REVIEW 1 major objections 5 minor 3 references
Polynomials whose nth powers have prescribed multiple-of-nth-degree coefficients
T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper claims that any chosen coefficient list can appear as the multiple-of-n coefficients in the nth power of a degree-m polynomial over a field extension.
desk verdict Clean coprime-case proof and a nice constructive n=2 argument, but the non-coprime reduction rests on a false gcd assertion, so the main theorem is unproved as written—though the gap looks easily repairable. 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 central object is the coefficient morphism $\varphi_{m,n}:\mathbb{A}_K^{m+1}\to\mathbb{A}_K^{m+1}$, together with the induced self-map of projective space $\mathbb{P}^m_K$; because $\varphi_{m,n}$ is homogeneous of degree $n$, scaling the input scales every prescribed coefficient by $\gamma^n$. The load-bearing calculation is the Jacobian matrix of this morphism at the vector $(1,0,\dots,0,1)$. When $\gcd(m,n)=1$, each row and each column of that matrix contains exactly one nonzero entry, namely $n\binom{n-1}{k}$ for a uniquely determined $k$, so the matrix is invertible. Invertibility makes the morphism étale at that point, hence étale on a Zariski-open neighbourhood; dominance follows by dimension, and the theorem that projective morphisms are closed then forces surjectivity.
What would settle it
Compute the image of the map $\varphi_{3,6}$ sending $(\alpha_0,\alpha_1,\alpha_2,\alpha_3)$ to the degrees $0,6,12,18$ coefficients of $(\alpha_0+\alpha_1x+\alpha_2x^2+\alpha_3x^3)^6$ over $\mathbb{C}$; since $\gcd(3,6)=3$ and $\gcd(4,6)=2$, the paper's reduction does not apply, so finding one quadruple not in the image would disprove Theorem 1 for this pair, while a witness for every quadruple would show the theorem holds despite the gap.
Extended reading notes
Core claim
The central claim is that the coefficient map $\varphi_{m,n}(\alpha_0,\dots,\alpha_m)$ returning the degrees $0,n,2n,\dots,mn$ coefficients of $(\alpha_0+\alpha_1x+\cdots+\alpha_m x^m)^n$ is surjective on $(m+1)$-tuples over any algebraically closed field satisfying the characteristic hypothesis. For $\gcd(m,n)=1$, the Jacobian at $(1,0,\dots,0,1)$ has exactly one nonzero entry in each row and column, so it is a scaled permutation matrix and the map is étale there; étaleness on an open set gives dominance, and closedness of morphisms from projective space upgrades dominance to surjectivity. The intended reduction for the non-coprime case invokes surjectivity of $\varphi_{m+1,n}$ to hit $(a_0,\dots,a_m,0)$, then argues the resulting polynomial cannot really have degree $m+1$. The theorem therefore says that arbitrary coefficient data at multiples of $n$ are realizable, and for $n=2$ the proof gives the polynomials explicitly by factoring a binomial polynomial into two conjugate factors and reassembling their coefficients with chosen signs.
Load-bearing premise
The non-coprime case rests on the assertion that whenever $\gcd(m,n)>1$ we must have $\gcd(m+1,n)=1$; the pair $m=3,n=6$ contradicts this, so the reduction in Corollary 5 does not go through.
Editorial extensions
If this is right
- For $n=2$, the proof is constructive: it factors $a_0+a_1x^2+\cdots+x^{2m}$ into two polynomials whose roots are opposite square roots of the same targets, then re-signs coefficients to build $f$; when the target polynomial has distinct roots and $K$ has characteristic not 2, this produces $2^{m+1}$ distinct polynomials $f$.
- The characteristic condition is exactly what the Frobenius reduction needs: for $n=p^s n'$ with $p>n'$, the theorem for $n'$ implies the theorem for $n$ by taking $p^s$-th roots of all prescribed coefficients.
- For coprime $m,n$, the coefficient map is étale over a dense open set, so the map has finite fibers generically; the paper suspects the number of preimages of a general target is exactly $n^{m+1}$.
- The result supplies the coefficient flexibility needed to prescribe the non-leading data of defining polynomials $y^p=f(x)$ in the construction of semistable models of superelliptic curves over mixed characteristic $(0,p)$.
Reading between the lines
- Beyond the paper, the non-coprime reduction fails for $(m,n)=(3,6)$, since $\gcd(3,6)=3$ while $\gcd(4,6)=2$; checking whether $\varphi_{3,6}$ is surjective would show whether the theorem itself survives this gap.
- Beyond the paper, the conjectural degree $n^{m+1}$ of $\varphi_{m,n}$ connects to the $n=2$ case: counting sign choices there gives $2^{m+1}$ solutions exactly when the input polynomial is squarefree, matching the conjectured count for generic inputs.
- Beyond the paper, the $n=2$ construction can be turned into an algorithm that works directly from the target coefficients by solving for roots and then choosing square roots, which may extend to other even exponents even where the general proof's reduction does not apply.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims (Theorem 1) that for any field K satisfying the stated characteristic hypothesis, any finite list a_0,...,a_m in K can be realized as the degree-jn coefficients of the nth power of a degree-m polynomial over a finite algebraic extension of K. The proof encodes the problem as surjectivity of the coefficient map φ_{m,n}, proves surjectivity when gcd(m,n)=1 via an invertible-Jacobian argument (Proposition 3 and Corollary 4), and then attempts to reduce the non-coprime case to the coprime case by replacing m with m+1 (Corollary 5). An independent constructive proof is given for n=2. As written, the proof of the main theorem is incomplete because the reduction in Corollary 5 rests on a false coprimality assertion.
Significance. If the gap in Corollary 5 is repaired, the theorem is a clean and useful result with applications to the author's work on semistable models of superelliptic curves. The algebraic-geometric approach is elegant, and the n=2 proof explicitly constructs 2^{m+1} solutions under genericity assumptions. The paper contains no fitted parameters or circular derivations; the cited standard references and the self-citation [1] are used appropriately. However, the central claim is not fully established for all non-coprime pairs in the submitted version.
major comments (1)
- [Corollary 5] The assertion "In that case, we certainly have gcd(m+1,n)=1" is false. For example, with m=3 and n=6 we have gcd(3,6)=3>1 but gcd(4,6)=2, so Corollary 4 cannot be applied to φ_{4,6}. Since the proof of Theorem 1 for non-coprime pairs depends entirely on this reduction, the theorem is not proved as written for pairs such as (m,n)=(3,6). The preceding reduction to n'<p in the characteristic-p case does not remove this obstruction: for p=0, the non-coprime case remains, and for p>0 one may still have gcd(m,n')>1. A repair is available: choose any integer M>m with gcd(M,n)=1, apply Corollary 4 to φ_{M,n} with target (a_0,...,a_m,0,...,0), and then force α_{m+1}=...=α_M=0 by iteratively using the vanishing of the degree-(jn) coefficients for j=m+1,...,M. This repair is not present in the manuscript, so the submitted proof is incomplete.
minor comments (5)
- [Abstract] The abstract reads "whose nth power whose degree-jn coefficient is equal to a_j"; this should be "whose nth power has degree-jn coefficient equal to a_j."
- [Paragraph after Eq. (1)] In the characteristic-p reduction, "the coefficient of degree-(jn'p^s=jn) of (f^{n'})^{pm}=f^n" should read "(f^{n'})^{p^s}=f^n"; the exponent "pm" appears to be a typo for "p^s."
- [Lemma 2] In items (ii) and (iii) of Lemma 2, the notation P^{m+1}_K should be P^m_K, since φ_{m,n} induces a morphism P^m_K → P^m_K.
- [Corollary 5] The displayed definition "f(x) := α_0x + ... + α_{m+1}x^{m+1}" omits the constant term; it should be "f(x) := α_0 + α_1x + ... + α_{m+1}x^{m+1}."
- [Remark 6] The sentence "if gcd(m,n)>1, then the Jacobian matrix is not invertible ... This implies ... φ_{m,n} does not ramify at (1,0,...,0,1)" appears to have the polarity reversed: non-invertibility of the Jacobian at a smooth point means the morphism ramifies (or at least is not étale) there.
Circularity Check
No significant circularity; the derivation is self-contained and the only self-citation is an independent, fully reproduced alternative proof.
full rationale
The paper's central argument defines a polynomial coefficient map φ_{m,n} and proves surjectivity via a Jacobian calculation (Proposition 3), étaleness and dominance (Corollary 4), and a dimension argument. None of these steps fits a parameter to the target coefficients or defines an object in terms of the conclusion; the target coefficients a_j enter only as the image point whose preimage is sought. The characteristic-p reduction via Frobenius is a standard descent and does not presuppose the target statement except for the smaller exponent n', which is later proved independently by the same geometric argument. The only self-citation is [1], used for the n=2 alternate proof; that proof is reproduced in full in the text, is explicitly described as 'completely independent', and is not used to establish Theorem 1 for general n. The external references [2] and [3] supply standard algebro-geometric facts (étale morphisms, closed images of projective morphisms), not the theorem itself. I therefore find no equation that reduces to itself, no fitted input renamed as a prediction, and no load-bearing self-citation chain. I note separately a correctness gap, not a circularity: Corollary 5 asserts 'In that case, we certainly have gcd(m+1,n)=1' after assuming gcd(m,n)>1, which is false (for example, m=3 and n=6 give gcd(3,6)=3 but gcd(4,6)=2), so the non-coprime case of Theorem 1 is unproved as written; this is a logical flaw in the reduction, not a circular derivation.
Assumptions & free parameters
assumptions (3)
- standard math The map φ_{m,n} is a morphism and morphisms from projective spaces have closed image (Shafarevich [3]).
- domain assumption Passing to an algebraic closure of K is permitted when seeking existence over a finite extension.
- ad hoc to paper If gcd(m,n)>1 then gcd(m+1,n)=1.
Cite this review
Pith. "Pith review of Polynomials whose nth powers have prescribed multiple-of-nth-degree coefficients." pith.science (2026). https://pith.science/paper/QQDHAJTH
@misc{pith2026250524013,
author = {Pith},
title = {Pith review of: Polynomials whose nth powers have prescribed multiple-of-nth-degree coefficients},
year = {2026},
howpublished = {\url{https://pith.science/paper/QQDHAJTH}},
note = {Machine review of arXiv:2505.24013}
}
abstract
We show under a mild hypothesis that given field elements $a_0, \dots, a_m \in K$, there always exists a degree-$m$ polynomial whose $n$th power whose degree-$jn$ coefficient is equal to $a_j$ for $0 \leq j \leq m$. We provide an alternate proof for the $n = 2$ case which is more constructive.
Reference graph
Works this paper leans on
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[1]
Clusters and semistable models of hyperelliptic curves in the wild case
Leonardo Fiore and Jeffrey Yelton. Clusters and semistable models of hyperelliptic curves in the wild case. arXiv preprint arXiv:2207.12490v4, 2023
work page Pith review arXiv 2023
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[2]
J. S. Milne. Algebraic geometry. Online lecture notes (v6.10), 2024
work page 2024
-
[3]
Basic algebraic geometry: Varieties in projective space
Igor Shafarevich. Basic algebraic geometry: Varieties in projective space. Springer, 1994
work page 1994
Reviewed August 7, 2026 · model on record in the stance chip above.
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