{"id":"afcffe68-5446-49b6-a8ab-198a54c848d7","arxiv_id":"1908.01076","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The number of trinomials vanishing on a finite set of nonzero complex numbers is bounded by an absolute constant unless the set lies in at most two root-of-unity classes, and algebraic inputs come with explicit degree and height bounds.","lead":"Given any finite set of nonzero complex numbers that is not one of the two obvious special shapes, only a bounded number of trinomials can vanish on all of them at once. When the numbers are algebraic, the paper also gives explicit upper bounds on the degree and coefficients of such trinomials, making the search effectively computable.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Height bound in Theorem 1.2 is not derived: the final estimate from (3.2) carries an extra h(Ω) factor that the stated constants cannot absorb.","rationale":"I read the paper in good faith and checked the main structural steps: the reduction to 6-term S-unit equations, Lemma 3.2, the use of Matveev's inequality, and the iterative bounds on m and n. These parts are coherent, and the effectiveness concern raised by the reader about the Evertse–Schlickewei–Schmidt theorem is not the real problem: the cited theorem does give an effective bound on the number of non-proportional primitive solutions. The actual soft spot is the very last displayed estimate. The proof bounds the degree m by e^{10d^2(h(Ω)+1)}, but the height of A and B via (3.2) is at most about 2m h(Ω), which contains an unbounded factor h(Ω) that the claimed bound does not have. The stated height bound in Theorem 1.2 is therefore not established as written. The fix is straightforward—replace the height bound by a slightly larger exponential in h(Ω) or add a factor h(Ω)+1—and the decidability claim survives, so the appropriate verdict is conditional acceptance rather than rejection. The reader's weakest assumption did not identify this issue, so my agreement with the reader is 'disagree'.","tokens_in":8756,"tokens_out":26813,"duration_ms":254022,"concrete_test":"Re-derive the final estimate in §4.3: from (3.2) bound h(A) and h(B) in terms of m, n, and h(Ω), then check whether any inequality in §4.1–4.2 implies h(A) ≤ 10^{65} e^{10d^2(h(Ω)+1)} without an extra h(Ω) factor. A direct calculation gives h(A) ≤ 2m h(Ω)+O(1), so the test is to find the missing sharper estimate. If none exists, amend Theorem 1.2 to h(A), h(B) ≤ 10^{70}(h(Ω)+1)e^{10d^2(h(Ω)+1)} or equivalently adjust the exponent to 11d^2(h(Ω)+1); the main finiteness and decidability argument still works.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 4.3 concludes that from (3.2) one gets h(A), h(B) ≤ 10^{65} e^{10d^2(h(Ω)+1)}. The standard height comparison from (3.2) gives h(A) ≤ (m+n)(h(α)+h(β)) + O(1) ≤ 2m h(Ω)+O(1), and h(B) is bounded similarly. Using the proved bound m ≤ 10^{60} e^{10d^2(h(Ω)+1)} yields h(A) ≤ 10^{61} h(Ω) e^{10d^2(h(Ω)+1)}. The extra factor h(Ω) is unbounded, so no absolute constant 10^{65} or 10^{70} can absorb it. This is not a numerical-constant quibble: because the exponent 10d^2 is fixed and h(Ω) is arbitrary, h(Ω)e^{10d^2 h(Ω)} is not ≤ C e^{10d^2 h(Ω)} for any absolute C. The statement of Theorem 1.2 therefore needs either an additional argument showing cancellations in (3.2) reduce the height of A and B, or a corrected bound such as 10^{70} e^{11d^2(h(Ω)+1)}. The decidability conclusion is unaffected once any effective height bound is supplied.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":8915,"tokens_out":15061,"duration_ms":129891,"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":[{"comment":"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.","section":"Section 4.3, Eq. (3.2)"}],"minor_comments":[{"comment":"The definition of ˜h(Ω) contains a stray brace: it should read ˜h(Ω) = max{h(α/β) : α, β ∈ Ω}.","section":"Section 4, introduction of ˜h"},{"comment":"The displayed exponents '10d^2ν 6' should read '10 d^2 ν^6' (the superscript formatting is lost in the text).","section":"Corollary 1.3"},{"comment":"The sentence 'Let θ is a complex algebraic number' should read 'Let θ be a complex algebraic number'.","section":"Proposition 2.1"},{"comment":"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.","section":"Section 3, after Theorem 3.1"}],"recommendation":"major_revision","confidential_remarks":"The height-bound gap is localized and repairable; the rest of the proof (degree bound, finiteness, reduction to S-unit equations) is sound. I would be willing to accept after the authors correct the statement of Theorem 1.2 and Corollary 1.3, e.g., by replacing the height bound with a bound containing an extra factor or a slightly larger exponent in the exponential."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the effective degree bound in Theorem 1.2 is real and new, and the proof is clean. Second, the height bound in that same theorem is not derived. The stress-test note is correct: the step 'from (3.2) we deduce h(A), h(B) ≤ 10^65 e^{...}' does not follow.\n\nThe concrete problem is in Section 4.3. The degree bound m ≤ 10^60 e^{10d^2(h(Ω)+1)} is fine. But the height of A from (3.2) satisfies h(A) ≤ (m+n)(h(α)+h(β)) + O(1) ≤ 4m h(Ω) + O(1). Substituting the degree bound gives h(A) ≤ 10^61 h(Ω) e^{10d^2 h(Ω)}. For fixed d, h(Ω) is unbounded, so no absolute constant 10^65 or 10^70 can absorb the extra factor. The same applies to h(B). So Theorem 1.2 as stated is false. This is not a minor numerical slip; the exponential form with 10d^2 in the height bound is too strong.\n\nThe fix is straightforward: replace the height bound by something like 10^70 e^{11d^2(h(Ω)+1)}. The decidability conclusion still holds. This should be a required revision.\n\nWhat the paper does well: it is honest that Theorem 1.1 is a combination of known results (Evertse–Győry–Stewart–Tijdeman, ESS, Amoroso–Viada), and the proof it gives is clean. Theorem 1.2's degree bound is the genuine new content. Lemma 4.3 is a nice geometric observation, used effectively. No circularity; deep theorems are imported as black boxes. The effectiveness inherited from ESS is standard and not a concern.\n\nWho should read it: number theorists working on effective Diophantine results for sparse polynomials. The finiteness and degree parts can be trusted. The height bound should not be cited until corrected.\n\nFor peer review: yes, it deserves refereeing, but the referee should insist on fixing the height estimate. In current form I would not accept.","headline":"Effective degree bound is solid, but the height bound in Theorem 1.2 is not justified; the gap is real and fixable.","tokens_in":9579,"tokens_out":9005,"would_cite":false,"duration_ms":79113,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11C08","11J86","11J87"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["trinomials","sparse polynomials","Subspace Theorem","linear forms in logarithms","effective bounds","heights","algebraic numbers","decidability"],"falsifier":"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.","tokens_in":8430,"feed_emoji":"🔢","tokens_out":10828,"duration_ms":97379,"temperature":0.7,"pith_summary":"Fix a finite set $\\Omega$ of nonzero complex numbers. A trinomial here is a polynomial $X^m+AX^n+B$ with $B\\neq 0$. The paper shows that unless $\\Omega$ can be covered by two 'classes' in which every ratio is a root of unity—the only case where infinitely many such trinomials clearly exist—the number of trinomials vanishing at all points of $\\Omega$ is bounded by an absolute constant. When $\\Omega$ consists of algebraic numbers generating a degree-$d$ field, every such trinomial has degree at most $10^{60}e^{10d^2(h(\\Omega)+1)}$ and height at most $10^{70}e^{10d^2(h(\\Omega)+1)}$. Because these bounds are explicit, the finite list of all trinomials vanishing at $\\Omega$ can in principle be determined by computation.","feed_headline":"Three unrelated roots leave room for only finitely many trinomials","feed_subtitle":"When the roots are algebraic, explicit degree and height bounds make listing all such trinomials a finite computation.","key_machinery":"The argument's engine is reduction of a trinomial vanishing at $\\alpha,\\beta,\\gamma$ to a six-term multiplicative equation. Writing the determinant equation\n$$\\det\\begin{pmatrix}\\$\\alpha$^m&\\$\\alpha$^n&1\\\\ \\$\\beta$^m&\\$\\beta$^n&1\\\\ \\gamma^m&\\gamma^n&1\\end{pmatrix}=0$$\nproduces 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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the effective bound on non-proportional primitive solutions of $x_1+\\cdots+x_s=0$ in a multiplicative group; this is the engine behind Theorem 1.1.","marker":"[3]"},{"why":"Provides the explicit lower bound for linear forms in logarithms of algebraic numbers used to prove the quantitative degree and height estimates in Theorem 1.2.","marker":"[6]"},{"why":"Contributes the proof strategy for Theorem 1.1, which the authors combine with the newer results [1,3].","marker":"[2]"},{"why":"Gives the quantitative refinement of the linear-equations bound used when the authors mention an explicit numerical constant for Theorem 1.1.","marker":"[1]"}],"fun_headline_variants":["Three root classes cap the number of vanishing trinomials","Unrelated roots allow only finitely many trinomials","Effective bounds tame the search for vanishing trinomials","Algebraic roots make trinomial enumeration finite","Trinomial count has an absolute constant bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Three root classes cap the number of vanishing trinomials","Unrelated roots allow only finitely many trinomials","Effective bounds tame the search for vanishing trinomials","Algebraic roots make trinomial enumeration finite","Trinomial count has an absolute constant bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000467,"raw_usage":{"total_tokens":2266,"prompt_tokens":819,"completion_tokens":1447,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":435,"completion_tokens_details":{"reasoning_tokens":1373}},"tokens_in":435,"tokens_out":1447,"duration_ms":13758,"temperature":1.0,"reasoning_tokens":1373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:26:22.198652+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Evertse, H","cited_arxiv_id":null,"evidence_quote":"Supplies the effective bound on non-proportional primitive solutions of $x_1+\\cdots+x_s=0$ in a multiplicative group; this is the engine behind Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit lower bound for linear forms in logarithms of algebraic numbers used to prove the quantitative degree and height estimates in Theorem 1.2."},{"cited_title":"Evertse, K","cited_arxiv_id":null,"evidence_quote":"Contributes the proof strategy for Theorem 1.1, which the authors combine with the newer results [1,3]."},{"cited_title":"Amoroso and E","cited_arxiv_id":null,"evidence_quote":"Gives the quantitative refinement of the linear-equations bound used when the authors mention an explicit numerical constant for Theorem 1.1."}],"review_version":1}