{"id":"5b5fbc22-f32d-4641-a19d-8b170bf96f4f","arxiv_id":"1909.00135","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New upper bounds on how many monic integer polynomials of degree n and height H share a fixed field discriminant, plus improved lower bounds for distinct discriminants from trinomials.","lead":"This paper counts how many monic integer polynomials of fixed degree and bounded coefficients can produce the same discriminant for the number field they generate. The authors prove new upper bounds for these counts and show that simple trinomials already generate many distinct field discriminants, improving estimates from 2010.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exceptional-degree fix is a sketch: (1.2) for n = u^2 or u^2+1 depends on Hering/Smith steps not proved in the paper; Lemma 3.3's appeal to [9, Section 5] is incomplete.","rationale":"The central claim of the paper is Theorem 1.1, and its first bound (1.2) depends on the determinant method as developed in Dietmann's earlier paper [9]. The paper's own Lemmas 2.2 and 2.3 appear mathematically correct for all n ≥ 3 under the stated conditions, including the exceptional degrees, because the problematic u=1 case is excluded by the non-square hypotheses. This suggests the Section 6 appendix might not be strictly necessary for the current theorem. However, Lemma 3.3 is proved only by reference to '[9, Section 5]' with substitutions, and the paper never demonstrates step-by-step that the substitutions cover the exceptional degrees. The appendix explicitly labels itself as a quick fix of an error in [9], and it invokes two external theorems (Hering [13], Smith [31]) without stating their precise hypotheses or proving the uniformity required in the determinant-method counting. Since the manuscript itself flags this missing support, the burden of proof is on the authors to make the exceptional-degree case fully rigorous. I found no other load-bearing concern: Lemma 2.7 and the square-sieve argument appear sound, Lemma 3.2 is a correct reduction to the u=1 case, and the trinomial results are plausible. Thus the reader's CONDITIONAL verdict is appropriate; my stress-test does not push the verdict in a new direction, but it does reinforce the need for a complete exceptional-degree proof.","tokens_in":16765,"tokens_out":55333,"duration_ms":417894,"concrete_test":"Write out a complete proof of Lemma 3.3 for the exceptional cases n=9 and n=10 with squarefree u=2, following the amended [9, Section 5]. For n=9, verify explicitly that the bad a1 values for each fixed (a2,...,a8) are O(1) uniformly as claimed (state and prove the needed form of Hering [13, Satz 1]); for n=10, verify that the two-parameter family in (a3,a4) satisfies all conditions of [9, Corollary 1] or Smith [31] so that the curve z^2 = u Disc(...) over the relevant function field is irreducible and the point count gives H^{1/2+o(1)} in the relevant variables. If any step uses an unproved external result, the test fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.3, which yields the generic bound (1.2), is proved by 'working through [9, Section 5]' with Lemmas 2.2, 2.3, 3.1, 3.2 in place of [9, Lemmas 5,6,8,11]. The appendix concedes that [9, Lemmas 5 and 6] (and hence Lemma 8) are false for n = u^2 and n = u^2+1 with odd u, and repairs them only in a sketch. For n = u^2 the repair uses Hering [13, Satz 1] to claim only O(1) 'bad planes' per fixed coefficient tuple, uniform in those coefficients; this uniformity is exactly what is needed but is not stated or proved. For n = u^2+1 the repair shifts to a different pair of coefficients (a_{n-3}, a_n) and invokes Smith [31] for the Galois group of trinomials; the required analogue of the determinant-method point count on the resulting irreducible curves is not spelled out. If either step is incomplete, the exceptional n are left without the advertised exponent H^{n-2+√2+o(1)} for the infinitely many good squarefree u (u≥2, since u=1 is excluded by the hypothesis). Note that Lemma 2.2 may already cover the c0=0 case for u>1, but the paper does not explicitly integrate this with the counting in [9, Section 5], so the reader cannot verify completeness.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies N_n(H,\\Delta), the number of monic irreducible degree-n integer polynomials with coefficients bounded by H whose root field has discriminant \\Delta. The main result, Theorem 1.1, gives two uniform upper bounds: a generic bound H^{n-2+\\sqrt{2}+o(1)} for square-free parts u not satisfying either square condition, and a universal bound H^{n-n/(2n-1)}(\\log H)^{(3n-2)/(2n-1)} in all cases. The proofs combine a determinant-method argument modeled on Dietmann [9] and a square-sieve argument built on new character-sum estimates (Lemma 2.7). The paper also proves an improved bound for trinomials (Theorem 1.4), T_n(A,B,C,D;s) \\le A(A+B+C+D)^{o(1)}, with corollaries on the number of distinct quadratic fields and field discriminants. An appendix corrects errors in [9, Lemmas 5, 6, 8] for degrees n = u^2 or u^2+1 with odd u.","tokens_in":38,"tokens_out":25429,"duration_ms":243141,"significance":"If the determinant-method component is made fully rigorous, this is a substantial contribution: it provides the first uniform upper bounds for N_n(H,\\Delta), together with a lower bound on the number of distinct discriminants of fields generated by height-H polynomials. Lemma 2.7, which saves a factor p over the trivial or Weil-bound estimate for mixed character sums with polynomial discriminants, is of independent interest. Theorem 1.4 improves Shparlinski's trinomial result unconditionally and without the restriction A \\ll B^{2-\\varepsilon}. The square-sieve part of the paper is self-contained and convincing. The main weakness is that the determinant-method bound (1.2) for the exceptional degrees n = u^2 and n = u^2+1 rests on a sketched correction in the appendix rather than on a complete proof; the same is true for the uniformity-over-u claim in Lemma 3.3.","major_comments":[{"comment":"The amendment to [9, Lemmas 5 and 6] for degrees n = u^2 or n = u^2+1 with odd u is presented only as a sketch. For n = u^2, the assertion that Hering [13, Satz 1] yields only O(1) 'bad planes', uniformly in the fixed coefficients a_1,\\ldots,a_{n-2}, is not proved; the uniformity is exactly what the determinant-method counting requires. For n = u^2+1, the paper shifts to the coefficient pair (a_{n-3}, a_n) and invokes Smith [31] for the Galois group of the trinomial X^n + aX^{n-3} + b, but the analogue of the determinant-method point count on the resulting irreducible curves is not spelled out. Since Lemma 3.3 and hence Theorem 1.1 claim (1.2) for all n \\ge 3, these exceptional degrees currently lack a complete proof. Please expand the appendix into a full argument or restrict the statement of (1.2) to n not of these exceptional forms.","section":"Section 6 (Appendix) and Lemma 3.3"},{"comment":"The proof of Lemma 3.3 consists of saying 'we can now work through [9, Section 5]' and 'in exactly the same way as in [9] we derive the desired result.' This is not a self-contained proof: it does not verify that the determinant-method estimates, including the treatment of the additional factor u in equation (3.4), carry through uniformly for all square-free u satisfying the hypothesis. In particular, the height of the curve z^2 = u Disc(...) depends on u, and the Bombieri\\textendash Pila type point count must be shown to absorb this dependence into the o(1) term. A detailed derivation of the uniform bound T_n(H,u) \\le H^{n-2+\\sqrt{2}+o(1)} is necessary for the uniformity claim in Theorem 1.1.","section":"Section 3.1, Lemma 3.3"},{"comment":"The proof of Lemma 2.2 states that the leading monomial u(-1)^{(n-1)(n-2)/2}(n-1)^{n-1} c_0^n A_0^n 'cannot be a square in \\mathbb{Q}[A_0] as |u|(n-1)^{n-1} is no square.' This reasoning is not correct for odd n, because c_0^n can supply the missing square factors; for example, when n is odd and u c_0 is chosen to be a square, the coefficient may become a rational square. The lemma remains true in that case because D(A_0) has odd degree and hence cannot be a square, but the proof must be rewritten with an explicit parity split (even n handled by the non-square coefficient, odd n by the degree argument). Since Lemma 2.2 is used in Lemma 3.3, this is a load-bearing correction.","section":"Section 2.1, Lemma 2.2"}],"minor_comments":[{"comment":"The set T_n(H,u) is defined only for square-free integers u \\ge 1, but in the proof of Theorem 1.1 it is applied to the square-free part of \\Delta, which may be negative. Please clarify that u is replaced by |u|, or redefine T_n(H,u) to take any nonzero square-free u with |\\Delta(f)| = r^2 u.","section":"Sections 3.1 and 3.2, notation"},{"comment":"The transition from Theorem 1.4 to the lower bound S_n(A,B,C,D) \\ge B(A+B+C+D)^{o(1)} is asserted with 'As in [29]', but the paper does not state or prove the needed estimate on the number of reducible trinomials X^n + aX + b in the rectangle. Please add an explicit reducibility bound or a more detailed reference to [29].","section":"Section 4, Corollary 1.5"},{"comment":"The introduction says 'For C \\ge 1 and A \\ll B^{2-\\varepsilon} ... we can sharpen this as follows,' but the statement of Theorem 1.4 does not include the condition A \\ll B^{2-\\varepsilon}. The proof in fact does not use such a condition, so the introductory phrase should be reworded to avoid the impression that the theorem is conditional on it.","section":"Introduction, Theorem 1.4"},{"comment":"In the optimization step, the choice of z is announced as z = H^{n/(2n-1)}(\\log H)^{-(n-1)/(2n-1)}, and the paper says 'we obtain the desired result' without showing the algebra that yields the log exponent (3n-2)/(2n-1). Displaying this short computation would improve readability.","section":"Section 3.2, Lemma 3.4"}],"recommendation":"major_revision","confidential_remarks":"The paper's main ideas are strong, and the square-sieve part (Section 3.2) and the trinomial section are essentially complete. The determinant-method bound (1.2) is the weak point: it depends on an appendix that is an acknowledged sketch and on a non-self-contained appeal to [9, Section 5]. The authors should be asked to provide a full proof of the exceptional-degree amendment and of the uniformity-over-u counting, or to restrict the scope of (1.2). The paper is openly self-critical about the need to correct [9], which is to its credit, but the correction needs to be complete for the central theorem to be considered proven."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you work on counting fields by discriminant. The genuinely new pieces are: the first upper bounds for N_n(H,Δ), the improved trinomial bound (Theorem 1.4, exponent better by 3/2), and Lemma 2.7, a character sum that saves p rather than p^{1/2}. The authors also correct an error in Dietmann's 2013 paper. That is good faith and worth credit.\n\nThe square-sieve half (Lemma 3.4 → (1.3)) and the trinomial section are solid. Lemma 2.7 is elegant: average over the affine transformations f ↦ f_{u,v} and then apply Deligne to the highest form. The argument checks out. The determinant-method half is where the conditional verdict is right. Lemma 3.3 is not self-contained: it says 'work through [9, Section 5]' and the appendix only sketches the repair for the exceptional degrees n = u^2 or u^2+1 with odd u. For n = u^2, the fix invokes Hering's Satz 1 to get O(1) bad planes per fixed coefficient tuple, uniform in the coefficients; that uniformity is exactly what is needed and is not proved here. For n = u^2+1, the paper shifts to the coefficient pair (a_{n-3}, a_n) and invokes Smith for the Galois group of trinomials, but the analogue of the determinant-method point count on the resulting irreducible curves is not spelled out. If either step has a hidden non-uniformity, the advertised exponent H^{n-2+√2+o(1)} is unproven for infinitely many good u.\n\nMinor: Lemma 2.2's proof says 'For c1 = 0' where it evidently means c0 = 0; the argument itself is fine and this is a typo, not a real gap. The citation pattern is appropriate: the heavy reliance on [9] is the issue, but it is openly disclosed and the paper is correcting that prior work, not hiding behind it.\n\nWho this is for: people working on counting fields by discriminant or on discriminants of trinomials. I can see citing Theorem 1.4 and Corollary 1.7. The paper deserves a serious referee, not a desk reject: the new results are real, the sound parts are sound, and the appendix is likely repairable. The referee should push the authors to prove the uniformity claims in the Hering step and to finish the n = u^2+1 case with full details.","headline":"Genuinely new bounds in arithmetic statistics, with one honest but under-proved repair: the exceptional-degree case in the appendix needs full details before (1.2) is fully established.","tokens_in":17667,"tokens_out":4155,"would_cite":true,"duration_ms":520386,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R29","11R32","11N36","11L40","11C08"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every fixed degree $n \\ge 3$, the number of monic irreducible polynomials of height $H$ whose root field has a prescribed discriminant $\\Delta$ is at most $H^{n-2+\\sqrt{2}+o(1)}$ in the generic case; almost all such polynomials have…","keywords":["field discriminants","polynomial discriminants","square-free part","determinant method","square sieve","character sums","trinomials","number fields"],"falsifier":"For a degree $n=9$ or $n=10$, compute the number of coefficient vectors $(a_2,\\ldots,a_{n-1})$ in $[-H,H]^{n-3}$ for which $Z^2 - u\\operatorname{Disc}(X^n+\\cdots)$ fails to be absolutely irreducible; if for some square-free $u$ satisfying the non-square conditions this number grows faster than $H^{n-3}$, or the associated solution count exceeds $H^{n-2+\\sqrt{2}+\\varepsilon}$, Lemma 3.3 and Theorem 1.1 would be false. A more direct check is to verify the appendix's asserted finiteness of bad rational specialisations for $n=9$ and $n=10$, since that step is left as a sketch.","tokens_in":16532,"feed_emoji":"🔢","tokens_out":12064,"duration_ms":87235,"temperature":0.7,"pith_summary":"Fix a degree $n \\ge 3$ and a nonzero integer $\\Delta$. This paper counts monic irreducible integer polynomials of height $H$ whose roots generate a number field of discriminant $\\Delta$, and proves bounds that hold uniformly in $\\Delta$. In the generic case, when the square-free part $u$ of $\\Delta$ makes neither $|u|(n-1)^{n-1}$ nor $|u| n^n$ a square, the count is at most $H^{n-2+\\sqrt{2}+o(1)}$; otherwise it is at most $H^{n-n/(2n-1)}(\\log H)^{(3n-2)/(2n-1)}$. Uniformity over $\\Delta$ implies that almost all polynomials of height $H$ generate fields with discriminant at least $H^{2-\\sqrt{2}+o(1)}$, and the same counting machinery improves the known bound for trinomials with a prescribed square-free part of the discriminant.","feed_headline":"Almost all degree-n polynomials make huge field discriminants","feed_subtitle":"Uniform counting bounds show few polynomials share one discriminant — at most H^(n−2+√2) in the generic case.","key_machinery":"The load-bearing object is the square-free part of the polynomial discriminant, counted through the Diophantine equation $z^2 = u \\operatorname{Disc}(f)$, which by Lemma 2.1 has the same square-free part as the field discriminant. Two tools do the counting. The determinant method bounds integral points on the hypersurface $z^2 = u \\operatorname{Disc}(\\ldots)$, using irreducibility lemmas that hinge on the two non-square conditions on $|u|(n-1)^{n-1}$ and $|u|n^n$, plus a fibration lemma controlling solutions with two free coefficients. The square-sieve method instead averages the Jacobi symbol $(\\operatorname{Disc}(f)/pq)$ over polynomials; a character-sum lemma shows the sum over monic polynomials of degree $n$ over $\\mathbb{F}_p$ is $O(p^{n-1})$, a full power of $p$ better than the usual square-root saving, by averaging over the transformations $f_{u,v}(X)=u^n f(u^{-1}(X+v))$ and applying the geometric character-sum bound. Balancing the two estimates gives the two regimes of Theorem 1.1.","core_discovery":"The central discovery is a pair of complementary counting estimates for the square-free part of polynomial discriminants. Writing $u$ for the square-free part of $\\Delta$, Lemma 3.3 bounds the number of irreducible polynomials $f$ of height $H$ with square-free part $u$ by $H^{n-2+\\sqrt{2}+o(1)}$ whenever $|u|(n-1)^{n-1}$ and $|u|n^n$ are not squares; Lemma 3.4 bounds the same quantity by $H^{n-n/(2n-1)}(\\log H)^{(3n-2)/(2n-1)}$ without any restriction on $u$. Since $\\operatorname{Disc}(f)$ and the field discriminant $\\Delta(f)$ have the same square-free part, these bounds immediately give Theorem 1.1. The paper also proves an analogous saving for trinomials $X^n+aX+b$: with $n \\equiv 1 \\pmod 4$ and $a$ in a box away from zero, the number of pairs with a given square-free part of the discriminant is at most $H^{o(1)}$ times the number of $a$-values, yielding at least $H^{1+o(1)}$ distinct quadratic fields from trinomials up to height $H$.","pith_inferences":["The uniformity over $\\Delta$ suggests that small discriminants are not the main source of polynomials; a natural next step is to prove the authors' expectation that the total number of distinct discriminants from all degree-$n$ polynomials of height $H$ is $H^{n+o(1)}$.","The exceptional condition in Theorem 1.1 singles out discriminants for which the top term of $u\\operatorname{Disc}(f)$ can be a square as a polynomial in a coefficient; this looks like an intrinsic arithmetic obstruction, and one could test whether counts in the exceptional case actually reach the larger bound.","The $p$-saving character-sum estimate behind the square-sieve argument is independent of the rest of the paper and may apply to other coefficient-polynomial arithmetic functions, such as counting square-free values in multi-parameter families without an ABC-type assumption."],"forward_implications":["Almost all monic irreducible polynomials of degree $n$ and height $H$ generate number fields whose discriminant has size at least $H^{2-\\sqrt{2}+o(1)}$.","The total count of polynomials with field discriminant $\\Delta$ summed over $|\\Delta|\\le D$ is at most $D H^{n-2+\\sqrt{2}+o(1)} + D^{1/2} H^{n-n/(2n-1)+o(1)}$, so each small discriminant is generated by comparatively few polynomials.","For trinomials $X^n+aX+b$ with $n\\equiv 1\\pmod 4$ and $1\\le a,b\\le H$, at least $H^{1+o(1)}$ distinct quadratic fields $\\mathbb{Q}(\\sqrt{\\Delta_n(a,b)})$ occur.","The number of distinct quadratic fields $\\mathbb{Q}(\\sqrt{\\Delta_n(a,b)})$ with $|\\Delta_n(a,b)|\\le \\Delta$ is at least $\\Delta^{1/(n-1)+o(1)}$, improving the previous exponent by a factor of $3/2$.","The number of polynomials whose splitting field has a given discriminant $\\delta$ is at most $H^{n-n/(2n-1)+o(1)}$."],"supporting_citations":[{"why":"Supplies the determinant-method framework and the lemmas (5, 6, 8, 11) that the paper generalises to arbitrary square-free part $u$.","marker":"[9]"},{"why":"Provides the square sieve used in Lemma 3.4 to count polynomials whose discriminant has a given square-free part.","marker":"[11]"},{"why":"Supplies the character-sum and complete/incomplete summation technology underlying Lemmas 2.7 and 2.9.","marker":"[16]"},{"why":"Gives the transformation rule $\\operatorname{Disc}(f_{u,v}) = u^{n(n-1)}\\operatorname{Disc}(f)$ used in the averaging argument of Lemma 2.7.","marker":"[30]"},{"why":"Used in the appendix to bound the finitely many bad specialisations for the exceptional degrees $n=u^2$.","marker":"[13]"},{"why":"Used in the appendix to show that the relevant trinomial has symmetric Galois group over a function field, completing the repair.","marker":"[31]"},{"why":"Supplies the bound on solutions of the Pell-type equation needed in the proof of Theorem 1.4 for trinomials.","marker":"[19]"},{"why":"Provides the previous trinomial counts and the corollary structure that Theorem 1.4 and Corollary 1.7 improve.","marker":"[29]"},{"why":"Gives the discriminant formula for trinomials used to compute the leading coefficient in the appendix and in Section 5.","marker":"[33]"}],"fun_headline_variants":["Polynomials rarely share field discriminants","Many distinct discriminants from bounded-height polynomials","Counting polynomials by discriminant: tight bounds","Few polynomials per discriminant, many discriminants total","Heights and discriminants: sparse collisions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the generic bound for degrees $n$ of the form $u^2$ or $u^2+1$ with odd $u$ rests on an appendix that repairs earlier determinant-method lemmas only in sketch form, citing two auxiliary results without a complete proof; if that repair has a gap, bound (1.2) is unproved for those degrees.","fun_headline_variants_meta":{"raw":{"variants":["Polynomials rarely share field discriminants","Many distinct discriminants from bounded-height polynomials","Counting polynomials by discriminant: tight bounds","Few polynomials per discriminant, many discriminants total","Heights and discriminants: sparse collisions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000414,"raw_usage":{"total_tokens":2132,"prompt_tokens":933,"completion_tokens":1199,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":1133}},"tokens_in":549,"tokens_out":1199,"duration_ms":8758,"temperature":1.0,"reasoning_tokens":1133,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T06:02:19.313178+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a degree $n=9$ or $n=10$, compute the number of coefficient vectors $(a_2,\\ldots,a_{n-1})$ in $[-H,H]^{n-3}$ for which $Z^2 - u\\operatorname{Disc}(X^n+\\cdots)$ fails to be absolutely irreducible; if for some square-free $u$ satisfying the non-square conditions this number grows faster than $H^{n-3}$, or the associated solution count exceeds $H^{n-2+\\sqrt{2}+\\varepsilon}$, Lemma 3.3 and Theorem 1.1 would be false. A more direct check is to verify the appendix's asserted finiteness of bad rational specialisations for $n=9$ and $n=10$, since that step is left as a sketch.","supporting_citations":[{"cited_title":"Dietmann, ‘Probabilistic Galois theory’, Bull","cited_arxiv_id":null,"evidence_quote":"Supplies the determinant-method framework and the lemmas (5, 6, 8, 11) that the paper generalises to arbitrary square-free part $u$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the square sieve used in Lemma 3.4 to count polynomials whose discriminant has a given square-free part."},{"cited_title":"Iwaniec and E","cited_arxiv_id":null,"evidence_quote":"Supplies the character-sum and complete/incomplete summation technology underlying Lemmas 2.7 and 2.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the transformation rule $\\operatorname{Disc}(f_{u,v}) = u^{n(n-1)}\\operatorname{Disc}(f)$ used in the averaging argument of Lemma 2.7."},{"cited_title":"Hering, ‘Seltenheit der Gleichungen mit Affekt bei linearem Para meter’, Math","cited_arxiv_id":null,"evidence_quote":"Used in the appendix to bound the finitely many bad specialisations for the exceptional degrees $n=u^2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Used in the appendix to show that the relevant trinomial has symmetric Galois group over a function field, completing the repair."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the bound on solutions of the Pell-type equation needed in the proof of Theorem 1.4 for trinomials."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the previous trinomial counts and the corollary structure that Theorem 1.4 and Corollary 1.7 improve."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the discriminant formula for trinomials used to compute the leading coefficient in the appendix and in Section 5."}],"review_version":1}