REVIEW 1 major objections 4 minor 6 references
Trinomials with given roots
T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Beyond the trivial case of at most two root-of-unity classes, only boundedly many monic trinomials can vanish at a prescribed finite set of complex numbers, and for algebraic inputs the bounds are explicit enough to make the search…
desk verdict Effective degree bound is solid, but the height bound in Theorem 1.2 is not justified; the gap is real and fixable. 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 argument's engine is reduction of a trinomial vanishing at $\alpha,\beta,\gamma$ to a six-term multiplicative equation. Writing the determinant equation $$\det\begin{pmatrix}\$\alpha$^m&\$\alpha$^n&1\\ \$\beta$^m&\$\beta$^n&1\\ \gamma^m&\gamma^n&1\end{pmatrix}=0$$ produces a solution of $x_1+\cdots+x_6=0$ in the group generated by $\alpha,\beta,\gamma$ and $-1$. The fundamental effective theorem on linear equations in multiplicative groups bounds non-proportional primitive solutions of such equations, and a purely combinatorial lemma (Lemma 3.2) partitions the six indices so that two trinomials of the same 'type' would force proportional primitive solutions; counting types gives $10\kappa(3)^2+15\kappa(4)+\kappa(6)$. For Theorem 1.2, comparison of two determinant-form expressions for the coefficient $A$ yields a lower bound on $|\alpha/\beta|^{m-n}$; a circle-intersection lemma shows three roots of equal modulus force a root-of-unity quotient, and a lower bound for linear forms in logarithms converts the resulting inequalities into explicit degree and height bounds.
What would settle it
Find three nonzero complex numbers $\alpha,\beta,\gamma$ such that none of $\alpha/\beta,\alpha/\gamma,\beta/\gamma$ is a root of unity but infinitely many monic trinomials vanish at all three; this would refute Theorem 1.1. More narrowly, a numerical search could test the circle-intersection lemma directly: any complex trinomial with three distinct roots of equal modulus whose pairwise quotients are not roots of unity would disprove that lemma and block the proof of Theorem 1.2.
Extended reading notes
Core claim
The central discovery is a finiteness-and-effectivity theorem: if $\Omega\subset\mathbb{C}^\times$ splits into at least three equivalence classes, where $\alpha\sim\beta$ iff $\alpha/\beta$ is a root of unity, then the monic trinomials vanishing at $\Omega$ are finite in number, and the count is bounded by $10\kappa(3)^2+15\kappa(4)+\kappa(6)$, with $\kappa(s)$ the (effective) number of non-proportional primitive solutions of $x_1+\cdots+x_s=0$ in a multiplicative group. If $\Omega\subset\overline{\mathbb{Q}}$ generates a number field of degree $d$, then every such trinomial has degree $\le 10^{60}e^{10d^2(h(\Omega)+1)}$ and height $\le 10^{70}e^{10d^2(h(\Omega)+1)}$, so the problem of listing them is decidable. The proof also yields the algebraic-field analogue: for a single element $\alpha$ over a field $K$ with $[K(\alpha^k):K]\ge 3$ for every $k$, only boundedly many $K$-trinomials vanish at $\alpha$, and over a number field the same explicit bounds apply.
Load-bearing premise
The paper's effectiveness is inherited from a quoted deep theorem about solutions of linear equations in multiplicative groups, whose constant is not computed here; if that theorem were only known ineffectively, the degree and height bounds, and hence the decidability conclusion, would not follow.
Editorial extensions
If this is right
- For any finite set of algebraic numbers that falls into at least three root-of-unity classes, all monic trinomials vanishing on the set can be found by finite computation; the theorem gives explicit search bounds.
- Over a number field $K$, for an algebraic $\alpha$ satisfying $[K(\alpha^k):K]\ge 3$ for all $k$, only boundedly many trinomials in $K[X]$ vanish at $\alpha$, and all can be listed.
- The degree and height bounds are uniform in the field degree $d$ and the height $h(\Omega)$, so families of inputs with bounded $d$ and $h(\Omega)$ have trinomials of bounded size.
- The absolute constant in Theorem 1.1 does not depend on the field or the height; only the bound's existence, not its size, is used in the finiteness statement.
Reading between the lines
- The same partition-counting mechanism is likely to work for $\ell$-nomials: if $\Omega$ splits into more than $\ell-1$ equivalence classes, the number of monic $\ell$-nomials vanishing at $\Omega$ should also be finite and effectively bounded, although the paper only states the infinite-family side for this generalization.
- The explicit bounds are far too large for direct numerical search, but the theorem's real algorithmic content is decidability; a practical algorithm would likely bypass the bounds via diophantine approximation techniques.
- Because the proof uses only the ratio heights $\tilde h(\Omega)=\max h(\alpha/\beta)$, inputs whose elements are multiplicatively close have much smaller effective bounds than the stated $h(\Omega)$ form suggests; using a sharper separation inequality mentioned in the paper would tighten the exponents.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies monic trinomials X^m + A X^n + B (B ≠ 0) that vanish on a given finite set Ω ⊂ C^×, and on algebraic sets Ω ⊂ Q̄. Theorem 1.1 shows that if Ω splits into at least three classes modulo roots of unity, then the number of such trinomials is bounded by an absolute effective constant. Theorem 1.2 adds an effective height and degree bound when the elements of Ω generate a number field of degree d, implying decidability. Corollary 1.3 extends this to trinomials over a field K. The proofs reduce the problem to a six-term S-unit equation (Section 3) and apply Matveev's inequality to bound the exponents and heights (Section 4).
Significance. If the height bound is repaired, this is a solid, self-contained contribution that makes a finiteness statement fully effective. The reduction in Section 3 is clean, and Lemma 3.2 is an elegant combinatorial tool. The paper also gives explicit (if very large) constants, uses known deep results (ESS, Matveev) as black boxes without circularity, and honestly places the result relative to earlier work. The main value is in providing a quantitative, effective version of a known finiteness phenomenon.
major comments (1)
- [Section 4.3, Eq. (3.2)] The final step of Theorem 1.2, 'from (3.2) we deduce that h(A), h(B) ≤ 10^{65} e^{10d^2(h(Ω)+1)}', is not justified. Standard height estimates applied to A = -(α^m−β^m)/(α^n−β^n) give h(A) ≤ h(α^m−β^m)+h(α^n−β^n)+O(1) ≤ (m+n)(h(α)+h(β))+O(1) ≤ 2m h(Ω)+O(1). With the proved bound m ≤ 10^{60} e^{10d^2(h(Ω)+1)}, this yields h(A) ≤ 10^{61} h(Ω) e^{10d^2(h(Ω)+1)} (up to additive constants), and similarly for h(B). Since h(Ω) is unbounded, this cannot be absorbed into an absolute constant times e^{10d^2(h(Ω)+1)}. Thus the height bound in Theorem 1.2, and the corresponding height bound in Corollary 1.3, are not established by the given argument. The decidability conclusion is unaffected because any effective height bound suffices; for example, the present estimates combined with h(Ω) ≤ e^{h(Ω)} give h(A), h(B) ≤ 10^{70} e^{11d^2(h(Ω)+1)}. The theorem statement and Corollary 1.3 should be corrected accordingly, or a refined argument for h(A), h(B) must be supplied.
minor comments (4)
- [Section 4, introduction of ˜h] The definition of ˜h(Ω) contains a stray brace: it should read ˜h(Ω) = max{h(α/β) : α, β ∈ Ω}.
- [Corollary 1.3] The displayed exponents '10d^2ν 6' should read '10 d^2 ν^6' (the superscript formatting is lost in the text).
- [Proposition 2.1] The sentence 'Let θ is a complex algebraic number' should read 'Let θ be a complex algebraic number'.
- [Section 3, after Theorem 3.1] The phrase 'Γ in [1,3] corresponds to our Γ s' is confusing; it should say that the group Γ in [1,3] corresponds to our Γ, and that the rank r there corresponds to our r.
Circularity Check
No circularity: all bounds are derived from external effective theorems and height inequalities; no fitted parameter is relabeled as a prediction.
full rationale
The derivation chain is self-contained in the relevant sense. Theorem 1.1 counts suitable pairs (m,n) by bounding the number of primitive solutions of x1 + ... + x6 = 0 using the Evertse-Schlickewei-Schmidt theorem (Theorem 3.1), an external effective result, together with a purely combinatorial Lemma 3.2. No parameter is fitted to data, and no quantity is defined in terms of the claimed conclusion. Theorem 1.2 obtains degree and height bounds by applying Matveev's explicit lower bound for linear forms in logarithms (Theorem 4.1) and standard height inequalities to formulas (3.2) for the coefficients A and B. Although the paper cites prior work, including [2] for the overall strategy and [1,3] for quantitative versions of the Subspace Theorem, these are external results; there are no citations whose authors overlap with the present paper, and no uniqueness theorem from the authors is invoked to force a choice. The final height estimate in Section 4.3 may contain an arithmetic sharpness issue (the extra h(Omega) factor from the coefficient-height comparison), but that would be a correctness or gap issue, not circularity: the bound is still being deduced from the roots and from the already-proved degree bound rather than assumed. Accordingly, no circular step can be exhibited, and the honest finding is score 0.
Assumptions & free parameters
assumptions (3)
- standard math Evertse-Schlickewei-Schmidt theorem: the number of non-proportional primitive solutions of a1x1 + ... + asxs = 0, with all xi in a multiplicative group of finite rank r, is bounded by an effectively computable quantity depending only on r and s (Theorem 3.1).
- standard math Matveev's explicit lower bound for a nonzero linear form in logarithms of algebraic numbers (Theorem 4.1).
- standard math Standard absolute logarithmic height properties and the Liouville inequality (Section 2).
Cite this review
Pith. "Pith review of Trinomials with given roots." pith.science (2026). https://pith.science/paper/Y7HPHTBP
@misc{pith2026190801076,
author = {Pith},
title = {Pith review of: Trinomials with given roots},
year = {2026},
howpublished = {\url{https://pith.science/paper/Y7HPHTBP}},
note = {Machine review of arXiv:1908.01076}
}
read the original abstract
We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these trinomials.
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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