proves new degree-wise cases rather than the full conjecture: for equal generator degree d>2 it reaches degree d+2, and for sufficiently many variables it reaches degree 2d−1.

[1]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. PreprintFor ideals generated by generic forms of one degree d>2, the conjectured Hilbert function is proved through degree d+2; for sufficiently many variables, it is proved through degree 2d−1.[1]
  2. Peer reviewedPardue established equivalent formulations and linked the conjecture to weak reverse-lexicographic behavior of generic initial ideals.[4]
  3. Peer reviewedThe conjecture is known for at most three variables; the three-variable case is due to Anick.[2]
10 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusFröberg’s conjecture
Logical consequenceMoreno–Socías conjecture on generic initial ideals

Almost reverse-lexicographic behavior of the generic initial ideal implies Fröberg’s conjecture.

[3]
Related problemWeak and Strong Lefschetz properties

Lefschetz properties are used to study generic Hilbert functions and almost reverse-lexicographic initial ideals.

[5][3]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • computation · source linked; not reproduced by ProofAtlasMacaulay2 and MATLAB calculations for the 2026 degree-wise results

    Finite computations used for exceptional cases and sign-change checks in the cited partial theorem; not a verification of the full conjecture.

    [1][9]

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

11 mapped milestonesretained argument map

Browse all 11 mapped stages

  1. stage 1Almost-revlex witness reduction
  2. stage 2Primitive map and dual criterion
  3. stage 3Early and small primitive ranges
  4. stage 4Strong-Lefschetz-only witness fails centrally
  5. stage 5Pascal–Lefschetz determinant families
  6. stage 6Monic factor exchange
  7. stage 7Finite residual strip
  8. stage 8One-tail repair at arbitrary corank
  9. stage 9Rigid residual specializations separated from the full family
  10. stage 10Residual single-tail scan evidence
  11. stage 11Conormal closing frontier
Almost-revlex witness reductionThe current work organizes the full conjecture around one almost-revlex complete-intersection witness for each Artinian degree type.

Mapped research milestoneInitial research sequence

Research stage 1
Primitive map and dual criterionThe source turns the induction into a square primitive determinant and an equivalent small restricted multiplication map.

Mapped research milestoneInitial research sequence

Research stage 2
Early and small primitive rangesThe current work solves all q≤d blocks, the range below the first relation, every binary prefix, and every one- or two-column primitive block.

Mapped research milestoneInitial research sequence

Research stage 3
Strong-Lefschetz-only witness fails centrallyAn exact projected conic defeats g=L³ in a central block, while the displayed g=xyz determinant repairs that same block.

Mapped research milestoneInitial research sequence

Research stage 4
Pascal–Lefschetz determinant familiesExplicit long-factor, higher-jet, and single-tangent determinants solve infinite families and identify cross-string coupling as the remaining issue.

Mapped research milestoneInitial research sequence

Research stage 5
Monic factor exchangeExchanging retained and appended monic equations solves every block at or below the largest generator degree.

Mapped research milestoneInitial research sequence

Research stage 6
Finite residual stripGorenstein symmetry concentrates every remaining sorted block in 1≤s<(D−d)/2 and solves every extension with d≥D−2.

Mapped research milestoneInitial research sequence

Research stage 7
One-tail repair at arbitrary corankAn invertible conormal coefficient matrix is shown to let one monomial tail repair an entire corank-k primitive defect.

Mapped research milestoneInitial research sequence

Research stage 8
Rigid residual specializations separated from the full familyExact and computational witnesses narrow one split multiplier, one Lefschetz direction, pure transfers, and the rigid monomial prefix without implying failure of the full parameter family.

Mapped research milestoneInitial research sequence

Research stage 9
Residual single-tail scan evidenceA bounded scan reports nine-for-nine first-equation one-tail repairs, including one corank-two block, while the audit records material reproducibility gaps.

Mapped research milestoneInitial research sequence

Research stage 10
Conormal closing frontierThe remaining proof is separated into Weak Conormal Escape, the smallest-equation determinant, the conormal-kernel bound, stationary level contradiction, and the first δ-strips.

Mapped research milestoneInitial research sequence

Research stage 11

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

20 standing statements6 proposed statements11 mathematical milestones7 open questions1 narrowed routes3 conditional results9 completed special cases
Statements by mathematical role26 selected mapped statements
  • theorem candidate1 of 261
  • reduction6 of 266
  • equivalence1 of 261
  • lemma12 of 2612
  • negative result2 of 262
  • computational claim3 of 263
  • counterexample1 of 261
Selected mathematical clusters7 mathematical clusters
Conjecture and witness reductionThe exact Hilbert-series claim, the almost-revlex witness reduction, the primitive map, its dual form, and the unresolved literature audit.8 displayed rows
  • retained route statementFröberg’s conjecture
  • retained route statementAlmost-revlex witness reduction
  • retained route statementNormalized primitive-map inductionintermediate
  • retained route statementDual primitive criterionintermediate
  • Recorded relationshipThe current work adopts the almost-revlex complete-intersection witness theorem as the entry reduction for the full conjecture.reduces to · reported by source
  • Recorded relationshipGorenstein duality turns the square primitive map into restricted multiplication on the orthogonal complement of W_q.equivalent to · reported by source
  • ChallengeThe current work treats the Pardue/Moreno–Socías reduction as accepted but explicitly says its exact formulation and citation must be checked before publication. Until that check, it cannot serve as reviewed public proof authority.unsupported step · open
  • Research targetAudit the literature reductionopen
Solved ranges and determinant familiesEarly degrees, binary prefixes, one-/two-column blocks, low-conormal separation, long-factor Pascal families, and dominant-degree extensions.7 displayed rows · 1 route included
  • retained route statementUniform early-degree rangesspecial case
  • retained route statementOne- and two-primitive-column blocksspecial case
  • retained route statementBinary primitive theoremspecial case
  • retained route statementAmbient separation and low-conormal nontrivialityconditional
  • retained route statementLong-factor Pascal–Lefschetz familiesspecial case
  • retained route statementDominant-degree extension theoremspecial case
  • Active routeProjected-Veronese jets and Pascal–Lefschetz blocksActive secondary route with solved one-/two-column, binary, long-factor, and single-tangent families; the missing theorem is general prescribed multivariate jet nondefect with cross-string coupling.
Residual normalizationMonic factor exchange, the finite strip, the small dual determinant, and the explicit δ=3,4 frontier.9 displayed rows · 2 routes included
  • retained route statementMonic factor exchangeintermediate
  • retained route statementSorted residual-strip lemmaintermediate
  • retained route statementFinite residual frontierintermediate
  • retained route statementFirst two socle-deficit stripsintermediate
  • DerivationThe solved early and small primitive-count ranges remove q≤d, binary prefixes, and c_q≤2. Monic factor exchange places all remaining degrees above d, and Gorenstein symmetry plus unimodality gives 2s<D−d. The dual criterion then expresses each residual block on an H_A(s)-dimensional space.active reported
  • DerivationFor δ=3 or 4, the inequality 1≤s<δ/2 forces s=1. The dual criterion therefore has source and target dimension H_A(1)=n, yielding an explicit n×n problem.active reported
  • Research targetClose the δ=3 and δ=4 stripsopen
  • Active routeSorted residual single-tail conormal repairPrimary current route: combine degreewise openness, monic factor exchange, the residual strip, and one-tail repair; close the remaining blocks via Weak Conormal Escape, the small conormal-kernel bound, or the smallest-equation determinant.
  • Active routeAppended-quadratic special targetPromising special target supported by universal support, broad exact scans, and simpler conormal degrees; repeated-label noncancellation or a smallest-equation repair theorem is still required.
Conormal closing frontierThe conditional full-repair theorem and four open closure formulations: weak escape, smallest-equation determinant, small conormal kernel, and stationary level contradiction.14 displayed rows · 2 routes included
  • retained route statementSingle-monomial full-repair lemmaconditional
  • retained route statementWeak Conormal Escape
  • retained route statementSmallest-equation single-tail lemma
  • retained route statementSmall Conormal Kernel Lemma
  • retained route statementStationary level–Jacobian contradictionconditional
  • retained route statementCross-string conormal escape
  • DerivationIf the smallest-equation coefficient matrix is invertible in every residual block, the single-tail lemma repairs every defect. Degreewise openness assembles the primitive loci, the monic extension produces almost-revlex witnesses, and the literature reduction would imply Fröberg’s conjecture.active reported
  • DerivationWeak Conormal Escape supplies a rank-raising tangent at every alleged generic maximum defect. Thus every residual primitive locus is nonempty; the current work’s degreewise openness and extension chain then reaches the almost-revlex witness theorem and, subject to the literature reduction, the conjecture.active reported
  • Research targetProve Weak Conormal Escapein progress reported
  • Research targetProve smallest-equation full repairin progress reported
  • Research targetBound the conormal tensor kernelopen
  • Research targetProve cross-string couplingopen
  • Active routeSorted residual single-tail conormal repairPrimary current route: combine degreewise openness, monic factor exchange, the residual strip, and one-tail repair; close the remaining blocks via Weak Conormal Escape, the small conormal-kernel bound, or the smallest-equation determinant.
  • Active routeStationary level–Jacobian contradictionGeometric alternative: assume every tangent vanishes, construct the level quotient A/Ann(V), and contradict its residual degree inequalities and conormal/Jacobian containments.
Rigid-specialization obstructionsExact conic, Hall, and one-direction defects plus carefully scoped split-locus and pure-transfer evidence prevent repetition of over-rigid universal routes.14 displayed rows · 5 routes included
  • retained route statementStrong-Lefschetz cube conic collapsespecial case
  • retained route statementExact Hall obstruction at the rigid monomial prefixspecial case
  • retained route statementSingle split-multiplier defect evidencecomputational
  • retained route statementOne-Lefschetz-direction multiplier defectspecial case
  • Useful failureUse a single strong-Lefschetz power g=L^d as the universal primitive witness.reported failure
  • Useful failureRestrict the universal multiplier to one product g=ℓ₁⋯ℓ_d of linear forms.reported failure
  • Useful failureRestrict multipliers to k[L,z] and rely on Pascal total positivity within individual Lefschetz strings.reported failure
  • Useful failureUse only pure degree-transfer deformations x_i^{e_i}→x_i^{e_i}−t x_j^{e_i}.reported failure
  • Useful failureUse the almost-revlex monomial prefix as the universal primitive witness.reported failure
  • Refuted routeUniversal strong-Lefschetz powerRefuted as a universal primitive witness by the exact central projected-Veronese conic collapse; strong Lefschetz remains useful for ordinary ranks and complement dimensions.
  • Narrowed routeOne split multiplierNarrowed away from universal use by compelling but unreproduced (5⁴), d=2, q=8 rank evidence; higher secants and special split certificates remain viable.
  • Refuted routeMultipliers confined to k[L,z]Refuted as a universal route by an exact rank-5/6 block; future Pascal arguments must add off-diagonal cross-string coupling.
  • Useful but insufficientPure degree-transfer tailsToo rigid as a universal repair family in the tested equal-degree block; mixed monomial tails and the general socle-complement construction remain viable.
  • Refuted routeAlmost-revlex monomial prefix witnessRefuted as a universal residual witness by an exact Hall cut, while the same block succeeds at a generic-coordinate pure-power specialization.
Exact repairs and computation reportsThe determinant-54 central repair, corank-two one-tail repair, pure-power Hall-block witness, split-locus trials, and bounded residual scan, all retained at their stated unreproduced evidence level.9 displayed rows · 1 route included
  • retained route statementCentral xyz repairspecial case
  • retained route statementPure-power witness for the Hall-obstructed blockcomputational
  • ComputationDisplayed rational 6×6 primitive matrix for A₀=Q[x,y,z]/(x³,y³,z³), d=3, q=4, with multiplier g=xyz.The current work reports determinant 54, giving an exact nonzero witness conditional on the recorded matrix and arithmetic. · reported unreproduced
  • ComputationTwo-prime full-defect conormal calculation for the (4,4,4,4,4), d=5, q=7 residual block after deforming x₁⁴ by t x₄²x₅².The current work reports a 155×155 matrix of rank 153 at the pure-power fiber, invertible 2×2 conormal coefficient matrices over both primes, and nonzero full primitive determinants at t=1. · reported unreproduced
  • ComputationTwo-prime determinant computation at a Vandermonde-coordinate pure-power specialization for the Hall-obstructed (3,4,4,4,4), d=4, q=7 block.The current work reports determinant residues 783101 mod 1,000,003 and 427109 mod 1,000,033. · reported unreproduced
  • ComputationForty finite-field rank trials for one split quadratic in the (5⁴), d=2, q=8 block, plus unrestricted and two-split comparison multipliers.All one-split trials had rank 84/85; the current work reports full rank 85 for an unrestricted quadratic and for a sum of two independently split quadratics. · reported unreproduced
  • ComputationReported residual-strip single-tail scan over sorted types with 3≤n≤5, 2≤e_i≤5, max e_i≤d≤5, and primitive matrix size at most 160.The current work reports 295 residual blocks, nine pure-power defects, and successful first-equation one-tail full repair for all nine over both primes; one defect had corank two. · reported unreproduced
  • Research targetRegenerate computational evidenceopen
  • Active routeSorted residual single-tail conormal repairPrimary current route: combine degreewise openness, monic factor exchange, the residual strip, and one-tail repair; close the remaining blocks via Weak Conormal Escape, the small conormal-kernel bound, or the smallest-equation determinant.
Structured backup routesReverse Bott/fusion and curvilinear Toeplitz retain exact local toolkits but are paused behind unresolved global multiplicity or noncancellation theorems.2 displayed rows · 2 routes included
  • Route held in reserveReverse Bott / fusion / deformed Clebsch–GordanStructured backup retaining a solved local projector, fusion reference basis, deterministic carries, and local Hermite/Toeplitz blocks; the global quotient-corrected multiplicity theorem remains open.
  • Route held in reserveCurvilinear generic Toeplitz determinantViable backup with an explicit curvilinear complete intersection and one determinant per type, but no uniform generic carry/block noncancellation theorem.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeAt a suitable maximum-rank pure-power specialization in every residual block, prove that one first-equation coefficient matrix is invertible.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Every residual degree type is covered.
  • The specialization is maximum rank and the selected coefficient matrix is proven invertible.
  • The resulting tail remains monic and the full-repair implication is audited.

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Fröberg’s Conjecture · ready to start

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Research contextPrepared context for any AI agent

Does the Hilbert series of an ideal generated by generic homogeneous forms always equal the positive truncation of the expected product formula?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references10 cited works · next context review by Nov 2, 2026

The mathematical context was checked on Aug 2, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
  2. 2
    Algebraic Stories from One and from the Other Pocketspeer reviewed result · accessed Aug 2, 2026
  3. 3
  4. 4
    Generic sequences of polynomialspeer reviewed result · accessed Aug 2, 2026
  5. 5
  6. 6
    Questions and conjectures on extremal Hilbert seriespeer reviewed result · accessed Aug 2, 2026
  7. 7
    An inequality for Hilbert series of graded algebrasoriginal source · accessed Aug 2, 2026
  8. 8
    Fröberg conjectureencyclopedia · accessed Aug 2, 2026
  9. 9
  10. 10
    An inequality for Hilbert series of graded algebrasauthoritative webpage · accessed Aug 2, 2026

Important qualifications

  • The scoped search did not verify a public proof-assistant formalization of this exact conjecture; formalizationResources is therefore intentionally empty.
  • The Wikipedia entry is recorded only as evidence that a dedicated page exists, not as authority for mathematical status.
  • Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.

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