REVIEW 2 major objections 5 minor 22 references
Law of large numbers for the discriminant of random polynomials
T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read For random polynomials, the discriminant obeys a sharp law of large numbers: with high probability $|\Delta(f_n)| = n^{2n}e^{-{\sf D}_* n(1+o(1))}$ for an explicit universal constant ${\sf D}_* \approx 5.92947$.
desk verdict Genuinely useful proof architecture and a nice symmetrized Mahler-measure representation, but the explicit constant D* is wrong: a factor-4 slip in the conditional-variance computation changes D* from roughly 5.93 to about 1.34. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing identity is a symmetrized representation of the discriminant (Claim 2 and Lemma 5): for a degree-$n$ polynomial $P$ with simple roots and no root on the unit circle, $$\log|\$\Delta$(P)| = \sum_{|\$\alpha$|<1}\log|P'(\$\alpha$)| + \sum_{|\$\alpha$|>1}\log\left|\frac{P'(\$\alpha$)}{\$alpha^{{n-2}}$}\right| + (n-2)\$int_0^{1}$ \log|P($e^{{2\pi i\theta}}$)|\,d\$\theta$ .$$ The third term is the logarithmic Mahler measure. For the random polynomial $f_n$, the reciprocal polynomial $z^n f_n(z^{-1})$ has the same law as $f_n$, so the two derivative sums are equal in distribution (Claim 6); this distributional reciprocal symmetry is what cancels the non-universal contribution of roots far from the unit circle. Proposition 3 shows the Mahler measure concentrates at $-\gamma/2$, and Proposition 4 shows each derivative sum concentrates at $c_*$; the latter is the technical bulk, carried by (i) removing the annulus away from the unit circle and the contribution of roots where $|f'_n|$ is atypically small, using companion-paper lemmas; (ii) a net of mesh $n^{-1-\beta}$ in the near-circle annulus $\{1-\log^3 n/n \le |z| \le 1\}$, where the linear approximation at net points predicts the roots; (iii) a Berry-Esseen Gaussian comparison showing the net sum is concentrated and close in law to the Gaussian coefficient case; and (iv) an exact Kac-Rice computation of the Gaussian mean, whose limiting density $\Psi(t)$ carries the same radial prefactor $1/t^2 - 1/\sinh^2 t$ as the expected root density of Kac polynomials. The constant assembles as $-{\sf D}_* = -\gamma/2 + 2c_*$, with $c_* = 1-\gamma + \int_0^\infty \Phi(t)\,dt$.
What would settle it
Run $f_n$ with standard Gaussian or Rademacher coefficients for $n$ large (say $10^4$ or beyond) and compute $(1/n)(\log|\Delta(f_n)| - 2n\log n)$ over many samples: the law of large numbers predicts convergence to approximately $-5.92947$, so a stable deviation would refute the theorem. A sharper target is the annulus bound behind Proposition 4: for any sub-Gaussian law with $P(\xi=0)=0$, the probability that $f_n$ has a root in $\{1-\log^3 n/n \le |z| \le 1\}$ with $|f'_n(\alpha)| \le n^{5/4}/\log^4 n$ is claimed to tend to zero (Claim 15); measuring this probability directly and finding it bounded away from zero would break the argument even if the final constant happened to agree.
Extended reading notes
Core claim
The paper's central discovery is that the logarithm of the discriminant of a random Kac polynomial concentrates on a scale far below its size, with a limiting value composed of exactly computable pieces. Theorem 1 states that $\frac{1}{n}(\log|\Delta(f_n)| - 2n\log n)$ converges in probability to $-{\sf D}_*$, where ${\sf D}_* = \frac{\gamma}{2} - 2(1-\gamma) - 2\int_0^\infty \Phi(t)\,dt \approx 5.92947$, with $\gamma$ Euler's constant and $\Phi$ the explicit function in (8). The constant assembles linearly: the Mahler measure of $f_n$ contributes $-\gamma/2$ per degree (Proposition 3), and each of the two derivative sums — over roots with $|\alpha|<1$ and their reciprocal partners with $|\alpha|>1$ — contributes $c_* = 1-\gamma + \int_0^\infty \Phi(t)\,dt$ (Proposition 4). The equality in law of the two sums (Claim 6) is what cancels the non-universal, distribution-dependent contribution of roots far from the unit circle, leaving a universal constant. The authors also record that the no-atom-at-zero assumption is inessential: the asymptotic holds except on the event, of asymptotic probability $P(\xi=0)^2$, that the polynomial has a double root and the discriminant is exactly zero.
Load-bearing premise
The load-bearing premise is that two lemmas from the companion paper hold for every sub-Gaussian coefficient law: a root very close to the unit circle is only rarely accompanied by an abnormally small derivative, and the limiting infinite power series almost surely has no double zero; the paper cites these lemmas rather than proving them, and if either fails for some admissible law, the concentration of the derivative sum and the value of ${\sf D}_*$ collapse.
Editorial extensions
If this is right
- With high probability $|\Delta(f_n)| = n^{2n}e^{-{\sf D}_* n(1+o(1))}$: the discriminant is enormous, and its logarithm is concentrated on a window of size $o(n)$ around $2n\log n - {\sf D}_* n$.
- The $2n\log n$ leading term is universal across all mean-zero, variance-one sub-Gaussian coefficient laws, correcting the linear-growth heuristic from earlier numerics on discriminants of random integer polynomials.
- Relaxing $P(\xi=0)=0$: except on the event, of asymptotic probability $P(\xi=0)^2$, that $f_n$ has a double root — where $\Delta = 0$ by definition — the same asymptotic holds.
- The proof's Gaussian comparison also gives a template for the next order: the authors note that for Gaussian coefficients the variance of $\log|\Delta(f_n)|$ is expected to grow linearly, with asymptotically normal fluctuations.
- The paper frames the theorem as a modest justification for numerical observations that discriminants of random integer polynomials are typically enormous and concentrated, the regime in which the Galois group is generically not the alternating group.
Reading between the lines
- A testable extension the paper does not pursue: the rate of convergence to $-{\sf D}_*$ is governed in the proof by the net error $n^{-\beta/5}$, so coefficient laws with a near-atom at zero — legal sub-Gaussian laws with tiny $P(|\xi|<\varepsilon)$ — should show visibly slower approach to the limit, which one could check numerically.
- The same symmetrized decomposition, Mahler measure plus two law-equal derivative sums, should transfer to other resultant-type quantities, such as the resultant of two independent random Kac polynomials or the discriminant of a polynomial with a planted root, yielding explicit exponential scales of the same shape.
- Because the limiting density $\Psi$ carries the Kac-polynomial radial root-density prefactor $1/t^2 - 1/\sinh^2 t$, the constant ${\sf D}_*$ is effectively an integral over root statistics; joint with the companion root-separation results, this suggests the next-order fluctuations of $\log|\Delta|$ should be expressible through the same Gaussian process and likely of order $n^{1/2}$ for Gaussian c
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies random Kac polynomials f_n(z)=∑_{k=0}^n ξ_k z^k with i.i.d. mean-zero, variance-one sub-Gaussian coefficients and proves a law of large numbers for the logarithmic discriminant: (1/n)(log|Δ(f_n)|−2n log n) converges in probability to −D_* for an explicit universal constant D_*>0, equivalently |Δ(f_n)|=n^{2n}e^{−D_*n(1+o(1))}. The proof combines a symmetrized representation of the discriminant (Lemma 5 and Claim 6), a concentration result for the Mahler measure (Proposition 3), a lengthy reduction of the derivative sum over roots to a sum over a net (Section 4), Gaussian comparison via Berry–Esseen (Section 7), and a Kac–Rice computation of the limiting expectation (Section 5). The main theorem is reduced to Propositions 3 and 4.
Significance. If correct, the result would be a valuable and non-obvious universal law of large numbers for the discriminant of random Kac polynomials, with an explicit constant; it would also provide a quantitative justification for numerical observations in the irreducibility literature. The paper contains a clean reduction of the main theorem to two concentration statements, and the symmetrized discriminant representation is an attractive idea. However, the explicit constant computation in Section 5.3 contains a factor-four error that changes the value of D_*, and several load-bearing estimates are imported without proof from the authors' companion preprint. These issues are fixable, and the overall approach is defensible, so the paper merits a major revision rather than rejection.
major comments (2)
- [§5.3, Eqs. (51)–(52), Claim 26] The conditional variance of G'_n(t) given G_n(t)=0 is computed incorrectly. Since G_n(t)=n^{-1/2}∑ γ_k e^{-tk/n}, one has E|G'_n(t)|^2=s''_n(t)/4 and E[G_n(t)\,\overline{G'_n(t)}]=s'_n(t)/2, so the Schur complement is (s''_n-(s'_n)^2/s_n)/4, not s''_n-(s'_n)^2/s_n. Consequently Eq. (46) should read Ψ(t)=(1/t^2−1/sinh^2 t)(log(S(t)/4)+1−γ)/8, and both c_* in Eq. (9) and D_* in Eq. (10) must be recomputed. The numerical value D_*≈5.92947 stated in Section 1 is therefore not the value proved by the manuscript. This is a load-bearing error in the statement of Theorem 1, although the general structure of the proof may survive with a corrected constant.
- [§3–§4, Claims 13, 15, Lemma 14] The concentration estimates that exclude roots where the derivative is small are imported from the companion preprint [19] without statements of the imported results. In particular, Claim 13 and Claim 15 are direct consequences of [19, Lemma 4.3], and Lemma 14 uses [19, Claim 3.5]; Theorem 11, used in Lemma 10, is [19, Theorem 1.3]. Because these estimates are essential for reducing the derivative sum to the net sum in Proposition 4, the manuscript is not self-contained and the referee cannot verify Proposition 4 from the present text alone. The authors should either state the needed results precisely, with proofs or precise references to statements in [19], or move the necessary portions of [19] into an appendix.
minor comments (5)
- [Abstract and Theorem 1] Once the constant is corrected, the abstract and introduction should be updated so that the advertised numerical value matches the result actually proved.
- [Notation around Eq. (47)] The quantity \tilde{s}_n(t) used in Eqs. (51)–(52) is not given a displayed definition; defining it explicitly as \tilde{s}_n(t)=s''_n(t)-(s'_n(t))^2/s_n(t) before using it would improve readability.
- [Eq. (8) and Claim 26] The relation between Φ in Eq. (8) and S(t) in Eq. (46) is only implicit; writing Φ(t)=(1/t^2−1/sinh^2 t)\log S(t) explicitly would help the reader track the constant computation.
- [Proof of Claim 16] The Erdős–Turán bound is quoted with a factor 2/π that is not explained; a reference to the exact form used would be helpful.
- [Throughout] There are several typographical issues, including 'discrimiant' and the broken word 'POL YNOMIALS' in the title; these should be corrected in the final version.
Circularity Check
No circularity: D* is computed from a Gaussian Kac–Rice limit, not fitted or defined by the conclusion; same-author companion results are independent inputs, not presuppositions of Theorem 1.
full rationale
Walking the derivation chain: Theorem 1 is assembled from Lemma 5 (exact discriminant/Mahler-measure identity), Claim 6 (reciprocal symmetry makes the inside/outside root sums equal in law), Proposition 3 (Mahler-measure concentration), and Proposition 4 (concentration of the normalized derivative root sum). Proposition 3 is proved in Section 6 by Gaussian comparison and sub-Gaussian tail estimates; Proposition 4 is proved in Sections 3–5 via a net argument, Berry–Esseen comparisons, and a Gaussian Kac–Rice computation. No step defines the target constant in terms of the conclusion: c* in Eq. (9) is not fitted to any data; it is obtained by passing to the Gaussian ensemble, applying the Kac–Rice formula (Eq. 45), and evaluating the limiting Gaussian conditional variance in Claim 26. Eq. (52), together with the integral identity in the proof of Lemma 23, yields c* = 1 − gamma + integral of Phi, and D* is then simply -gamma/2 + 2c* as in Eq. (10). The proof uses only the stated assumptions (i.i.d. mean-zero, variance-one sub-Gaussian coefficients, no atom at zero), not the conclusion of Theorem 1. The only load-bearing same-author input is the companion paper [19] (Theorem 1.3, Lemma 4.3, Corollary 1.2), used to control double zeros and atypically small derivatives. Those are separate root-separation results for the same Kac ensemble and do not presuppose the discriminant law of large numbers or the value of D*. Heavy reliance on a companion paper may be a completeness or correctness risk, but it is not circularity. The alleged factor-4 arithmetic inconsistency in Eqs. (51)–(52) is likewise a correctness concern, not a circularity concern.
Assumptions & free parameters
free parameters (1)
- β (net mesh exponent) =
10^{-3}
assumptions (11)
- standard math Jensen's formula (5) for log |P(re^{2πiθ})|
- standard math Rouché's theorem
- standard math Berry-Esseen theorem for sums of independent random vectors [7, Cor. 17.2]
- standard math Salem-Zygmund maximal inequality [15, Chapter 6, Theorem 1]
- standard math Erdős-Turán equidistribution theorem [10]
- domain assumption Kac-Rice formula for Gaussian analytic functions [2, Theorem 6.4]
- domain assumption Cook-Nguyen universality of the minimum modulus on the unit circle [9, Theorem 1.2]
- domain assumption [19, Corollary 1.2]: P(fn has a double root) tends to (P(ξ0=0))^2
- domain assumption [19, Theorem 1.3]: f∞ has no double zero in the unit disk almost surely
- domain assumption [19, Lemma 4.3]: probability bounds for roots in annuli with small |f'_n|
- domain assumption [19, Claim 3.5]: small-ball estimate for (fn(z), f'_n(z))
Cite this review
Pith. "Pith review of Law of large numbers for the discriminant of random polynomials." pith.science (2026). https://pith.science/paper/SKE26OXP
@misc{pith2026250612206,
author = {Pith},
title = {Pith review of: Law of large numbers for the discriminant of random polynomials},
year = {2026},
howpublished = {\url{https://pith.science/paper/SKE26OXP}},
note = {Machine review of arXiv:2506.12206}
}
abstract
Let $f_n$ be a random polynomial of degree $n$, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $\Delta(f_n)$ denote the discriminant of $f_n$, that is $\Delta(f_n) = A^{2n-2}\prod_{i < j} (\alpha_j - \alpha_i)^2$ where $A$ is the leading coefficient of $f_n$ and $\alpha_1,\ldots\alpha_n$ are its roots. We prove that with high probability $$|\Delta(f_n)| = n^{2n} e^{-{\sf D}_\ast n(1+o(1))}$$ as $n\to \infty$, for some explicit universal constant ${\sf D}_\ast>0$. A key step in the proof is an analytic representation for the logarithm of the discriminant, which captures both the distributional reciprocal symmetry of the random roots and the cancellations this symmetry induces.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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