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Discriminants of Fields Generated by Polynomials of Given Height

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 1909.00135 v2 pith:KMXA23TL submitted 2019-08-31 math.NT

Rainer Dietmann , Alina Ostafe , Igor E. Shparlinski This is my paper · ORCID
classification math.NT MSC 11R2911R3211N3611L4011C08
keywords fielddiscriminantspolynomialsquare-freepartdeterminantmethodsquaresievecharactersumstrinomialsnumberfields
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $H$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $H^{n-n/(2n-1)+o(1)}$.

Reading between the lines

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

  • 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.
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Editorial analysis

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 6 (Appendix) and Lemma 3.3] 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.
  2. [Section 3.1, Lemma 3.3] 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.
  3. [Section 2.1, Lemma 2.2] 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.
minor comments (4)
  1. [Sections 3.1 and 3.2, notation] 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.
  2. [Section 4, Corollary 1.5] 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].
  3. [Introduction, Theorem 1.4] 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.
  4. [Section 3.2, Lemma 3.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central bounds are proved by determinant-method and square-sieve arguments; self-citations are to published prior work, and the appendix explicitly repairs the cited flawed lemmas rather than assuming them.

full rationale

The paper contains no step in which a claimed prediction is equivalent by construction to its input, no fitted parameter is renamed as a prediction, and no load-bearing conclusion is forced by a definition. Theorem 1.1's two bounds are obtained from Lemma 3.3 (determinant method) and Lemma 3.4 (square sieve), each proved from lemmas developed in Sections 2 and 3. Lemma 3.3 is the most self-citation-dependent step: its proof says "We can now work through [9, Section 5]" and "in exactly the same way as in [9] we derive the desired result," citing Dietmann's earlier paper [9]. This is a genuinely load-bearing self-citation, but it is not circular: [9] is a published, parameter-free prior result with stated assumptions that do not include the target theorem, and the present paper does not merely assume [9] — Section 6 explicitly acknowledges that [9, Lemmas 5 and 6] (and consequently Lemma 8) are false for degrees n = u^2 or n = u^2+1 with odd u, and repairs them via Hering [13] and Smith [31]. That repair is presented only as a sketch, which is a real proof-completeness/correctness limitation for the exceptional n, but it is not circularity. The square-sieve bound (1.3) is derived in the text from Lemmas 2.6–2.9 via Heath-Brown's square sieve, and the trinomial bound (1.4) is reduced to a Pellian equation and a standard divisor/point-counting estimate, again with no fitted parameter or prediction-by-construction. The appendix's admission of an omitted-proof sketch should be weighed as a correctness risk, not as evidence that the derivation reduces to its own inputs.

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

The central claim rests on standard analytic number theory tools (Deligne bound, square sieve, PNT) and on prior results by the authors and others. The most fragile items are the un-reproved external results used in the Section 6 correction. No fitting parameters or invented entities are introduced.

assumptions (4)
  • standard math Deligne's bound on exponential sums over finite fields (Iwaniec-Kowalski [16, Section 11.11]) applies to the two-variable sums in Lemma 2.7 because the leading homogeneous form is nonsingular.
    Used in the proof of Lemma 2.7 to obtain (2.11), the O(p) bound for the inner exponential sum.
  • domain assumption Dietmann's [9, Corollary 1 and Lemma 11] are correct as needed for Lemmas 3.1 and 3.2.
    Lemma 3.1 is deduced from [9, Corollary 1] with Lemmas 2.2-2.3; Lemma 3.2 reduces to [9, Lemma 11]. These are prior published results not reproved here. The appendix corrects only [9, Lemmas 5, 6, 8], so Corollary 1 and Lemma 11 are assumed sound.
  • domain assumption Hering's Satz 1 [13] and Smith's theorem [31] as invoked in the Section 6 amendment are correct.
    The repair for n = u^2 and n = u^2+1 relies on these external results to control the number of bad coefficients; the amendment is sketched and does not prove them.
  • standard math Prime number theorem for dyadic intervals: π(2z)-π(z) ≫ z/log z.
    Used in Lemma 3.4 to control the square sieve's main term and the choice of z.

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Pith. "Pith review of Discriminants of Fields Generated by Polynomials of Given Height." pith.science (2026). https://pith.science/paper/KMXA23TL

@misc{pith2026190900135,
  author       = {Pith},
  title        = {Pith review of: Discriminants of Fields Generated by Polynomials of Given Height},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KMXA23TL}},
  note         = {Machine review of arXiv:1909.00135}
}
abstract

We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant, improving previous results of I. E. Shparlinski (2010).

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