{"id":"ccdffbc2-57fb-41e3-8adc-86d39300d5a0","arxiv_id":"1908.07947","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"New infinite families and explicit asymptotic densities of monogenic trinomials x^n + A x^m + A with non-squarefree discriminant, plus a conditional construction for A != B.","lead":"For trinomials of the form x^n + A x^m + A, this paper proves exactly how many A up to X make the polynomial generate a monogenic number field, even when the discriminant is not squarefree. It gives new infinite families, several with small Galois groups, and a new counting method for linked squarefree values.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 is false as stated at n=2: the counted set includes odd A whose quadratic has squarefree discriminant.","rationale":"The paper's main theorem has a genuine boundary counterexample. For n=2, m=1, kappa=1 and t=2, Corollary 3.3 characterizes monogenicity as A and 4-A being squarefree; A=5 satisfies this, and by Swan's formula Delta(f)=5 is squarefree, so the polynomial is monogenic with squarefree discriminant. The proof of Theorem 1.1 counts this A because the asymptotic (1.2) has no parity or congruence restriction excluding odd A. This is not a criticism of Theorem 2.2 or the JKS criterion; it follows from the paper's own reduction. The fix is straightforward (state n>=3, or for n=2 require A even), and the n>=3 asymptotic arguments appear internally consistent: conditions (3.6) check out and the Euler product in (3.14) yields (1.2). I therefore do not reject the paper's core construction, but the theorem as stated is false, so the verdict should remain conditional with an explicit correction rather than unchanged. I disagree with the reader's choice of the most load-bearing assumption: the dependency on JKS is external and was not shown to fail, whereas the n=2 case is an internal contradiction in the stated theorem.","tokens_in":17353,"tokens_out":27392,"duration_ms":250153,"concrete_test":"Run the Table-2 computation for n=2, m=1, X=10000: enumerate squarefree A with |4-A| squarefree and split by parity. A=5 is in this list and gives Delta=5, a counterexample to the printed theorem. Counting only even A gives the monogenic quadratics with non-squarefree discriminant, and its leading constant should be compared with (1.2) after removing the odd-A contribution. If the statement is corrected by requiring A even when (n,m)=(2,1), the even-only main term should match; the uncorrected (1.2) will overshoot by the odd-A density. This single check settles whether the theorem is false as written.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Take n=2, m=1. Then kappa=1, t=2, and Corollary 3.3, which the proof of Theorem 1.1 invokes verbatim, says f(x)=x^2+Ax+A is monogenic iff A and D=4-A are squarefree. For A=5, A is squarefree and D=-1 is squarefree, so f is monogenic; Swan's discriminant formula gives Delta(f)=A^2-4A=5, which is squarefree. Thus A=5 is counted by the asymptotic (1.2), which counts all squarefree A with A-4 squarefree, but it fails the theorem's 'non-squarefree discriminant' conclusion. The proof of Theorem 1.1 concludes 'By Corollary 3.3, the proof of the theorem is complete' without enforcing Corollary 3.3's non-squarefree sufficient conditions, which require t>=3 when m=1, or A even. Hence Theorem 1.1 is false as stated for n=2, and the same issue enters Theorem 1.2 when m=1 and t=2. The theorem and proof need a parity restriction for n=2, or the statement should be changed to n>=3.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs infinite families of monogenic trinomials f(x)=x^n+A x^m+A with non-squarefree discriminant for n≥2 and m a proper divisor of n, and gives asymptotic counts for A≤X. The proofs combine the Jakhar–Khanduja–Sangwan criterion for monogeneity of trinomials, Swan's discriminant formula, and squarefree sieve estimates (Prachar, Helfgott–Hooley–Pasten). A separate result, conditional on the abc conjecture for number fields when n/m≥5, treats the case A≠B. The paper also proves a Galois-group bound showing these families can have Galois groups much smaller than S_n.","tokens_in":17466,"tokens_out":16834,"duration_ms":147036,"significance":"If the main theorems are correct, the paper provides genuinely new infinite families of monogenic trinomials with non-squarefree discriminant and small Galois groups, with explicit densities. The proofs are detailed and the central technique—reducing monogeneity to squarefree conditions on A and an associated linear form D—is transparent and reproducible. The use of the JKS criterion is appropriate, and the analytic number theory inputs are standard. However, the main counting theorems are false as stated for n=2, m=1, because the asymptotic includes odd A for which the discriminant is squarefree. This is a load-bearing error in the paper's principal claims, though it appears to be repairable by restricting to n≥3 or by imposing a parity condition for n=2.","major_comments":[{"comment":"Theorem 1.1 is false as stated for n=2, m=1. In that case κ=1, t=2, and Corollary 3.3 gives that f(x)=x^2+Ax+A is monogenic iff A and D=4-A are squarefree, while the discriminant is non-squarefree only if A is even. The proof of Theorem 1.1 applies Theorem 3.8 to count all squarefree A with A-4 squarefree, making no parity restriction. For A=5, A and 4-A are squarefree, so f is monogenic, and Swan's formula gives Δ(f)=5, which is squarefree; yet A=5 is counted by the asymptotic (1.2). Thus the counted set is not contained in the set of trinomials with non-squarefree discriminant. The statement and proof need either the restriction n≥3, or, for n=2,m=1, the additional condition that A is even together with a corresponding adjustment of the asymptotic.","section":"Theorem 1.1 and §4.1"},{"comment":"The same defect appears in Theorem 1.2 when n=2, m=1. Then κ=1, the congruence A≡-1 (mod κ²) is vacuous, and Proposition 3.5 again only guarantees non-squarefree discriminant for even A when t=2 and m=1. The proof counts all squarefree A with A-4 squarefree, including odd values such as A=5 with Δ(f)=5. Consequently, the asymptotic (1.3) overcounts the stated family. The theorem requires the same correction as Theorem 1.1: either restrict to n≥3 or add the appropriate parity condition.","section":"Theorem 1.2 and §4.2"}],"minor_comments":[{"comment":"In the proof of Theorem 1.1, the application of Theorem 3.8 uses the variable y=a=A/κ and therefore counts a up to X/κ, but the paper does not explicitly say that Theorem 3.8 is applied with X replaced by X/κ; the notation is understandable but would benefit from clarification.","section":"§4.1"},{"comment":"The transition from the first line of (3.10) to the second line replaces the upper limit √F(X) with X0 and writes O(X/X0) for the tail; this is correct after summing X/d² over d>X0, but the reader must fill in that step. A sentence explaining the tail bound would improve readability.","section":"Theorem 3.8 proof, Eq. (3.10)"},{"comment":"The phrase 'non-squarefree discriminant' is occasionally used where 'monogenic with non-squarefree discriminant' is meant; the distinction matters for the n=2 issue, and aligning terminology with the theorem statements would prevent confusion.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The false n=2 case is a simple but substantive oversight in the statements of Theorems 1.1 and 1.2. It is repairable by restricting the theorems to n≥3 (or by adding an evenness condition for n=2), so I do not recommend rejection. The rest of the manuscript appears sound, and the methods are solid. The authors should also ensure that the abstract and introduction are updated consistently once the n=2 issue is resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The stress-test is right. Theorem 1.1 is false as stated for n=2, m=1. Take A=5: f(x)=x^2+5x+5 is monogenic because A and 4−A are both squarefree, but the discriminant is 5, which is squarefree, so it should not be counted. The proof of Theorem 1.1 invokes Corollary 3.3, whose non-squarefree clause requires t≥3 for m=1 or A even; neither holds for odd A. The same issue enters Theorem 1.2 when n=2, m=1. So the theorem statements overclaim for n=2. The fix is easy: restrict to n≥3 or add a parity condition for n=2.\n\nThat is a real flaw, but it is localized. For n≥3 the arguments seem to hold up. The JKS-criterion checks in Lemma 3.1 and Corollary 3.3 are careful, and the sieve asymptotic in Theorem 3.8 is standard and well executed. The numerical tables match the main terms closely, which is good evidence the constants are right. The small-Galois-group families from Proposition 1.4 are genuinely new and distinct from earlier S_n/A_n trinomial families. This is a useful contribution to the monogenic trinomial literature.\n\nTwo minor soft spots: the finite verification for primes 2,3,5,7 in Proposition 3.5 is asserted with a table but not fully shown, and Proposition 1.4 is presented as a sketch. Both are minor compared with the n=2 bug.\n\nRecommendation: send it to a serious referee, but ask the authors to fix the n=2 case before publication. I would not cite the current version; after the correction, I would.","headline":"The main theorem overclaims at n=2, but the construction for n≥3 is solid and worth a serious referee.","tokens_in":18102,"tokens_out":4628,"would_cite":false,"duration_ms":43214,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R04","11R09","11R32","12F12"],"pacs":[],"model":"deepseek-v4-flash","headline":"A formula counts monogenic trinomials with non-squarefree discriminant.","keywords":["monogenic polynomials","trinomials","non-squarefree discriminant","Galois group","index of an algebraic integer","squarefree values","asymptotic density","arithmetic progressions"],"falsifier":"Enumerate, for a fixed pair $(n,m)$ with $m$ a proper divisor of $n$, all $A\\le X$ satisfying the conditions of Corollary 3.3, and compute the index $[O_K:\\mathbb{Z}[\\theta]]$ directly for the first few trinomials it predicts to be monogenic. If any computed index exceeds 1, the characterization behind Theorem 1.1 fails. Alternatively, compare the exact count at a much larger $X$ than the paper's tables against the main term in (1.2); a discrepancy larger than the stated $O(X^{3/4})$ term would falsify the asymptotic.","tokens_in":17036,"feed_emoji":"🧮","tokens_out":14271,"duration_ms":103990,"temperature":0.7,"pith_summary":"The paper shows that being monogenic does not force the discriminant to be squarefree. For every $n\\ge 2$ and every proper divisor $m$ of $n$, it constructs infinite families of trinomials $f(x)=x^n+Ax^m+A$ that generate the full ring of integers of their number field even though their discriminants are not squarefree, and it counts them: the number of admissible $A\\le X$ is asymptotic to $X/(\\kappa\\zeta(2))$ times two explicit local factors, where $\\kappa$ is the squarefree kernel of $m$. A companion result counts the case $A\\equiv -1\\pmod{\\kappa^2}$ when $\\kappa\\mid 210$, and a third theorem produces infinitely many $B$ for trinomials $x^n+Ax^m+B$ with $A\\ne B$, conditionally on the abc conjecture for number fields when $t=n/m\\ge 5$. Because the Galois group of these trinomials has order at most $\\varphi(m)m^t t!$, the families can have Galois groups much smaller than the symmetric group $S_n$.","feed_headline":"A formula counts monogenic trinomials with non-squarefree discriminant","feed_subtitle":"For polynomials x^n + A x^m + A, it counts how many A up to X are monogenic and shows small Galois groups appear.","key_machinery":"The load-bearing identity is the discriminant formula (Theorem 2.1), which expresses $\\Delta(f)$ for $f(x)=x^n+Ax^m+B$ in terms of $n,m,A,B$, together with the five-condition criterion (Theorem 2.2) that decides, from the coefficients alone, which prime divisors of $\\Delta(f)$ divide the index $[O_K:\\mathbb{Z}[\\theta]]$. The paper's Lemma 3.1 makes one of the five clauses vacuously true by requiring $\\gcd(A,B)$ to be divisible by the squarefree kernel $\\kappa$ of $m$, so that monogenicity reduces to the simultaneous squarefreeness of $B$ and the reduced quantity $D=(t^tB^{t-1}+(1-t)^{t-1}A^t)/\\gcd(A,B)^{t-1}$. For $A=B$, this collapses to the condition that $A$ and $D=t^t+(1-t)^{t-1}A$ be squarefree. Counting such pairs is then handled by a new asymptotic (Theorem 3.8) for squarefree values of a linear polynomial in an arithmetic progression, built on the classical asymptotic for squarefree integers in arithmetic progressions.","core_discovery":"Fix $n\\ge 2$ and a proper divisor $m$ of $n$, write $t=n/m$, and let $\\kappa$ be the squarefree kernel of $m$. The paper proves that, up to an $O(X^{3/4})$ error, the number of $A\\le X$ with $A\\equiv 0\\pmod{\\kappa}$ for which $f(x)=x^n+Ax^m+A$ is monogenic with non-squarefree discriminant is $$\\frac{X}{\\kappa\\zeta(2)}\\prod_{p\\mid\\kappa}\\left(1-\\frac{1}{p+1}\\right)\\prod_{p\\nmid t(t-1)\\kappa}\\left(1-\\frac{1}{$p^{2}$-1}\\right).$$ The characterization behind the count is that, under these hypotheses, monogenicity is equivalent to $A$ and $D=t^t+(1-t)^{t-1}A$ both being squarefree. The same squarefree-pair mechanism, with extra coprimality checks limited to the primes $2,3,5,7$, yields the companion count for $A\\equiv -1\\pmod{\\kappa^2}$. A separate construction with $A\\ne B$ gives infinitely many $B$ making $x^n+Ax^m+B$ monogenic, unconditional for $2\\le t\\le 4$ and conditional on the abc conjecture for number fields for $t\\ge 5$; when $m\\ge2$, infinitely many of those also have non-squarefree discriminant.","pith_inferences":["A natural extension, not pursued in the paper, is to replace the residue conditions $A\\equiv0\\pmod{\\kappa}$ or $A\\equiv-1\\pmod{\\kappa^2}$ by other residue classes modulo powers of $\\kappa$; the same squarefree-pair counting method should yield analogous Euler-product constants whenever the corresponding coprimality calculation can be completed.","The small-Galois-group bound suggests these trinomials could serve as explicit generators for number fields with prescribed small Galois groups and non-squarefree discriminant, but the paper does not itself construct such fields.","When $t(t-1)$ has many small prime factors, the asymptotic constant is close to $1/(\\kappa\\zeta(2))$, hinting that almost every admissible $A$ yields a monogenic trinomial; checking exact counts at larger $X$ than the tables provide would be a direct numerical test."],"forward_implications":["For $m=1$, the condition $A\\equiv 0\\pmod{\\kappa}$ is vacuous, so Theorem 1.1 gives a complete classification of monogenic polynomials of the form $x^n+Ax+A$ with $A\\ge 2$, essentially all of which have non-squarefree discriminant.","For $n=2^k t$ and $m=2^k$ with $t\\ge 2$, the results combine to give a complete classification of monogenic trinomials $x^{2^k t}+A x^{2^k}+A$ with $A\\ge 2$, all with non-squarefree discriminant.","Every resulting field has Galois group of order at most $\\varphi(m)m^t t!$, so taking $n$ large with $m$ commensurate with $n$ makes the Galois group far smaller than $S_n$.","For fixed $A$ satisfying the stated divisibility condition, there are infinitely many $B$ such that $x^n+Ax^m+B$ is monogenic; when $m\\ge 2$, infinitely many of those also have non-squarefree discriminant, with the $t\\ge5$ cases conditional on the abc conjecture for number fields."],"supporting_citations":[{"why":"Supplies the five-condition criterion, stated as Theorem 2.2, that every monogenicity verification in the paper reduces to.","marker":"[20]"},{"why":"Supplies the discriminant formula for $x^n+Ax^m+B$ used to compute $\\Delta(f)$ and to define the reduced factor $D$.","marker":"[35]"},{"why":"Provides the asymptotic for squarefree integers in an arithmetic progression that underlies Corollary 2.4 and Theorem 3.8.","marker":"[31]"},{"why":"Establishes the squarefree-values-at-primes asymptotic for cubic polynomials, giving the unconditional range $2\\le t\\le4$ in Theorem 1.3.","marker":"[16]"},{"why":"Gives the abc-conditional asymptotic for squarefree values of polynomials of degree at least four at prime arguments, used for the $t\\ge5$ part of Theorem 1.3.","marker":"[30]"}],"fun_headline_variants":["New infinite monogenic trinomial families with non-squarefree discriminant","Asymptotic count for monogenic trinomials with non-squarefree discriminant","New monogenic trinomial families: non-squarefree discriminant, small Galois groups","Monogenic trinomials: new families, non-squarefree discriminants, small Galois groups","Formula yields asymptotics for monogenic trinomials with non-squarefree discriminant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the quoted five-condition test fully and correctly decides when a trinomial generates the full ring of integers of its number field; every claimed monogenic family depends on that test being complete. A secondary external assumption, used only for part of one theorem, is the abc conjecture for number fields.","fun_headline_variants_meta":{"raw":{"variants":["New infinite monogenic trinomial families with non-squarefree discriminant","Asymptotic count for monogenic trinomials with non-squarefree discriminant","New monogenic trinomial families: non-squarefree discriminant, small Galois groups","Monogenic trinomials: new families, non-squarefree discriminants, small Galois groups","Formula yields asymptotics for monogenic trinomials with non-squarefree discriminant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001813,"raw_usage":{"total_tokens":7138,"prompt_tokens":952,"completion_tokens":6186,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":6083}},"tokens_in":568,"tokens_out":6186,"duration_ms":498485,"temperature":1.0,"reasoning_tokens":6083,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:53:36.484886+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate, for a fixed pair $(n,m)$ with $m$ a proper divisor of $n$, all $A\\le X$ satisfying the conditions of Corollary 3.3, and compute the index $[O_K:\\mathbb{Z}[\\theta]]$ directly for the first few trinomials it predicts to be monogenic. If any computed index exceeds 1, the characterization behind Theorem 1.1 fails. Alternatively, compare the exact count at a much larger $X$ than the paper's tables against the main term in (1.2); a discrepancy larger than the stated $O(X^{3/4})$ term would falsify the asymptotic.","supporting_citations":[{"cited_title":"Jakhar, S","cited_arxiv_id":null,"evidence_quote":"Supplies the five-condition criterion, stated as Theorem 2.2, that every monogenicity verification in the paper reduces to."},{"cited_title":"Swan Factorization of polynomials over finite fields , Pacific J","cited_arxiv_id":null,"evidence_quote":"Supplies the discriminant formula for $x^n+Ax^m+B$ used to compute $\\Delta(f)$ and to define the reduced factor $D$."},{"cited_title":"Prachar, ¨Uber die kleinste quadratfreie Zahl einer arithmetischen R eihe, Monatsh","cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic for squarefree integers in an arithmetic progression that underlies Corollary 2.4 and Theorem 3.8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the squarefree-values-at-primes asymptotic for cubic polynomials, giving the unconditional range $2\\le t\\le4$ in Theorem 1.3."},{"cited_title":"Pasten, The ABC conjecture, arithmetic progressions of primes and s quare- free values of polynomials at prime arguments , Int","cited_arxiv_id":null,"evidence_quote":"Gives the abc-conditional asymptotic for squarefree values of polynomials of degree at least four at prime arguments, used for the $t\\ge5$ part of Theorem 1.3."}],"review_version":1}