A bound for the Waring rank of the determinant via syzygies

Mats Boij; Zach Teitler

REVIEW 3 major objections 4 minor 50 references

A bound for the Waring rank of the determinant via syzygies

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Pith's one-line read Waring rank of the 3x3 determinant is at least 15

desk verdict A real improvement on a benchmark rank bound via a genuinely new syzygy method; the proof hangs on one uncertified Macaulay2 computation that should be shipped before publication. read the letter →

arxiv 1908.08896 v2 pith:FU2HTYY6 submitted 2019-08-23 math.AG math.AC

Mats Boij , Zach Teitler This is my paper · ORCID

classification math.AGmath.AC MSC 15A2115A6914N1513D02

keywords WaringrankdeterminantpermanentapolaridealsyzygiesBettinumberscactuslowerbound

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the Waring rank of the $3\times 3$ determinant is at least 15, improving the known lower bound from 14 to 15 while the best upper bound remains 18. The proof is carried by a new technique: instead of studying apolar ideals directly, it counts syzygies of those ideals, specifically the graded Betti number $\beta_{5,6}$. A general result shows any zero-dimensional scheme of degree 13 in $\mathbb{P}^8$ with no linear forms has $\beta_{5,6} \ge 140$, while for every linear form $\ell$ the apolar ideal of $\det_3 - \ell^3$ has $\beta_{5,6} < 140$; the two inequalities force every such $\det_3 - \ell^3$ to have rank at least 14, and hence $\det_3$ itself cannot be a sum of 14 cubes. The same argument establishes that the symmetric cactus rank of the $3\times 3$ permanent is at least 14, up from the previously known 10.

What carries the argument

The load-bearing object is the graded Betti number $\beta_{5,6}(T/I)$, the number of minimal generators in homological degree 5 and internal degree 6 of the minimal free resolution of $T/I$. The argument rests on three inputs. First, the Apolarity Lemma equates Waring rank $r$ with the existence of a reduced zero-dimensional scheme of degree $r$ whose ideal is apolar to the form. Second, a low-degree-strand Betti inequality shows that if $I \subseteq J$ and $J$ contains no linear form, then $\beta_{i,i+1}(T/I) \le \beta_{i,i+1}(T/J)$ for all $i$; this lets small Betti numbers of an apolar ideal obstruct the existence of small apolar point ideals. Third, a consecutive-cancellations result gives the universal lower bound $\beta_{5,6}(T/I) \ge 140$ for every saturated ideal $I$ of degree 13 without linear forms. For the shifted determinant, the apolar ideal is described by 36 explicit quadratic generators in a universal family $\mu\det_3 - \lambda(x_1+x_5+x_9)^3$, and a computed free resolution of the universal ideal has $\beta_{5,6}=135$, which survives specialization by a flatness argument. Thus the mechanism is: universal lower bound 140 for degree-13 ideals versus universal upper bound 135 for apolar ideals of $\det_3-\ell^3$.

What would settle it

Recompute, in an independent computer algebra system or by a hand-checked Gröbner basis, the minimal free resolution of $T/(\mu\det_3 - \lambda(x_1+x_5+x_9)^3)^\perp$ for one explicit pair (e.g., $\mu=\lambda=1$) and check whether $\beta_{5,6}$ is 140 or larger; a value $\ge 140$ would invalidate the proof of the main theorem.

Watch

Extended reading notes

Core claim

The central claim is that $\operatorname{rk}(\det_3) \ge 15$: the $3\times 3$ determinant cannot be written as a sum of 14 or fewer cubes of linear forms. The proof proceeds by contradiction. For any linear form $\ell$, the apolar ideal $(\det_3 - \ell^3)^\perp$ has graded Betti number $\beta_{5,6} < 140$, and by the Apolarity Lemma a rank-13 expression of $\det_3 - \ell^3$ would produce an apolar ideal of a 13-point scheme, whose $\beta_{5,6}$ is at least 140. Hence $\operatorname{rk}(\det_3 - \ell^3) \ge 14$ for every $\ell$, so if $\det_3$ had rank 14, removing one cube would leave a rank-13 polynomial, a contradiction. The same comparison shows $\operatorname{crk}(\operatorname{per}_3) \ge 14$. The paper also gives a new proof of the previously known bounds $\operatorname{rk}(\det_3) \ge 14$, $\operatorname{crk}(\det_3) \ge 14$, and $\operatorname{rk}(\operatorname{per}_3) \ge 14$. The novelty is the use of syzygies of the apolar ideal, rather than the ideal's generators or Hilbert function, to bound rank.

Load-bearing premise

The entire bound rests on a computer-generated free resolution of a universal apolar ideal (reported with $\beta_{5,6}=135$) and on the flatness argument that this resolution stays exact after substituting any nonzero $\mu$ and any $\lambda$; if either the computation or the flatness is wrong, the rank bound could fail.

Editorial extensions

If this is right

The Waring rank of $\det_3$ lies between 15 and 18, narrowing the previously known interval by one.
The symmetric cactus rank of $\operatorname{per}_3$ is at least 14, improving the prior lower bound of 10.
Syzygies of apolar ideals are an effective lower-bound tool for Waring and cactus rank, yielding new proofs of the rank-14 bounds for $\det_3$ and $\operatorname{per}_3$.
Any polynomial of the form $\det_3 - \ell^3$ has Waring rank at least 14 for every linear form $\ell$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to apply the same $\beta_{5,6}$ comparison to other forms with known apolar ideals, such as larger determinants or permanents, where the required Betti-number computations may still be feasible.
Because the universal lower bound 140 comes from a consecutive-cancellations result, the method depends on the Hilbert function of the points; varying the degree or the ambient dimension should yield similar thresholds, potentially giving new rank bounds for other forms.
If the flatness step is verified independently (or fails), the method's scope changes: the syzygy obstruction would then only apply to smoothable schemes, and non-smoothable Gorenstein schemes would need separate treatment.
The paper's remark that $\beta_{5,6}$ for the universal family is 135 rather than 100 suggests the actual value for specializations may often be 100; if that is true, a sharper analysis of the family could yield $\operatorname{rk}(\det_3) \ge 16$ from the same framework.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The headline is simple: this paper improves the lower bound for the Waring rank of the 3x3 determinant from 14 to 15, and it does so with a method—syzygies of the apolar ideal—that is genuinely new to rank lower bounds. The same machinery gives cactus rank at least 14 for the 3x3 permanent, up from 10. If the computation at the core of Proposition 15 checks out, this is a clean, real result.

The paper deserves credit for the structure it builds. The reduction of the problem to a finite list of cases by normalizing the subtracted cube is elegant. The bound β5,6 ≥ 140 for degree-13 ideals via Peeva's consecutive cancellations is a nice argument, cleanly presented. The explicit description of the apolar ideal for µ det3 − λ(tr)^3 in Proposition 14 is a concrete piece of work that appears correct as far as I can see, and the Gorenstein symmetry argument tying the Hilbert function to (1,9,9,1) is sound.

The soft spot is exactly where the stress test points. Proposition 15 rests on an unscripted Macaulay2 computation of the Betti table of the universal ideal H~, giving β5,6 = 135. That single integer is what makes β5,6 < 140, and the whole theorem depends on it. The flatness argument is sketched but not fully detailed, and it leans on the constancy of the Hilbert function, which itself relies on a 'direct computation' in Proposition 14. None of this is disqualifying—computations this size are standard in commutative algebra and the authors are experienced enough that I'd trust them—but the paper as written is not reproducible at its most crucial step. The Macaulay2 code should be included or linked; without it, the referee can only re-run the computation or take it on faith.

The literature review is thorough and accurate, and the limitations are honestly acknowledged in the remarks on trying to push to 16. This is not a case of overclaiming; the authors know exactly where the method stops.

Who should read it: anyone working on Waring rank, apolarity, or the complexity of the determinant. The method is likely to be useful beyond this instance.

My recommendation: send it to peer review. The central argument is sound, the bound is a genuine improvement, and the computational gap is fixable with the code attached. I'd want the scripts before accepting, but this is exactly the kind of paper a good referee should see.

Referee Report

3 major / 4 minor

Summary. The paper proves that the Waring rank of the 3x3 determinant is at least 15, improving the previously known lower bound of 14. The method is based on syzygies of apolar ideals, which the authors introduce for this purpose. They show that any one-dimensional saturated homogeneous ideal of degree 13 containing no linear form has Betti number β5,6 ≥ 140 (Proposition 6), while the apolar ideal of det3 satisfies β5,6 = 100 and for every linear form ℓ the apolar ideal of det3 − ℓ^3 satisfies β5,6 < 140 (Propositions 13 and 15). This rules out a 13-term decomposition of det3 − ℓ^3, giving rk(det3 − ℓ^3) ≥ 14 for all ℓ and hence rk(det3) ≥ 15. The same argument proves the symmetric cactus rank of the 3x3 permanent is at least 14. The paper also gives a new proof of the known bound rk(det3) ≥ 14 and reviews prior bounds.

Significance. If the computational steps are correct, this is a meaningful advance on a longstanding problem: it is the first improvement over 14 for the Waring rank of det3. The syzygy-of-apolar-ideal method is a novel tool in this area and is likely to be applicable to other forms. The paper is clearly written and gives a careful reduction of the linear-form perturbation to three normal forms via the action of SL3 × SL3. Proposition 6 uses Peeva's consecutive cancellation theorem and a neat h-vector enumeration, and the authors include Macaulay2 code for the lex-segment computations in that proposition. However, the decisive Betti table in Proposition 15 is not accompanied by a reproducible script, and Proposition 14 contains an unexpanded 'direct computation.' These gaps prevent the reader from fully verifying the main theorem from the manuscript alone.

major comments (3)

[§3.4, Proposition 15] The central inequality β5,6(T/F^⊥) < 140 is established by a Macaulay2 computation of the Betti table of the universal ideal H~ in T~ = T[µ,λ], reporting β5,6 = 135. No code, log, or certificate is provided, so this decisive numerical assertion is not independently checkable from the manuscript. The authors should include the Macaulay2 script (or a certified free resolution) used to obtain the table, and ideally an independent verification (for example via a different algorithm or a Betti number certificate). Without this, the proof of Theorem 17 is incomplete.
[§3.4, Proposition 14] The proof that H = F^⊥ depends on the statement: 'One computes directly that the first 35 generators of H generate a codimension 2 space of cubics, while (F^⊥)_3 has codimension 1.' This is a load-bearing assertion: it is what allows the conclusion that H agrees with F^⊥ in degree 3 and hence that the listed quadrics generate F^⊥. The computation is not shown, and it is not a trivial consequence of the listed generators. Please provide the actual computation, for example a basis of (F^⊥)_3 and the dimension of the span of the products T_1 · (span of the first 35 quadrics), or a reproducible code snippet.
[§3.4, Proposition 15] The flatness/specialization argument is too terse. The sentence 'We can choose a constant monomial cobasis for the family of ideals F(µ,λ)^⊥ ... so that the family is locally free' requires justification: one must show that a single set of monomials maps to a basis of every fiber algebra T/F(µ,λ)^⊥, and that this yields a locally free sheaf over the chosen base Spec k[µ±1,λ]. While the constant Hilbert function (1,9,9,1) is a standard starting point, the authors should spell out the argument or cite a precise theorem for why the family is flat, and why exactness of the specialized complex follows. This step is essential for transferring the computed β5,6 = 135 to arbitrary µ,λ with µ ≠ 0.

minor comments (4)

[§3.3, proof of Proposition 12] The sentence 'Multiply the ith row of A by a^{-1}_{i,i} and the dth row of A by a_{i,i}, for i from 1 to k or d-1, whichever is less' is unclear and likely contains a typo; please rephrase the normalization step.
[§3.1, Betti tables] The Betti tables would be easier to read if the convention (β_{i,j} appears in column i, row j−i) were stated explicitly before the first table.
[Abstract and §2.2] The abstract uses 'symmetric cactus rank' while the body defines 'cactus rank' and then states that all ranks are symmetric; please unify the terminology.
[§3.4, Proposition 13] The two equalities β5,6 = 100 in Proposition 13 are stated as 'a direct computation in Macaulay2' without code; please include the script for these computations as well, or at least specify the exact input.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rank bound is derived from independent Betti-number computations, not from the target inequality or from a fitted input.

full rationale

The paper's central claim, rk(det3) >= 15, is proved by a chain of independent algebraic steps: Proposition 6 gives a lower bound beta_{5,6} >= 140 for degree-13 ideals via Peeva's consecutive-cancellation theorem and explicit lex-segment Betti tables; Theorem 4 and the Macaulay2 computations give beta_{5,6}(T/det3^perp)=100 and beta_{5,6}(T/per3^perp)=116; Proposition 14 identifies the apolar ideal of F = mu det3 - lambda(x1+x5+x9)^3; Proposition 15 computes the Betti table of a universal ideal H~ in Macaulay2 and uses flatness to conclude beta_{5,6}(T/F^perp) <= 135 < 140; Theorem 17 then combines these bounds with the subtraction trick to prove rk(det3) >= 15. At no point is the target bound rk(det3) >= 15 used as an assumption, nor is any parameter fitted to the quantity being predicted. The numerical values 100, 116, 135, and 140 are computed or derived from stated commutative-algebra theorems, not imposed to force the conclusion. The paper does cite prior work coauthored by Teitler, notably [13] for indecomposability of det_d and per_d and [13, Proposition 1.6] for excluding a degree-4 minimal generator, and [22, 39] for earlier lower bounds. These are published, independently stated results used as building blocks; they do not define the new rank bound and are not invoked as a substitute for the present derivation. The Macaulay2 computations in Propositions 13, 14, and 15 are not accompanied by code or certificates, and the flatness argument in Proposition 15 relies on the constant Hilbert function (1,9,9,1); these are legitimate reproducibility and correctness concerns, but they are not circularity. The derivation does not reduce to its own inputs by construction, so no circular step is present.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The argument imports standard theorems (apolarity lemma, Peeva's cancellation theorem, Macaulay bounds, Shafiei's generator description, Gorenstein symmetry, flatness criteria) and adds unscripted Macaulay2 computations. There are no fitted constants and no invented algebras beyond the usual apolar construction; the polynomial families F(µ,λ) are normalizations of det3 minus a cube, not new entities.

assumptions (8)

standard math Apolarity Lemma (Lemma 1): Waring rank equals minimal degree of a reduced zero-dimensional apolar scheme.
Used throughout Section 3 to translate rank questions into ideal containment and Betti numbers; cited to [5,49,34].
standard math Peeva's consecutive cancellations theorem: Betti numbers of T/I are obtained from the lex-segment ideal with the same h-vector by consecutive cancellations.
Core of Proposition 6, which establishes β5,6 >= 140 for degree 13 ideals; cited to [41].
standard math Macaulay h-vector bounds: after the h-vector entry drops to at most the degree, the entries are nonincreasing.
Enumerates the five possible h-vectors of degree 13 ideals in Prop 6; cited to [9, Theorem 4.2.10].
domain assumption Shafiei's theorem giving explicit generators of det⊥d and per⊥d.
Used as a black box to know the apolar ideals and their Hilbert functions; from [46, Theorem 4].
domain assumption If G⊥ has a minimal generator of degree deg(G)+1 then G is a power of a linear form.
Used in Proposition 14 to rule out degree-4 generators of F⊥; cited to [13, Proposition 1.6].
standard math Gorenstein symmetry of apolar algebras gives Hilbert function (1,9,9,1) for a concise cubic in 9 variables.
Used in Prop 14 to identify the 36 quadrics as the entire degree-2 part of F⊥.
domain assumption Correctness of Macaulay2 computations of graded Betti numbers and free resolutions.
Unscripted computer outputs in Props 13 and 15 determine the decisive inequality β5,6 < 140; this is a verification burden.
standard math Flatness criterion: constant Hilbert function on the base implies flatness of the family of apolar algebras, so specialization preserves exactness.
Used in Prop 15 to argue the computed complex specializes to a resolution of each T/F(µ,λ)^⊥; cited to [30, Theorem III.9.9].

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Pith. "Pith review of A bound for the Waring rank of the determinant via syzygies." pith.science (2026). https://pith.science/paper/FU2HTYY6

@misc{pith2026190808896,
  author       = {Pith},
  title        = {Pith review of: A bound for the Waring rank of the determinant via syzygies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FU2HTYY6}},
  note         = {Machine review of arXiv:1908.08896}
}

abstract

We show that the Waring rank of the $3 \times 3$ determinant, previously known to be between $14$ and $18$, is at least $15$. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the cactus rank of the $3 \times 3$ permanent is at least $14$.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alexander and A

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Alessandra Bernardi, Enrico Carlini, Maria Virginia Cat alisano, Alessandro Gimigliano, and Alessan- dro Oneto, The hitchhiker guide to: Secant varieties and tensor decompo sition, arXiv:1812.10267 [math.AG], Dec 2018. 1

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Alessandra Bernardi, Joachim Jelisiejew, Pedro Macias M arques, and Kristian Ranestad, On polynomials with given Hilbert function and applications , Collect. Math. 69 (2018), no. 1, 39–64. 3 16 MATS BOIJ AND ZACH TEITLER

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Mats Boij, Artin level modules , J. Algebra 226 (2000), no. 1, 361–374. 11

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