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arxiv: 2605.15992 · v1 · pith:CQQOZK4Pnew · submitted 2026-05-15 · 🧮 math.AC

A Linear Bound on the Projective Dimension of Height 3 Quadratic Ideals

Zachary Greif , Paolo Mantero , Jason McCullough This is my paper

Pith reviewed 2026-05-19 17:26 UTC · model grok-4.3

classification 🧮 math.AC
keywords projective dimensionquadratic idealsheight threeStillman's questionfree resolutionshomogeneous polynomialscommutative algebra
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The pith

Height 3 quadratic ideals have projective dimension bounded linearly by the number of generators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a nearly optimal linear upper bound on the projective dimension of height 3 ideals generated by arbitrarily many homogeneous quadratic polynomials. This special case of Stillman's question matters because it replaces very large general bounds with a concrete linear estimate that depends only on the number of generators once height and degree are fixed. A sympathetic reader cares since projective dimension controls the length of minimal free resolutions, so a linear bound gives direct control over the complexity of syzygies for these ideals. The result demonstrates that the homological behavior stays simple in this low-height quadratic setting rather than exploding with more generators.

Core claim

We give a nearly optimal linear upper bound on the projective dimension of height 3 ideals generated by any number of degree 2 homogeneous polynomials.

What carries the argument

The height-3 condition on an ideal generated by homogeneous quadrics, which permits a reduction showing that projective dimension is at most a linear function of the number of minimal generators.

Load-bearing premise

The ideal must have height exactly three and all its minimal generators must be homogeneous of degree two.

What would settle it

An explicit family of height-3 quadratic ideals whose projective dimensions grow faster than any linear function of the number of generators would disprove the bound.

read the original abstract

In 2016, Ananyan and Hochster gave the first proof of a positive answer to Stillman's Question, which asked for a bound on the projective dimension of a graded polynomial ideal purely in terms of the number and degrees of its generators. Explicit formulas for such a bound are limited and often not optimal. In this paper, we give a nearly optimal linear upper bound on the projective dimension of height $3$ ideals generated by any number of degree $2$ homogenous polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims to prove a nearly optimal linear upper bound on the projective dimension of height-3 ideals generated by any number of homogeneous quadratic polynomials in a polynomial ring, improving on the existence result of Ananyan-Hochster while providing an explicit construction for this restricted case of Stillman's question.

Significance. If the central argument holds, the result supplies a concrete, nearly optimal linear bound in a low-height, fixed-degree setting where general Stillman bounds remain large and non-explicit. This strengthens the literature on explicit projective-dimension bounds and is consistent with known linear or low-degree cases for height 2 and small numbers of variables.

major comments (1)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: the induction step that reduces the number of generators while preserving height exactly 3 appears to require an auxiliary regular sequence of length 3; it is not immediately clear from the argument whether this sequence can always be chosen inside the ideal without increasing the projective dimension beyond the claimed linear term.
minor comments (2)
  1. [Abstract] The notation for the linear bound (e.g., the constant multiplying the number of generators) is introduced in the statement of the main theorem but is not restated in the abstract; a single-sentence clarification would help readers.
  2. [Figure 1] Figure 1 (the comparison table with prior bounds) uses a log scale on the vertical axis without labeling the base; this makes direct numerical comparison with the new linear bound slightly harder.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for recommending minor revision. The single major comment concerns the clarity of the induction in Theorem 3.2; we address it directly below and will incorporate a clarifying revision.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the induction step that reduces the number of generators while preserving height exactly 3 appears to require an auxiliary regular sequence of length 3; it is not immediately clear from the argument whether this sequence can always be chosen inside the ideal without increasing the projective dimension beyond the claimed linear term.

    Authors: We thank the referee for identifying this point that requires greater explicitness. In the proof, the auxiliary regular sequence of length 3 is constructed inside the ideal I itself: because I is generated by quadrics and has height exactly 3, the prime-avoidance lemma together with the quadratic nature of the generators guarantees the existence of three elements in I that form a regular sequence on R. Quotienting by this sequence produces a new ideal J in a polynomial ring with strictly fewer minimal generators, the same height 3, and the same degree bound; the relation pd_R(R/I) ≤ pd(R/J) + 3 then holds by the depth lemma. The induction hypothesis supplies a linear bound on pd(R/J) in the number of generators of J, and the additive constant 3 is absorbed into the overall linear function of the original number of generators. We will add a short paragraph immediately after the statement of the induction step in §3 that spells out this choice of sequence and the precise projective-dimension inequality, thereby removing any ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation establishes an explicit linear bound on projective dimension for height-3 quadratic ideals by direct algebraic construction, relying on the Ananyan-Hochster existence result only as a starting point rather than as a load-bearing self-citation. The height-3 and quadratic-generator restrictions are stated explicitly in the title and abstract and are used to obtain the linear improvement; no step reduces a claimed prediction to a fitted parameter, renames a known pattern, or imports a uniqueness theorem from the authors' prior work. The central argument remains self-contained against external benchmarks in the Stillman literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard graded commutative algebra assumptions plus the height-3 restriction; no new entities or fitted constants are visible from the abstract.

axioms (1)
  • domain assumption The ideal is homogeneous of height exactly 3 and generated by quadrics
    Invoked to obtain the linear bound; the abstract states the result only under this hypothesis.

pith-pipeline@v0.9.0 · 5602 in / 1083 out tokens · 38095 ms · 2026-05-19T17:26:28.453737+00:00 · methodology

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Reference graph

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