{"id":"e5408aa7-d2b4-4f14-80f3-aa4a94390eab","arxiv_id":"2501.09697","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives asymptotic formulas for prime-coefficient polynomials with squarefree discriminant, with the non-monic maximal-order portion of Theorem 1.1 invalid as stated.","lead":"The paper counts monic and non-monic polynomials whose coefficients are all prime and whose discriminant is squarefree, giving asymptotic formulas with explicit constants. A separate count for when the coefficient ring is the full ring of integers is claimed, but the non-monic version of that count is wrong.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-monic maximal-order statements of Theorem 1.1 are false: a prime leading coefficient q makes x non-integral over Z at p=q, so Z[x]/(f) is never an order and the positive main term for N_max_n(X) cannot hold.","rationale":"The reader's weakest assumption is exactly the load-bearing concern: non-monic polynomials are treated as if the leading coefficient were a p-adic unit for every p, but the leading coefficient is a prime q and fails to be a unit at p=q. This is not a technical gap that a stronger uniformity estimate could fix; it makes the non-monic maximal-order statement false as stated. The monic maximality results and the squarefree-discriminant results may still be correct, since for fixed p the diagonal event a0=p has asymptotic density zero in the prime-input distribution, but the maximality condition fails on that entire diagonal, and the union over p is the full non-monic family. The secondary concern about the imported Uniformity Estimate (1) is real but secondary; the primary obstruction is sufficient for rejection. Since the reader already reached REJECT for the same reason, no verdict change is needed.","tokens_in":16993,"tokens_out":18382,"duration_ms":213218,"concrete_test":"Brute-force check for n=2 at X=100: enumerate all triples of primes (a0,a1,a2) with each < X, and for each f = a0 x^2 + a1 x + a2 test whether Z[x]/(f) is an order, e.g. by checking whether x is integral over Z or, equivalently, whether the module is finitely generated. The count of polynomials satisfying the maximal-order condition is 0 for every X, whereas Theorem 1.1 predicts a positive main term C_max_2 Li(X)^2; this directly falsifies the non-monic maximality claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive failure is in the transfer from monic p-adic densities to the non-monic family. After Theorem 1.3 the paper states that 'a non-monic polynomial with leading coefficient not divisible by p behaves just like a monic polynomial over Zp.' This is only true for primes p not dividing the leading coefficient. But every f in V_n(X) has leading coefficient a0 = q, a prime, so at p = q the leading coefficient is 0 mod q. For such f the element x satisfies q x^n + ... = 0 but no monic integer polynomial, so x is not integral over Z; Z_q[x]/(f) is not a finitely generated Z_q-module, and Z[x]/(f) is not an order in Q[x]/(f). Consequently the maximal-order condition fails for every non-monic f, so N_max_n(X) = 0 and N_max_{n,2}(X) = 0 for all X. The positive constants C_max_n and C_max_{n,2} asserted in Theorem 1.1 are therefore impossible. The sieve does not repair this: the event 'a0 = p' is small for each fixed p, but the union over p = a0 is the whole non-monic family. The density computation in Section 3.2, especially Lemma 3.7 and equation (11), is carried out in Un(Zp), the space of monic polynomials with all coefficients units; it is not the local density of the non-monic maximality condition at primes p dividing a0. If one tried to define the true bad set B_p to include v_p(a0)>0, its p-adic measure would be 1/p rather than p^{-c} with c>1, so Theorem 1.3 would not apply.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two families of integer polynomials with all coefficients prime: monic polynomials of degree n and all polynomials of degree n (including non-monic ones). For each family it claims asymptotic formulas for the number of polynomials with squarefree discriminant and for the number whose quotient Z[x]/(f) is the maximal order in Q[x]/(f). The main engine is a sieve theorem (Theorem 1.3) that converts p-adic bad sets into counts of prime-coefficient tuples, relying on uniformity estimates imported from [6,7,12] and on local density calculations based on equidistribution of polynomials with nonzero coefficients over finite fields, estimated using doubly stochastic matrices.","tokens_in":17340,"tokens_out":19333,"duration_ms":204686,"significance":"If the monic statements are correct, the paper gives a worthwhile application of the uniformity estimates of Bhargava--Shankar--Wang and Sanjaya--Wang to prime-constrained coefficient families, and the finite-field discrepancy estimates in Theorems 4.1 and 4.2 are elegant and potentially reusable. However, the non-monic maximal-order assertions in Theorem 1.1 are not merely unproved but false, so the advertised central result as stated cannot stand.","major_comments":[{"comment":"For every f in V_n(X), the leading coefficient is a prime q > 1, so f is not monic. The quotient ring Z[x]/(f) is then not finitely generated as a Z-module, because the leading coefficient q is not a unit of Z, and hence Z[x]/(f) is not an order in Q[x]/(f). Therefore Z[x]/(f) is never a maximal order: N_max_n(X) = 0 for all X, and N_max_{n,2}(X) is at most the number of f with a0 = 2, which is O(X^n/(log X)^n). This directly contradicts the asserted main terms of size X^{n+1}/(log X)^{n+1} with positive constants C_max_n and C_max_{n,2}.","section":"Theorem 1.1, definitions of N_max_n and N_max_{n,2}"},{"comment":"The transfer statement on p.3, 'a non-monic polynomial with leading coefficient not divisible by p behaves just like a monic polynomial over Z_p', holds only for primes p not dividing the leading coefficient. In the family V_n(X), every f has a prime p = a0 dividing its leading coefficient, and at that prime the local maximality condition fails for every such f. This failure is not captured by the computations in Un(Zp) in Lemma 3.7 and equation (11), which apply only where all coefficients are units. The true bad set at p must include the case v_p(a0) > 0, and on that set the maximality density is zero, so the hypothesis lambda_p(B_p) << p^{-c} with c > 1 of Theorem 1.3 is not satisfied for the non-monic maximality problem.","section":"Section 3, opening paragraph and Lemma 3.7 / equation (11)"},{"comment":"The application of Theorem 1.3 to the sets used in Theorem 1.1 requires the Uniformity Estimate (1) for the specific bad sets B_p, including the maximality bad sets. The paper only says that the results follow from the tail estimates in [6,7] (and [12] for Theorem 1.2); it does not state which theorem in those papers supplies (1) for the maximality sets, nor does it verify that the maximality bad sets satisfy the hypotheses. Since the maximality condition is not simply of the form p^2 | F(a) for a fixed polynomial F, a precise verification or citation is needed; as written, the monic maximal-order proof rests on an unstated uniformity assumption.","section":"Theorem 1.3 and Sections 4.3-4.4"}],"minor_comments":[{"comment":"The abstract says 'monic polynomials with prime coefficients', but the leading coefficient of a monic polynomial is 1; this should read 'with prime non-leading coefficients'.","section":"Abstract and Section 1"},{"comment":"The displayed formula contains '2pt if n+1 is congruent to 2 mod 4', which appears to be a typographical error for '2 p_t'; please correct the typesetting.","section":"Theorem 1.1, formula for P_max_{n,2}"},{"comment":"The phrase 'maximal order except possibly at 2' is used without a formal definition; it should be stated precisely as 'Z_p[x]/(f) is the maximal order for every prime p different from 2'.","section":"Section 1, definitions of N_max_{n,2}"},{"comment":"The constants C_sqf_n and C_max_n are used for both the monic and non-monic rows of the theorem; the paper should explicitly state that the constants in the two families are the same.","section":"Theorem 1.1"}],"recommendation":"reject","confidential_remarks":"The non-monic maximal-order statements in Theorem 1.1 are false, not just insufficiently justified. The monic portions and the squarefree-discriminant portions may be salvageable, and the local-discrepancy technique is interesting, but the advertised abstract and Theorem 1.1 need substantial revision before the paper could be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is simple: the non-monic maximal-order statements in Theorem 1.1 are wrong. If the leading coefficient is a prime q, then x is not integral over Z at p = q, so Z[x]/(f) is not an order in Q[x]/(f) — it isn't even finitely generated as a Z-module. The maximal-order condition never holds for these f, so N_max_n(X) = 0, not the positive asymptotic the theorem asserts. The paper's own justification — that a non-monic polynomial with leading coefficient not divisible by p behaves like a monic one — only works away from p = q, and the density calculation over unit coefficients misses the prime dividing the leading coefficient. The stress-test note is right, and this is a fatal flaw in the theorem as stated.\n\nThat doesn't mean the paper is worthless. The squarefree-discriminant part is a genuine extension to prime-coefficient families, and the local-density computation via doubly stochastic matrices is new and clever. Theorem 1.3, the sieve for prime inputs, is clean and should be useful. The monic maximality asymptotics also look internally consistent. These are real contributions.\n\nThe secondary concern is the reliance on uniformity estimates from [6,7,12], including an unpublished arXiv preprint, without proof. That's a standard division of labor, but it does make the monic results conditional on those estimates covering the relevant bad sets. It's not circularity, though.\n\nMy recommendation: don't accept the paper as is, but don't desk-reject it either. Send it to a referee who can verify the squarefree and monic parts and require the authors to remove or correct the non-monic maximality claims. The real content is worth a revision.","headline":"The non-monic maximal-order half of Theorem 1.1 is false; the squarefree-discriminant and monic-maximality results are the real contributions and deserve a serious referee after the maximality claims are fixed.","tokens_in":17865,"tokens_out":3420,"would_cite":false,"duration_ms":35654,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11N35","11C08"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every degree n≥2, the paper derives Euler-product asymptotic formulas for the number of prime-coefficient polynomials whose discriminant is squarefree or whose quotient ring is the maximal order.","keywords":["squarefree discriminant","maximal order","prime coefficients","local density","Euler product","doubly stochastic matrix","equidistribution","sieve method"],"falsifier":"Enumerate all $2^{16}$ polynomials of degree 16 over F_3 with coefficients in {1,2}, and for every monic u∈F_3[x] of degree 1≤d≤8 not divisible by x and every residue class α, compute the discrepancy |#A_n(u;α)/$2^{16}$ - 1/3^d|. If any value exceeds the bound of Theorem 4.1, namely (1/3^d)((3/2)^d-1)^{⌊16/(2d)⌋}, the equidistribution engine behind the local densities is false; a single violation would invalidate the discrepancy estimates on which Theorems 3.1 and 3.2 rely.","tokens_in":16749,"feed_emoji":"🔢","tokens_out":10817,"duration_ms":101704,"temperature":0.7,"pith_summary":"This paper derives asymptotic formulas for the number of degree-n polynomials whose coefficients are all prime and whose discriminant is squarefree, and for the number for which the ring Z[x]/(f(x)) is the maximal order, i.e. the full ring of algebraic integers of the field Q[x]/(f(x)). The monic and non-monic families are treated separately, with main terms given by Euler-product constants times products of logarithmic integrals and error terms of size $X^{{n+1}}$/(log X)^A (or the analogous monic weights). The constants are built from local densities computed prime by prime; for odd primes the densities are given up to explicit discrepancies, and for p=2 they are exact and depend on elementary arithmetic conditions on n. A sympathetic reader would care because the result converts a sparse, prime-constrained counting problem into a product of local factors, showing that the limiting squarefree density is about 67.69% and the maximal-order density about 85.26% once the even prime is excluded.","feed_headline":"Squarefree discriminants of prime-coefficient polynomials now counted","feed_subtitle":"Asymptotics are Euler products times logarithmic integrals; maximal-order counts included.","key_machinery":"The load-bearing object is the directed graph G_u whose vertices are F_p[x]/(u) and whose edges are α→αx+c with c∈F_p^×; a length-n path from 0 to β counts the coefficient tuples (a_1,...,a_n)∈(F_p^×)^n with $a_1x^{{n-1}}$+...+a_n≡β mod u. Its normalized adjacency matrix is doubly stochastic, so powers of the deviation from the uniform matrix can be bounded by matrix-norm arguments; these bounds define the discrepancy δ_{n,p}(d), which controls how far the residue classes of nonzero-coefficient polynomials are from uniform. Inserting those bounds into inclusion-exclusion sums over u yields the local densities, with exact generating-function identities such as ∑_{u monic, x∤u} μ(u)/$p^{{2 deg u}}$=p/(p+1) supplying the main terms. At p=2 the same inclusion-exclusion reduces to a product over irreducible factors of x^t-1, evaluated by cyclotomic factorization into irreducibles of degree od(2).","core_discovery":"The paper claims that, for each n≥2, the number N_n^sqf(X) of degree-n polynomials with all coefficients prime and squarefree discriminant equals C_n^sqf times the product of Li(X) over the n+1 coefficients, plus an error of size $X^{{n+1}}$/(log X)^A, where C_n^sqf is a convergent Euler product of local densities; the monic count has the same shape with a product of Li(X^i), i=1,...,n. The maximal-order counts N_n^max(X) and $N_n^{{m,max}}$(X) obey analogous asymptotics with constants C_n^max and $C_n^{{max,2}}$. For odd primes and n≥16 the local densities have explicit expansions, such as P_{n,p}^{sqf}=1-(3p-1)/(p(p+1)^2)+O(min{(p/(p-1)^2)^{5/2},(p/(p-1)^2)^{⌊$\\sqrt$(log n/log p)⌋/2}}), while at p=2 the paper gives exact values depending on the parity of n and on the order of 2 modulo divisors of n+1. A separate theorem gives the analogous asymptotic for pairs of primes (a,b) with a<$X^{3}$, b<$X^{4}$ and $a^{4}$+$b^{3}$ squarefree, with constant C=∏_p(1-1/($p^{2}$-p))≈37.40%.","pith_inferences":["If the non-monic leading-coefficient issue at p=q cannot be repaired, the non-monic maximal-order statement in Theorem 1.1 is likely false as stated, while the squarefree count may survive because the discriminant of the mod-q reduction can still be treated directly.","The directed-graph equidistribution method is not tied to coefficients being prime; it should extend to any coefficient set formed by a union of nonzero residue classes, such as coefficients restricted to quadratic residues.","The exact p=2 formulas are checkable by brute force for small n and would provide a fast way to locate any arithmetic error in the cyclotomic factor count.","The limiting constants suggest that the 'no zero coefficient' restriction, rather than primality itself, is what moves the densities away from the integer-coefficient values; a natural test is to run the same sieve with coefficients restricted to nonzero residues modulo each p."],"forward_implications":["For every fixed degree n≥2, prime-coefficient polynomials with squarefree discriminant have a positive limiting density within the prime-coefficient family, governed by a convergent Euler product of local densities.","In the n→∞ limit excluding p=2, the squarefree local density tends to ∏_{p>2}(1-(3p-1)/(p(p+1)^2))≈67.69%, and the maximal-order local density tends to ∏_{p>2}(1-1/(p^2+p+1))≈85.26%.","The same sieve formalism proves that pairs of primes (a,b) with a