{"id":"9c62cd96-d686-46cd-b986-a6f09d6c18d7","arxiv_id":"1908.09793","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For two families of trinomials, this paper proves explicit congruence criteria for monogeneity in degrees 5 and 6 and shows infinite monogenic subfamilies for every n greater than 2 with positive density.","lead":"This paper gives explicit conditions under which roots of two families of trinomials, x^n + a x + b and x^n + c x^(n-1) + d, generate the full ring of integers of their number fields, and proves infinite families with positive coefficient density. The practical use is a fast test for monogenic number fields in degrees 5 and 6 and beyond.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6 is stated without the gcd(c,n)=1 condition that its proof assumes; when gcd(c,n)>1, Prachar's theorem cannot be applied and the claimed density lower bound is unsupported.","rationale":"I read the paper as establishing necessary and sufficient conditions (Theorems 3.1–3.4) via the Montes algorithm for n=5 and 6 under a square-free discriminant-factor hypothesis, and as proving infinitude and density results for the families x^n+bx+b and x^n+cx^{n-1}+cd (Theorems 3.5 and 3.6). I checked the discriminant computations, the Newton polygon arguments, and the density derivations. The reader's weakest assumption identifies the single genuine gap: Theorem 3.6's proof assumes gcd(c,n)=1 without including it in the theorem statement, and Prachar's theorem indeed requires a unit residue class modulo n^n. When gcd(c,n)>1, the linear form M(d) is never square-free, so the proof's appeal to the positive square-free density D is invalid. The qualitative infinitude of monogenic fields for each n is not destroyed, because one can choose c to be a prime not dividing n, but the theorem as stated overclaims. The paper's acknowledgments of overlap with Jakhar–Khanduja–Sangwan and Jones–White are honest, and Section 6's lack of shipped code is a reproducibility limitation rather than a mathematical flaw. Since the reader's verdict is already CONDITIONAL and my read identifies the same concern with the same recommended fix, no verdict adjustment is needed.","tokens_in":15280,"tokens_out":36428,"duration_ms":373232,"concrete_test":"For the smallest shared-prime case n=3, c=3, compute the proportion of d in [-10^6, 10^6] for which g(x)=x^3+3x^2+3d is monogenic, using Dedekind's index criterion at all primes dividing the discriminant, and compare with B = 6/π^2(3/4 + 9/8) − 1 ≈ 0.140. Independently, a squarefree-sieve computation for the conditions d square-free, 3∤d, and 3d+4 square-free gives local density (3/4)(2/3)∏_{p≥5}(1 − 2/p^2) ≈ 0.415. If the computed proportion is below B, the theorem's stated lower bound is false as written; if it is above B, the statement may survive but the proof still needs a separate argument removing the gcd assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.6 (Section 5) begins with the assumption that c is square-free, c ≠ ±1, and gcd(c, n) = 1, but the theorem statement only requires c to be nonzero, not ±1, and square-free. The density argument applies Prachar's theorem (Theorem 5.1) to square-free values of M(d) = d n^n + (1−n)^{n−1} c^{n−1} in the residue class m = (1−n)^{n−1} c^{n−1} modulo n^n. Prachar's theorem requires gcd(m, n^n) = 1, which is exactly gcd(c, n) = 1. If p divides both c and n, then p divides m and n^n, and M(d) is divisible by p for every d; the square-free density D = (6/π^2)∏_{p|n} p^2/(p^2−1) used in the constant B is then 0, not the positive quantity inserted. Consequently, the lower bound B is not derived for allowed c such as n=3, c=3 or n=4, c=2, and the 'in particular' positive-density conclusion is not established in the stated generality. The rest of the paper, especially Theorems 3.1–3.5 and the Montes/Dedekind arguments, appears internally consistent; this is a statement–proof mismatch rather than a failure of the overall method.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies monogeneity of trinomials. For the families x^n+ax+b and x^n+cx^{n-1}+d with n=5,6 and a square-free quotient of the discriminant, Theorems 3.1–3.4 give necessary and sufficient congruence conditions on the coefficients for a root to generate a power integral basis, proved via the Montes algorithm and Ore's theorem. For general n>2, Theorems 3.5 and 3.6 claim that the specialized families x^n+bx+b and x^n+cx^{n-1}+cd contain infinitely many monogenic examples, with explicit lower bounds on the natural density of the coefficients, using Prachar's theorem on square-free values in arithmetic progressions. The paper also remarks on the equivalence between the index part of the Montes algorithm and Dedekind's index criterion, and it includes computational data comparing the proved densities with observed monogeneity rates.","tokens_in":15463,"tokens_out":24282,"duration_ms":231351,"significance":"The n=5,6 congruence criteria are clean and appear to be a useful complement to the more general criteria of Jakhar–Khanduja–Sangwan, and the density analysis is a genuine addition to the existing literature on monogenic trinomials. The proofs of Theorems 3.1–3.5 are carefully structured around Ore's theorem and the Greenfield–Drucker discriminant formula, and the logical dependencies are transparent. The main weakness is Theorem 3.6, where the proof assumes a coprimality condition that is absent from the theorem statement; this is a statement–proof mismatch in a load-bearing result.","major_comments":[{"comment":"The statement of Theorem 3.6 allows any nonzero square-free c with c ≠ ±1, but the proof begins with the additional hypothesis gcd(c,n)=1. This assumption is essential: the density calculation applies Prachar's theorem (Theorem 5.1) to the residue class m = (1−n)^{n−1} c^{n−1} modulo n^n, and Prachar's theorem requires gcd(m, n^n)=1, which is equivalent to gcd(c,n)=1. If a prime p divides both c and n, then every integer M(d) = d n^n + (1−n)^{n−1} c^{n−1} is divisible by p, so the density of square-free values M(d) is 0, not the positive quantity (6/π²)∏_{p|n} p²/(p²−1) that is inserted into the constant B. Consequently the lower bound B is not derived for allowed parameters such as n=3, c=3 or n=4, c=2, and the 'in particular' positivity and infinitude conclusions are not established in the stated generality. The theorem should either add gcd(c,n)=1 to its hypotheses or provide a separate argument for the non-coprime case (for instance, by removing the primes dividing gcd(c,n) from the square-free requirement on M(d)).","section":"Section 5, Theorem 3.6 (statement p. 5; proof p. 11)"}],"minor_comments":[{"comment":"The title contains a typographical artifact: 'T rinomials' should be 'Trinomials'.","section":"Title"},{"comment":"The text 'See Section6' is missing a space and should read 'See Section 6'.","section":"Remark 3.7"},{"comment":"The density statements are formulated for natural density over N, but the theorems involve arbitrary integer coefficients; the authors should clarify that the densities are computed among positive values of the relevant parameter (or give a convention for handling negative integers).","section":"Sections 5 and 6"}],"recommendation":"major_revision","confidential_remarks":"The central issue is confined to Theorem 3.6. Theorems 3.1–3.5 and the computational section appear sound, and the paper's main method is not in question. The authors can likely repair the gap by adding the missing gcd(c,n)=1 hypothesis or by refining the density argument, so rejection would be disproportionate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, honest paper on monogeneity of trinomials. The n=5 and n=6 congruence criteria (Theorems 3.1-3.4) are explicit and genuinely more succinct than the general criteria of Jakhar-Khanduja-Sangwan, and the density lower bound for x^n + c x^(n-1) + c d in Theorem 3.6 is new, though flawed as stated. The Montes/Dedekind reasoning in Section 4 checks out, and the paper is transparent about overlap with Jones-White and JKS.\n\nThe main problem is Theorem 3.6. The theorem statement only requires c nonzero, not ±1, and square-free. But the proof starts by assuming gcd(c,n)=1. That assumption is load-bearing: the argument applies Prachar's theorem to square-free values of M(d)=d n^n + (1-n)^(n-1) c^(n-1) in the residue class modulo n^n, and that requires the class to be coprime to n^n. If a prime p divides both c and n, then p divides M(d) for every d, so the square-free density is zero, not the positive B in the statement. The 'in particular' positive-density claim is not established for e.g. n=3, c=3. This is a real gap but a local one: the fix is either to add gcd(c,n)=1 to the theorem (and then the proof goes through) or to handle shared primes separately. As stated, the theorem overclaims.\n\nEverything else is in better shape. Theorems 3.1-3.5 look internally consistent. The union-bound density argument in 3.5 is valid, and the honest statement in Remark 3.7 that the bounds are weak compared to computation is nice. The computational data in Section 6 is not backed by shipped code, so exact reproducibility is limited, but that does not affect the main proofs. The citation pattern is appropriate; the authors do not hide the parallel work of JKS and Jones-White.\n\nWho is this for? Number theorists working on Hasse's problem, especially those using Montes or needing explicit congruence tests. I'd send it to a referee, but the referee should be asked to verify Theorem 3.6 carefully. With the gcd condition added or the proof reworked, this is publishable. The core method and the degree 5/6 results are worth preserving.","headline":"A mostly careful Montes-algorithm analysis with one statement-proof mismatch in Theorem 3.6; the explicit n=5,6 criteria and density lower bounds are the real value.","tokens_in":16138,"tokens_out":2064,"would_cite":true,"duration_ms":19389,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R04"],"pacs":[],"model":"deepseek-v4-flash","headline":"For four trinomial families of degrees 5 and 6, the paper gives necessary and sufficient congruence conditions for monogeneity under a square-free discriminant hypothesis, and for every n>2 it proves that two trinomial families are…","keywords":["monogenic","power integral basis","trinomials","Montes algorithm","Newton polygon","discriminant","ring of integers","number field"],"falsifier":"Compute the index [O_K : Z[θ]] for an explicit trinomial in one of the degree-5 or 6 families whose discriminant factor is square-free but whose coefficients satisfy the theorem's congruence conditions; if any such polynomial has p dividing the index for p outside the listed primes, the classification fails. For Theorem 3.6, test n=4, c=2: the residue class (1-n)^(n-1)c^(n-1) modulo n^n has gcd 8 with the modulus, so Prachar's theorem does not apply, and the claimed positive density of d should be checked numerically or repaired.","tokens_in":14975,"feed_emoji":"🧮","tokens_out":4830,"duration_ms":47003,"temperature":0.7,"pith_summary":"This paper asks when a root of a trinomial generates the full ring of integers of its number field, a property called monogeneity. For the families $x^{5}$+ax+b, $x^{6}$+ax+b, $x^{5}$+$cx^{4}$+d, and $x^{6}$+$cx^{5}$+d, it gives necessary and sufficient congruence conditions on the coefficients, assuming a certain large factor of the discriminant is square-free. It further proves that the families x^n+bx+b and x^n+$cx^{{n-1}}$+cd contain infinitely many monogenic examples for every n>2, with positive lower bounds on the density of the coefficients. A sympathetic reader would care because monogenic fields are exactly those whose rings of integers admit a power integral basis, a classical and generally difficult classification problem.","feed_headline":"For every n>2, two trinomial families yield monogenic fields infinitely often","feed_subtitle":"Degree 5 and 6 cases get necessary-and-sufficient congruence tests; densities bound the coefficient sets.","key_machinery":"The central mechanism is Ore's theorem of the index, accessed through the Montes algorithm: for each prime p, one writes f in its φ-adic development for each irreducible factor φ of f modulo p, forms the principal φ-Newton polygon (the lower convex hull of the points (i, v_p(a_i(x)))), and defines the φ-index as the number of positive integer lattice points on or under that polygon. The theorem says p divides [O_K : Z[θ]] exactly when some φ-index is positive; the paper's Corollary 2.3 sharpens this to: p is harmless precisely when every principal polygon is one-sided. The discriminant formula for trinomials identifies the only primes that can appear, and the paper remarks that this polygon test is equivalent to Dedekind's index criterion.","core_discovery":"Under the square-free hypotheses, the primes that can divide the index [O_{Q(θ)} : Z[θ]] are exactly the primes dividing a specific gcd of coefficient terms, and monogeneity is decided by finitely many congruence conditions: for $x^{5}$+ax+b, conditions modulo 4 and 25 for primes 2 and 5; for $x^{6}$+ax+b, additional conditions modulo 9 for prime 3; for $x^{5}$+$cx^{4}$+d, square-freeness of d plus a condition modulo 25; for $x^{6}$+$cx^{5}$+d, square-freeness of d plus conditions modulo 4 and 9. The paper also proves that for every n>2, the family x^n+bx+b is monogenic for a set of b of density at least 21.58%, and that x^n+$cx^{{n-1}}$+cd is monogenic for a set of d of positive density when c has one prime factor or has two prime factors and is coprime to 6, with a larger lower bound in the latter case.","pith_inferences":["The missing gcd(c,n)=1 hypothesis in the statement of Theorem 3.6 is likely repairable: one could add congruence conditions treating primes dividing both c and n, in the style of Theorems 3.1-3.4, rather than relying on Prachar's theorem for coprime residue classes.","The congruence characterizations for n=5 and 6 suggest that for general n the analogous monogeneity conditions would take the form of p-adic congruence classes coming from one-sided Newton polygons, potentially yielding a full recursive classification for all n.","If the square-freeness events in the density proofs were independent, the paper's own computation shows the density lower bounds would rise to roughly 0.28-0.61 for x^n+bx+b and up to about 0.74 for x^n+cx^{n-1}+cd; testing the actual correlations numerically would clarify how much the current bounds understate the true densities.","The computational table indicates cases where the square-free hypotheses fail but the polynomial is still monogenic, so the true monogenic set is larger than the theorems capture; a refined analysis of when repeated discriminant factors fail to contribute to the index would extend the classification."],"forward_implications":["For the degree 5 and 6 families, monogeneity becomes a finite congruence check whenever the stated discriminant factor is square-free, so one can enumerate candidates for power integral bases algorithmically.","The families x^n+bx+b and x^n+cx^{n-1}+cd supply infinitely many monogenic number fields in every degree n>2, not just degrees 5 and 6.","The density bounds in Theorems 3.5 and 3.6 show that monogeneity is not a rare phenomenon in these coefficient families, even though the full monogenic set is not completely characterized.","The equivalence between one-sided Newton polygons and Dedekind's index criterion gives a geometric way to visualize and test a classical criterion, and the Montes algorithm carries strictly more information than the index criterion alone."],"supporting_citations":[{"why":"Supplies the closed-form discriminant formula for trinomials, which identifies the primes that can divide the index.","marker":"[11]"},{"why":"Ore's theorem of the index is the main tool: it relates Newton-polygon indices to the index of Z[θ] in the ring of integers.","marker":"[19]"},{"why":"Prachar's square-free distribution theorem gives the density of square-free values along arithmetic progressions, used to prove Theorems 3.5 and 3.6.","marker":"[20]"},{"why":"Dedekind's index criterion is shown in Remark 2.5 to be equivalent to the one-sided polygon test used throughout the paper.","marker":"[3]"}],"fun_headline_variants":["Trinomial monogeneity: infinite families for every n>2","Monogenic trinomials: densities and exact tests for n=5,6","New criteria for monogenic trinomial fields","For all n>2, trinomials give infinitely many monogenic fields","Necessary and sufficient conditions for degree 5,6 monogenic trinomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the stated square-free factor of the discriminant is indeed square-free (and, for Theorem 3.6, that gcd(c,n)=1, a condition used in the proof but missing from the theorem statement).","fun_headline_variants_meta":{"raw":{"variants":["Trinomial monogeneity: infinite families for every n>2","Monogenic trinomials: densities and exact tests for n=5,6","New criteria for monogenic trinomial fields","For all n>2, trinomials give infinitely many monogenic fields","Necessary and sufficient conditions for degree 5,6 monogenic trinomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000651,"raw_usage":{"total_tokens":2967,"prompt_tokens":905,"completion_tokens":2062,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":1982}},"tokens_in":521,"tokens_out":2062,"duration_ms":13689,"temperature":1.0,"reasoning_tokens":1982,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:02:06.915942+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the index [O_K : Z[θ]] for an explicit trinomial in one of the degree-5 or 6 families whose discriminant factor is square-free but whose coefficients satisfy the theorem's congruence conditions; if any such polynomial has p dividing the index for p outside the listed primes, the classification fails. For Theorem 3.6, test n=4, c=2: the residue class (1-n)^(n-1)c^(n-1) modulo n^n has gcd 8 with the modulus, so Prachar's theorem does not apply, and the claimed positive density of d should be checked numerically or repaired.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the closed-form discriminant formula for trinomials, which identifies the primes that can divide the index."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Ore's theorem of the index is the main tool: it relates Newton-polygon indices to the index of Z[θ] in the ring of integers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prachar's square-free distribution theorem gives the density of square-free values along arithmetic progressions, used to prove Theorems 3.5 and 3.6."},{"cited_title":"Dedekind","cited_arxiv_id":null,"evidence_quote":"Dedekind's index criterion is shown in Remark 2.5 to be equivalent to the one-sided polygon test used throughout the paper."}],"review_version":1}