Squarefree discriminants of polynomials with prime coefficients

Gian Cordana Sanjaya; Valentio Iverson; Xiaoheng Wang

REVIEW 3 major objections 4 minor 13 references

Squarefree discriminants of polynomials with prime coefficients

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.09697 v2 pith:Q3GJS6GL submitted 2025-01-16 math.NT

Valentio Iverson , Gian Cordana Sanjaya , Xiaoheng Wang This is my paper · ORCID

classification math.NT MSC 11N3711N3511C08

keywords squarefreediscriminantmaximalorderprimecoefficientslocaldensityEulerproductdoublystochasticmatrixequidistributionsievemethod

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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).

What would settle it

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.

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Extended reading notes

Core claim

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%.

Load-bearing premise

A single load-bearing premise runs through the argument: the imported uniformity estimate (1) applies to the maximal-order bad sets and, in the non-monic family, a prime leading coefficient q may be treated as a p-adic unit at every p including p=q.

Editorial extensions

If this is right

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<X^3, b<X^4 and a^4+b^3 squarefree are asymptotically C Li(X^3)Li(X^4) with C≈37.40%.
The equidistribution bounds imply that the nonzero-coefficient residue classes modulo any polynomial u are nearly uniform; in particular, for n≥16 and odd primes the discrepancy is small enough to yield the claimed local-density expansions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Formalized claims in Lean

Claim #1: 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 maxima

/-- @claim 1 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 maxima -/
def central_claim : Prop :=

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

That 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.

The 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.

My 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Theorem 1.1, definitions of N_max_n and N_max_{n,2}] 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 3, opening paragraph and Lemma 3.7 / equation (11)] 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.
[Theorem 1.3 and Sections 4.3-4.4] 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.

minor comments (4)

[Abstract and Section 1] 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'.
[Theorem 1.1, formula for P_max_{n,2}] 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 1, definitions of N_max_{n,2}] 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'.
[Theorem 1.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic constants are computed from self-contained local-density and equidistribution arguments, and the cited uniformity estimates are external theorems.

full rationale

The paper's derivation chain is not circular. Theorem 1.1 is obtained by applying the sieve statement Theorem 1.3, whose main constant is an Euler product of local densities. Those local densities are computed directly in Sections 3 and 4: P_sqf_n,p and P_max_n,p are expressed in terms of counts of lifts of polynomials in U_n(F_p), with the discrepancy delta_{n,p}(d) bounded by the doubly stochastic matrix arguments in Theorems 4.1 and 4.2. No parameter is fitted to the counts N(X) being predicted, and no displayed equation defines a local density in terms of the target asymptotic constant. The Uniformity Estimate (1) is imported from [6,7,12] as a stated external hypothesis; although those papers share authors with the present one, the cited results have independent published proofs and their assumptions do not include the target count, so they are real evidence rather than a self-citation loop. The passage asserting that a non-monic polynomial with leading coefficient not divisible by p behaves like a monic polynomial over Z_p is questionable for p dividing the leading coefficient and may invalidate the non-monic maximal-order statements, but that is a mathematical correctness concern, not a circular reduction: the paper does not define the maximal-order condition in terms of the counts it derives. Thus no circular step is established, and the score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The only fitted numbers are none; all constants are derived from Euler products. The paper assumes unproved uniformity estimates from prior work and, for non-monic maximality, a false local-global principle about the leading coefficient.

assumptions (4)

domain assumption Uniformity Estimate (1) holds for the discriminant and maximality bad sets B_p with some alpha, beta > 0 (cited from [6,7,12]).
Theorem 1.3 reduces the main theorems to this estimate; the paper does not prove or state how [6,7,12] deliver it. [7] is an unpublished preprint by a coauthor.
ad hoc to paper A non-monic polynomial with leading coefficient a prime q behaves, for the global maximal-order question, like a polynomial whose leading coefficient is a p-adic unit at every prime p.
This is used in Section 3.2 to transfer monic local densities to the non-monic family. It is false at p = q, where x is not integral over Z_q.
standard math Dedekind's criterion (Lemma 3.5, from [1]) characterizes maximality of Z_p[x]/(f).
Cited from Ash-Brakenhoff-Zarrabi; used in Section 3.2 to set up the inclusion-exclusion sieve for maximality.
standard math Siegel-Walfisz theorem for primes in arithmetic progressions uniformly for moduli up to a fixed power of log X.
Used in Section 2 for the small-range sum m <= (log X)^eta; constant C1 depends on N and eta.

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Cite this review

Pith. "Pith review of Squarefree discriminants of polynomials with prime coefficients." pith.science (2026). https://pith.science/paper/Q3GJS6GL

@misc{pith2026250109697,
  author       = {Pith},
  title        = {Pith review of: Squarefree discriminants of polynomials with prime coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q3GJS6GL}},
  note         = {Machine review of arXiv:2501.09697}
}

abstract

In this paper, we consider the family of monic polynomials with prime coefficients and the family of all polynomials with prime coefficients. We determine the number of $f(x)$ in each of these families having: squarefree discriminant; $\mathbb{Z}[x]/(f(x))$ as the maximal order in $\mathbb{Q}[x]/(f(x))$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 10 canonical work pages

[12]

G. C. Sanjaya and X. Wang, On the squarefree values of a4 + b3, Mathematische Annalen 386 (2023), 1237–1265

work page 2023
[1]

A. Ash, J. Brakenhoﬀ, and T. Zarrabi, Equality of Polynom ial and Field Discriminants, Experiment. Math. 16 (2007), 367–374

work page 2007
[2]

Bhargava and A

M. Bhargava and A. Shankar, Binary quartic forms having b ounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2) 181(1) (2015), 191-–242

work page 2015
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Bhargava and A

M. Bhargava and A. Shankar, Ternary cubic forms having bo unded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Ann. of Math. (2) 181(2) (2015), 587-–621

work page 2015
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Bhargava and A

M. Bhargava and A. Shankar, The average number of element s in the 4-Selmer groups of elliptic curves is 7, http://arxiv.org/abs/1312.7333

arXiv
[5]

Bhargava and A

M. Bhargava and A. Shankar, The average number of element s in the 5-Selmer groups of elliptic curves is 6, and the average rank is less than 1, http://arxiv.org/abs/1312.7859

arXiv
[6]

Bhargava, A

M. Bhargava, A. Shankar, and X. Wang, Squarefree values o f polynomial discriminants I. Invent. Math. 228(3) (2022), 1037–1073

work page 2022
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Bhargava, A

M. Bhargava, A. Shankar, and X. Wang, Squarefree values o f polynomial discriminants II, https://arxiv.org/pdf/2207.05592

arXiv

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M. Oller, The density of ADE families of curves having sq uarefree discriminant, https://arxiv.org/abs/2306.05961

arXiv
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G. C. Sanjaya, Density of Special Classes of Polynomial s with Squarefree Discriminants. 18
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K. Yamamura, Some analogue of Hilbert’s irreducibilit y theorem and the distribution of algebraic num- ber ﬁelds. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38(1) (1991), 99-–135. 19

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [12]

G. C. Sanjaya and X. Wang, On the squarefree values of a4 + b3, Mathematische Annalen 386 (2023), 1237–1265

work page 2023

[2] [1]

A. Ash, J. Brakenhoﬀ, and T. Zarrabi, Equality of Polynom ial and Field Discriminants, Experiment. Math. 16 (2007), 367–374

work page 2007

[3] [2]

Bhargava and A

M. Bhargava and A. Shankar, Binary quartic forms having b ounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2) 181(1) (2015), 191-–242

work page 2015

[4] [3]

Bhargava and A

M. Bhargava and A. Shankar, Ternary cubic forms having bo unded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Ann. of Math. (2) 181(2) (2015), 587-–621

work page 2015

[5] [4]

Bhargava and A

M. Bhargava and A. Shankar, The average number of element s in the 4-Selmer groups of elliptic curves is 7, http://arxiv.org/abs/1312.7333

arXiv

[6] [5]

Bhargava and A

M. Bhargava and A. Shankar, The average number of element s in the 5-Selmer groups of elliptic curves is 6, and the average rank is less than 1, http://arxiv.org/abs/1312.7859

arXiv

[7] [6]

Bhargava, A

M. Bhargava, A. Shankar, and X. Wang, Squarefree values o f polynomial discriminants I. Invent. Math. 228(3) (2022), 1037–1073

work page 2022

[8] [7]

Bhargava, A

M. Bhargava, A. Shankar, and X. Wang, Squarefree values o f polynomial discriminants II, https://arxiv.org/pdf/2207.05592

arXiv