REVIEW 3 major objections 5 minor 2 cited by
A matrix criterion and algorithmic approach for the Peterson hit problem: Part I
T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read A polynomial is 'hit' by Steenrod squares exactly when a certain linear system over the two-element field has a solution, reducing the hit problem to computing a matrix rank.
desk verdict A correct but largely tautological reformulation of the hit problem, carrying an unproven filter that may undermine the two headline numbers. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the matrix $M$, built from the images of all admissible Steenrod square operations on monomials of lower degree; the column space of $M$ is exactly the hit subspace. To keep $M$ finite and sparse, the algorithm uses only operators $\mathrm{Sq}^{2^j}$ whose source degree $d'$ passes the binary-addition test $\alpha(d' + k) \le k$, a filter rooted in the classical result that all polynomials in a degree with $\alpha(d + k) > k$ are hit. The paper represents $M$ sparsely and computes its rank over $\mathbb{F}_2$, so $\dim(Q\mathcal{P}_k)_d$ is obtained as a binomial coefficient minus a matrix rank.
What would settle it
Run the published algorithm, or an independent equivalent, at five variables and degree 64 and compare the computed rank with 812,695; the correction of 1694 to 1690 stands only if the rank is exactly 812,695. At smaller scale, checking that the degree-14 example polynomial is declared hit and that the output reproduces the input would confirm the matrix criterion in a directly checkable instance.
Extended reading notes
Core claim
The central claim is Theorem 2.1: with basis $B_d$ of monomials of degree $d$ and coordinate map $[\,\cdot\,]$, a polynomial $f$ is hit if and only if the linear system $Mc = [f]$ has a solution over $\mathbb{F}_2$, where each column of $M$ is the coordinate vector of $\mathrm{Sq}^{2^j}(m)$ for a monomial $m$ of degree $d - 2^j$. Because the column space of $M$ equals the hit subspace, $\dim((\mathcal{A}^+ \mathcal{P}_k)_d) = \operatorname{rank}(M)$, and $\dim(Q\mathcal{P}_k)_d = \binom{d+k-1}{k-1} - \operatorname{rank}(M)$. The author implements this by retaining only generators whose source degree $d'$ satisfies $\alpha(d' + k) \le k$, storing $M$ as a sparse matrix, and computing ranks over $\mathbb{F}_2$. The implementation yields the specific five-variable dimensions quoted above, correcting one published value and confirming another within its estimated range.
Load-bearing premise
The load-bearing premise is that the filtered list of Steenrod-square images used to build the matrix actually spans the whole space of hit polynomials, and that the computer implementation computes the matrix rank correctly.
Editorial extensions
If this is right
- Any future claim about the dimension of the hit space for $k$ variables at degree $d$ can be checked by a single matrix rank computation, turning the Peterson hit problem into a computational verification task rather than a purely manual one.
- The corrected value $\dim(Q\mathcal{P}_5)_{64} = 1690$ replaces the previously published $1694$, and $\dim(Q\mathcal{P}_5)_{128} = 1985$ pinpoints an exact value inside the earlier estimate $1984$–$1990$.
- Known hand-computed dimensions for five variables in generic degrees, such as $432$ at degree $15$ and $1117$ at degree $35$, are reproduced by the algorithm, giving those results independent computational confirmation.
- Within memory and time limits, the same algorithm applies to any number of variables, opening a systematic route to hit-problem data for $k \ge 6$.
Reading between the lines
- If the rank-based criterion is correct, the bottleneck shifts from mathematical impossibility to computational feasibility: any claimed hit-problem dimension could in principle be certified by exhibiting the matrix and its rank, making manual errors detectable by re-computation.
- The same matrix-criterion framework could be transferred to related modules over the Steenrod algebra, such as truncated polynomial algebras or invariant subrings, whenever a finite generating set for the image of the augmentation ideal is available.
- If the corrected value at degree 64 is confirmed by an independent implementation, the discrepancy with the earlier manual computation signals that other published dimensions obtained by hand in the same lineage may deserve re-checking.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents a linear algebra criterion for the Peterson hit problem: a homogeneous polynomial f in P_k^d is hit if and only if its coordinate vector [f] lies in the column space of a matrix M whose columns are coordinate vectors of Steenrod-square images of monomials of degree d. From this it derives a rank formula for the dimension of the hit space and, via Corollary 2.5, a formula for dim(QP_k)_d. The authors provide two SageMath implementations, one for testing whether a given polynomial is hit and one for computing hit-space dimensions, and they use the latter to report dim(QP_5)_64 = 1690 (correcting a value 1694 in Sum and Tai) and dim(QP_5)_128 = 1985 (within Sum and Tai's estimated range 1984 to 1990). The algorithms restrict attention to operations Sq^{2^j} and apply the filter alpha(d' + k) <= k on source degrees.
Significance. If the reported computations are correct, the paper provides a useful computational tool for a long-standing problem and produces falsifiable predictions about dimensions of QP_k. The paper also reproduces several previously known values and gives explicit hit decompositions, which is a concrete service to the community. However, the central theoretical statement is essentially a restatement of the definition of the hit subspace, and the correctness of the algorithms depends on two unproved spanning claims: that single operations Sq^{2^j} applied to monomials span the hit space, and that the alpha filter does not discard essential generators. The new values 1690 and 1985 are not independently verified, and no execution logs or archived code are provided. The contribution is therefore best understood as a potentially useful computational method whose correctness and headline numerical results are not yet established.
major comments (3)
- [Section 2, Theorem 2.3] Theorem 2.3 asserts, without proof, that the set H_d(P_k) = {Sq^{2^j}(m) | j >= 0, m a monomial of degree d - 2^j} is a spanning set for the hit subspace (A^+ P_k)_d. The Steenrod algebra is generated as an algebra by the operations Sq^{2^j}, but this does not imply that the image of the augmentation ideal A^+ on P_k is spanned by single operations Sq^{2^j} applied to monomials; an element of A^+ is a linear combination of composites of such operations, not necessarily of single squares. Since every dimension computed by the algorithms is rank(M) for a matrix whose columns come from this set, this unproved spanning assertion is load-bearing for all numerical results in Section 3.
- [Section 3.1 and 3.2, Algorithm 1 Step 7(b) and Algorithm 2 Step 7(b)] The algorithms discard all generator tasks for which alpha(d' + k) > k, where d' = d - 2^j is the source degree. No lemma or theorem in the paper justifies this filter. Wood's theorem shows that the source-degree space P^{d'}_k is entirely hit in those degrees, but it does not show that the images Sq^{2^j}(P^{d'}_k) in the target degree d are spanned by the remaining columns; removing columns can only shrink the column space. Consequently the rank computed from the filtered set may underestimate dim((A^+P_k)_d), which would make every value of dim(QP_k)_d reported via Corollary 2.5 an overestimate. This directly affects the headline correction dim(QP_5)_64 = 1690 and the new value dim(QP_5)_128 = 1985.
- [Section 3.2, Tables 3 and 4, and Remark 3.4] The two new numerical claims, dim(QP_5)_64 = 1690 and dim(QP_5)_128 = 1985, are not independently confirmed. The paper provides no execution logs, no checksums, no archived implementation, and no independent implementation in another system; the code listings in the paper are the only evidence for these numbers. Agreement with known small-degree values tests the pipeline at much smaller matrix sizes and does not validate the large sparse rank computations that produce the new values. Given the unproved spanning and filtering claims, these two numbers are not yet verified results.
minor comments (5)
- [Table 4, row s = 3] The displayed formula for dim(QP_5)_8 incorrectly uses 35480 where the corresponding Table 3 entry is 321; the intended computation is binom(12,4) - 321 = 174, but the printed arithmetic is wrong.
- [Corollary 2.4] The statement that there are 'infinitely many' distinct coefficient vectors is incorrect: the coefficient space is finite-dimensional over F2, so if the null space has dimension r then there are exactly 2^r solutions. The conclusion that representations are generally non-unique remains valid.
- [Introduction and Abstract] The abstract's 'degree 2^7' and the introduction's 'degree 27' are inconsistent notation; both should be typeset as 2^7 to avoid confusion.
- [Section 3.3] The text refers to 'LISTING 2.3' where Listing 2 is meant; the numbering in this section does not match the actual listing labels.
- [Section 5] The claim that Part II, together with Part I, 'has substantially achieved a complete solution to the Peterson hit problem' is not supported by anything proved in this manuscript; the algorithmic reduction to rank computation does not by itself constitute a complete solution, and the Section 5 material is not accompanied by complete algorithms or proofs.
Circularity Check
The advertised matrix criterion restates the definition of the hit subspace; the numerical dimensions are genuine rank computations and are not fitted, so the circularity is partial.
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self definitional
[Theorem 2.1, Section 2; the same structure recurs in Theorem 2.3]
"Let H_d(P_k) = {Sq^i(g_j)| i>0, and g_j is a monomial of degree d−i} be a generating set for the subspace of hit polynomials (A^+P_k)_d. Construct a matrix M whose columns are the coordinate vectors [h] for each generator h∈ H_d(P_k) ... a polynomial f∈ P^d_k is a hit polynomial if and only if the linear system of equations Mc = [f] has a solution."
Two paragraphs earlier, the paper defines a polynomial as hit exactly when it lies in the span of the elements Sq^i(g) for i>0, i.e. in the span of precisely the set H_d(P_k). The theorem then asserts that membership in that span is equivalent to solvability of a linear system whose columns are coordinate vectors of that same set. Because the coordinate map is a vector-space isomorphism, this is true by construction and carries no independent Steenrod-algebra content; the proof only applies the isomorphism and the definition of span. Theorem 2.3 repeats the pattern: H_d is declared to be a spanning set, and rank(M)=dim(A^+P_k)_d follows directly.
full rationale
The central advertised criterion, Theorem 2.1, is definitional: 'hit' was defined as lying in the span of the Steenrod squares of lower-degree polynomials, and the matrix M is built from exactly those generators. The criterion is a valid algorithmic encoding of the definition, but it is not a derived characterization. The reported dimensions, including dim(QP5)_64 = 1690 and dim(QP5)_128 = 1985, are not fitted to Sum and Tai's estimates; they come from explicit rank computations and are checked against several independent manual results, so they have real computational content. The implementation's alpha filter is a separate concern: no proof is given that the filtered set H'_d(P_k) still spans the hit space, so the computed rank could in principle underestimate the true hit-space dimension. That is a correctness gap or unproved assumption, but it is not a circular reduction. The self-citations in the paper are used only as corroboration of outputs, not as the load-bearing justification of the linear-algebra criterion. Overall, the circularity is partial and localized to the 'criterion' being a restatement of the definition, hence a score of 6.
Assumptions & free parameters
free parameters (2)
- PARALLEL_TASK_THRESHOLD (Listing 1) =
10000
- PARALLEL_TASK_THRESHOLD (Listing 2) =
5000
assumptions (3)
- domain assumption Wood's theorem: if alpha(d + k) > k, then every polynomial of degree d is hit.
- standard math The Steenrod algebra is generated as an F2-algebra by the squares Sq^{2^j}, so every Sq^i is a sum of composites of these squares.
- standard math Unstable condition: Sq^k(f) = 0 when k > deg(f).
Cite this review
Pith. "Pith review of A matrix criterion and algorithmic approach for the Peterson hit problem: Part I." pith.science (2026). https://pith.science/paper/G73LAJWV
@misc{pith2026250618392,
author = {Pith},
title = {Pith review of: A matrix criterion and algorithmic approach for the Peterson hit problem: Part I},
year = {2026},
howpublished = {\url{https://pith.science/paper/G73LAJWV}},
note = {Machine review of arXiv:2506.18392}
}
abstract
The Peterson hit problem in algebraic topology is to explicitly determine the dimension of the quotient space $Q\mathcal P_k = \mathbb F_2\otimes_{\mathcal A}\mathcal P_k$ in positive degrees, where $\mathcal{P}_k$ denotes the polynomial algebra in $k$ variables over the field $\mathbb{F}_2$, considered as an unstable module over the Steenrod algebra $\mathcal{A}$. Current approaches to this problem still rely heavily on manual computations, which are highly prone to errors due to the intricate nature of the underlying calculations. To date, no efficient algorithm implemented in any computer algebra system has been made publicly available to tackle this problem in a systematic manner. Motivated by the above, in this work, which is considered as Part I of our project, we first establish a criterion based entirely on linear algebra for determining whether a given homogeneous polynomial is "hit". Accordingly, we describe the dimensions of the hit spaces. This leads to a practical and reliable computational method for determining the dimension of $Q\mathcal{P}_k$ for arbitrary $k$ and any positive degrees, with the support of a computer algebra system. We then give a concrete implementation of the obtained results as novel algorithms in \textsc{SageMath}. As an application, our algorithm demonstrates that the manually computed result presented in the recent work of Sum and Tai [15] for the dimension of $Q\mathcal{P}_5$ in degree $2^{6}$ is not correct. Furthermore, our algorithm determines that $\dim(Q\mathcal{P}_5)_{2^{7}} = 1985,$ which falls within the range $1984 \leq \dim(Q\mathcal{P}_5)_{2^{7}} \leq 1990$ as estimated in [15].
Forward citations
Cited by 2 Pith papers
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At rank 6 and degree 36 the source of Singer's algebraic transfer is 2-dimensional while the target is 1-dimensional, so the transfer cannot be injective and Singer's conjecture is false.
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Computational Approaches to the Singer Transfer: Preimages in the Lambda Algebra and $G_k$-Invariant Theory
A lambda-algebra linear-algebra algorithm gives explicit Singer-transfer preimages for c0, d0, p0, a claimed refutation of a published d0 proof, and an automated G4-invariant basis tool.
Reference graph
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Reviewed August 15, 2026 · model on record in the stance chip above.
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