REVIEW 2 major objections 5 minor 8 references
A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero Conjecture
T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The paper proves that a prime is bad for the Casas-Alvero conjecture in degree $n$ exactly when it divides one of finitely many explicitly defined integers.
desk verdict A useful finite description of bad primes as divisors of gcds of minors, but the bound's proof has a false coefficient count and the main theorem leans entirely on Ghosh's unpublished equivalence. 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 mechanism is the matrix $M_T$, one for each $T\in\{1,\ldots,n\}^{n-1}$, together with the integer $J_T$ defined as the gcd of its maximal minors. The rows of $M_T$ are all products $G_{T,i}x^\alpha$ with $|\alpha|=d-i$, where $d=(n^2-3n+4)/2$ and $G_{T,i}$ is the image of the $i$-th elementary symmetric function under the involution $\Phi_{j_i}$. Macaulay's theorem turns the radical-ideal condition of the regular-sequence reformulation into the statement that the degree-$d$ monomials lie in the ideal generated by the $G_{T,i}$, which is equivalent to $M_T$ having full column rank. The maximal minors are integers, and their gcd $J_T$ therefore records, in one number per $T$, exactly which primes make the rank drop.
What would settle it
Take a small degree such as $n=5$ or $n=7$, where the Casas-Alvero conjecture and the bad-prime list are already known; compute $J_T$ for every $T$ by exact integer linear algebra. If some listed bad prime divides no $J_T$, Theorem 3.1 is false; if some listed bad prime reaches or exceeds the bound in (6), Corollary 3.2 is false.
Extended reading notes
Core claim
The central claim is Theorem 3.1. For each $T=(j_1,\ldots,j_{n-1})$ in $\{1,\ldots,n\}^{n-1}$, let $G_{T,i}$ be the $i$-th elementary symmetric polynomial in $n-1$ variables after the coordinate change $\Phi_{j_i}$; form the matrix $M_T$ whose rows are all products $G_{T,i}x^\alpha$ of total degree $d=(n^2-3n+4)/2$ and whose columns are the monomials of that degree. Let $C=\binom{\frac{n^2-n}{2}}{n-2}$ and let $J_T$ be the gcd of all $C\times C$ minors of $M_T$. Then a prime $p$ is bad for degree $n$ if and only if $p\mid J_T$ for some $T$, equivalently $p\mid \operatorname{lcm}_T J_T$. The proof passes through a theorem of Macaulay: the ideal generated by the $G_{T,i}$ contains all monomials of degree $d$ exactly when $M_T$ has full column rank, and the regular-sequence reformulation of the Casas-Alvero conjecture identifies a rank drop with failure of the conjecture. The corollary, assuming $CA_{n,0}$ holds, puts every bad prime below $C!\prod_{i=1}^{n-1}\binom{i+n-2}{n-2}^{\binom{d-i+n-2}{n-2}}$.
Load-bearing premise
The load-bearing premises are that the previously established equivalence between the conjecture and a regular-sequence condition is correct, and that the combinatorial count used in the determinant bound is accurate; if either fails, the characterization of bad primes or the upper bound is unsupported.
Editorial extensions
If this is right
- For each fixed degree $n$, the set of bad primes is finite and computable: factor the finitely many integers $J_T$ (equivalently, their lcm).
- If $CA_{n,0}$ holds, every bad prime is smaller than the closed-form bound in Corollary 3.2, so the exceptions for that degree form an explicitly bounded finite list.
- Combined with the known propagation result, a good prime for degree $n$ settles all degrees $np^\ell$; the remaining work in the conjecture is concentrated in finite base-case checks.
- The criterion gives a direct modular test: reduce $M_T$ modulo a candidate prime $p$ and check whether its rank drops below $C$.
Reading between the lines
- A natural extension, not pursued in the paper, is to search for a single prime $p$ that divides none of the $J_T$ for a fixed $n$; such a prime would be good for $n$, and by the known equivalence the conjecture in characteristic zero would follow for that degree without a large symbolic computation.
- For moderate degrees, computing all $J_T$ may be more practical than solving polynomial systems; the next open degree mentioned in the paper, $n=20$, is a natural testbed.
- The bound in (6) is very large, and Remark 3.4 sketches a sharper one; comparing the actual prime divisors of the $J_T$ with the bound on degrees where bad-prime lists already exist would show how much room the bound leaves.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper gives a finite, determinantal description of the set of bad primes for a fixed degree n in the Casas-Alvero problem. The main theorem (Theorem 3.1) asserts that p is a bad prime for n if and only if p divides the gcd J_T of the maximal minors of a certain matrix M_T, for some tuple T in {1,...,n}^{n-1}, or equivalently p divides the lcm of the J_T over all T. Under the assumption that the Casas-Alvero conjecture holds in degree n, the paper derives an explicit upper bound for such bad primes (Corollary 3.2). The proofs use Macaulay's theorem to convert a radical ideal condition into a statement about the rank of a matrix, and an imported equivalence of Soham Ghosh (Conjecture 1.6) that connects the Casas-Alvero conjecture to the regularity of the sequences (G_{T,1},...,G_{T,n-1}).
Significance. If the imported equivalence from Ghosh's preprint is correct and if the proof of Lemma 3.3 is repaired, the paper provides a genuinely explicit description of the bad primes and, conditionally on CA_{n,0}, a quantitative upper bound that is computable in any fixed degree. This is a useful tool for the strategy of proving the Casas-Alvero conjecture degree-by-degree. The reduction to maximal minors of an explicit matrix is elegant and potentially practical. However, the significance is currently conditional: the whole framework rests on an unpublished equivalence in [6], and the proof of the upper bound contains a false coefficient count. The paper's contribution is therefore not fully established as written.
major comments (2)
- [§1, Conjecture 1.6, and §3, Theorem 3.1] The main theorem depends entirely on the equivalence imported from [6, Proposition 5.2], stated as Conjecture 1.6. The authors neither prove nor sketch this equivalence, nor do they state precisely which hypotheses (algebraically closed field, characteristic, degree) are needed. Since Theorem 3.1 is obtained by applying Macaulay's theorem to this equivalence, a gap or missing hypothesis in [6] would invalidate the central claim of the paper, not merely the bound. The manuscript should either prove Conjecture 1.6, or state it explicitly as an assumption and present Theorem 3.1 and Corollary 3.2 as conditional on that assumption.
- [§3, Lemma 3.3] The proof of Lemma 3.3 contains a false statement about the coefficient of the monomial x_{j_i}^i in G_{T,i}. Direct expansion of Phi_{j_i}(sigma_i) shows that each of the binom(n-1,i) subsets of size i contributes exactly once to x_{j_i}^i, so the coefficient is binom(n-1,i), not binom(i+n-2,n-2). For n=4 and i=2 the coefficient is 3, while the lemma claims 6. Thus the proof of the determinant bound in Lemma 3.3 is not correct as written. Since Corollary 3.2 depends on this lemma, the upper bound is not established by the manuscript in its current form. The bound may be salvageable because binom(n-1,i) <= binom(i+n-2,n-2), but the proof must be revised and the actual maximal coefficient must be determined correctly.
minor comments (5)
- [§3, Corollary 3.2] The notation C_{n,0} is used in Corollary 3.2, whereas the paper elsewhere uses CA_{n,0}; these should be made consistent.
- [§3, first sentence] There is a typographical artifact 'GT, , . . .' in the sentence introducing the polynomials; it should read G_{T,1}, ..., G_{T,n-1}.
- [§1, Notation] The symbol T is used both for the set {1,...,n}^{n-1} and for an element (j_1,...,j_{n-1}) of that set. This overloading is confusing; using a calligraphic symbol for the set would improve readability.
- [Abstract and §3] The abstract uses the symbol d in the displayed upper bound without defining it; the definition d = (n^2-3n+4)/2 appears only later in Section 3. Please define d in the abstract or avoid its use there.
- [§3, Lemma 3.3] The claim that x_{j_i}^i is 'repeated more than any other monomial' is not supported, and the case j_i = n is problematic because x_n is not among the variables x_1,...,x_{n-1}. The lemma should be restated with a precise statement about the coefficients after grouping like terms.
Circularity Check
No significant circularity: the paper applies external results (Ghosh's equivalence and Macaulay's theorem) to derive a new determinant criterion for bad primes, without fitting, renaming, or self-citation.
full rationale
The central derivation is not circular. Theorem 3.1 combines Ghosh's external Proposition 5.2, which equates the Casas-Alvero conjecture with a regular sequence condition, with Macaulay's theorem, which converts that regularity condition into the maximal-rank condition on the integer matrix M_T. The resulting characterization of bad primes as divisors of the gcd of maximal minors is a genuine reduction, not a restatement of the definition of a bad prime. There is no fitted input called a prediction, no self-citation used as load-bearing evidence, and no ansatz smuggled in under the authors' own prior work. The paper's dependence on Ghosh's unpublished preprint is a correctness and verification risk: if that equivalence has hidden gaps, Theorem 3.1 collapses. The disputed coefficient count in Lemma 3.3 is likewise a proof error, not a circularity, since the claimed bound is used as an estimate rather than being assumed into the conclusion. Under the stated rules, neither an external unproved dependency nor a local mathematical flaw constitutes circular reasoning, so the score is 0.
Assumptions & free parameters
assumptions (3)
- domain assumption Ghosh's equivalence (Conjecture 1.6 iff CA_n) for all fields
- standard math Macaulay's theorem (Theorem 2.1)
- domain assumption Reduction to algebraically closed fields
Cite this review
Pith. "Pith review of A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero Conjecture." pith.science (2026). https://pith.science/paper/4AX2DORV
@misc{pith2026241113967,
author = {Pith},
title = {Pith review of: A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero Conjecture},
year = {2026},
howpublished = {\url{https://pith.science/paper/4AX2DORV}},
note = {Machine review of arXiv:2411.13967}
}
abstract
The Casas--Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree $n$, compile a list of bad primes for that degree (namely, those primes $p$ for which the conjecture fails in degree $n$ and characteristic $p$) and then deduce the conjecture for all degrees of the form $np^\ell$, $\ell\in \mathbb{N}$, where $p$ is a good prime for $n$. In this paper we give an explicit description of the set of bad primes in any given degree $n$. In particular, we show that if the conjecture holds in degree $n$ then the bad primes for $n$ are bounded above by $\binom{\frac{n^2-n}2}{n-2}!\prod\limits_{i=1}^{n-1}\binom{i+n-2}{n-2}^{\binom{d-i+n-2}{n-2}}$.
Reference graph
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Reviewed August 12, 2026 · model on record in the stance chip above.
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