Bounds for the maximum modulus of polynomial roots with nearly optimal worst-case overestimation

Prashant Batra

REVIEW 1 major objections 3 minor 17 references

Bounds for the maximum modulus of polynomial roots with nearly optimal worst-case overestimation

T0 review · 1 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read An explicit coefficient-only formula bounds every polynomial's largest root within 5 percent of the theoretical floor once the degree reaches 11.

desk verdict A useful near-optimal root bound that survives a numerical typo in the proof; worth a referee after a one-line fix. read the letter →

arxiv 2411.16385 v1 pith:TOLT5NQA submitted 2024-11-25 math.NA cs.NA

Prashant Batra This is my paper · ORCID

classification math.NAcs.NA MSC 65H0415A1812D10

keywords upperboundsforpolynomialrootsaprioriCassiniovalsrootmodulusworst-caseoverestimationcompanionmatrixvanderSluisthresholdFujiwarabound

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 constructs an explicit, coefficient-only upper bound $\Gamma(p)$ for the largest modulus of a root of any monic complex polynomial of degree $n\ge3$, and proves that its worst-case relative overestimation satisfies $\Gamma(p)/\mu(p)\le1.4655\,n$, where $\mu(p)$ is the true largest root modulus. Since any absolute root bound---one using only the coefficient moduli---has worst-case overestimation at least about $1.442\,n$, the new bound is nearly optimal: for $n\ge11$ it is within 5 percent of the theoretical floor, and for $n\ge85$ within 2 percent. The point is that the optimal Cauchy bound is not available as a closed algebraic expression, while $\Gamma(p)$ is a simple formula in the coefficients that can be evaluated immediately. This supplies a guaranteed, choice-free quality assessment that does not depend on selected test polynomials.

What carries the argument

The central object is the family of similarity-scaled companion matrices $C(t)=S(t)C_F S(t)^{-1}$ with $S(t)=\operatorname{diag}(1,t,\ldots,t^{n-1})$, whose eigenvalues are exactly the roots of $p$. The Ostrowski–Brauer theorem places those eigenvalues in the union of Cassini ovals $O_{i,j}(t)=\{z\in\mathbb{C}:|z-c_{ii}(t)|\,|z-c_{jj}(t)|\le r_i(t)r_j(t)\}$, with $r_i(t)$ the reduced row sums of $C(t)$. The parameter is fixed at $t=\tau$ so that the row-sum factors $(1+|a_{k-1}|/\tau^{n+1-k})$ are bounded by $2$, $3$, and $3.15$ according to $k$, letting each oval be enclosed in an explicit circle. Vieta's coefficient estimates then convert those radii into the numerical constant $1.4655$ after dividing by $n\mu(p)$.

What would settle it

Take the extremal family of monic polynomials that attains the lower threshold for $n\ge3$, compute $\mu(p)$ numerically, and evaluate the explicit formula $\Gamma(p)$; if $\Gamma(p)/\mu(p)>1.4655\,n$ for any $n$, Theorem 3.1 is false. The same check can be run as a random search over coefficient profiles because both quantities are directly computable.

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Extended reading notes

Core claim

Let $p(z)=z^n+\sum_{i=0}^{n-1}a_i z^i$ be monic and let $\mu(p)$ be the largest modulus of its roots. With $$\tau=\max\left\{\sqrt[3]{\frac{|a_{n-3}|}{2.15}},\sqrt[4]{\frac{|a_{n-4}|}{2}},\sqrt[5]{|a_{n-5}|},\ldots,\sqrt[n]{|a_0|}\right\},$$ the paper defines $$\Gamma(p)=\max\left\{\sqrt{3.15}\sqrt{\$tau^{2}$+\max\{|a_{n-2}|,2\$tau^{2}$\}},\ \frac{|a_{n-1}|+\sqrt{|a_{n-1}|^2+4(\$tau^{2}$+\max\{|a_{n-2}|,2.15\$tau^{2}$\})}}{2}\right\}$$ and proves $\mu(p)\le\Gamma(p)$ for every monic polynomial of degree $n\ge3$. Theorem 3.1 then bounds the worst-case relative overestimation by $\Gamma(p)/\mu(p)\le1.4655\,n$; the proof uses the estimates $|a_{n-k}|\le\binom{n}{k}\mu(p)^k$ to reduce the coefficient profile to the one number $\varphi=\tau/(n\mu(p))\le0.4264$ and then evaluates the four circle radii that enclose the Cassini ovals. Because the lower threshold (2) of the paper says no absolute root bound can do better than about $1.442\,n$, this explicit $\Gamma(p)$ reaches within 5 percent of optimal for $n\ge11$ and within 2 percent for $n\ge85$.

Load-bearing premise

The explicit formula is only a proven upper bound if the standard eigenvalue-inclusion theorem for Cassini ovals really does trap every root of the scaled companion matrix, together with the row-sum inequalities used to shrink the ovals.

Editorial extensions

If this is right

For any monic polynomial of degree $n\ge3$, $\Gamma(p)$ is an explicit bound computed directly from the coefficient moduli, so it can be evaluated in linear time without locating any roots.
The worst-case overestimation of the largest root modulus is at most $1.4655\,n$, which for $n\ge11$ is within 5 percent of the best possible factor for any absolute root bound, and for $n\ge85$ within 2 percent.
The same bound overestimates the Cauchy bound by at most $\sqrt{9.45}\approx3.07$, and the combined bound $\min\{\Gamma(p),F(p)\}$ stays within twice the Cauchy bound while keeping the near-optimal overestimation of $\mu(p)$.
Because the quality measure is the worst case over all polynomials of fixed degree, the guarantee is choice-free and independent of coefficient distributions or test examples.

Reading between the lines

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

The same mechanism---diagonal scaling of a companion matrix followed by Cassini oval enclosure---could plausibly produce explicit, near-optimal spectral bounds for other structured eigenvalue problems, since the scaling parameter can be tuned to the entry magnitudes.
The constants $2$, $2.15$, $3.15$, and the intermediate estimates $c_k$ are chosen for a short proof; a more careful optimization, especially for small degrees, could lower the constant $1.4655$, and computing exact worst-case ratios for $n=3,\ldots,10$ would test its sharpness.
Worst-case optimality says nothing about typical performance; on coefficient distributions with a few dominant coefficients, $\Gamma(p)$ may be much tighter than $1.4655\,n$, and a numerical benchmark suite would quantify that gap.

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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

Prashant Batra's note is a serious candidate for publication, but the proof of the main theorem has a numerical typo that must be corrected before the claim stands. The specific value c5 as printed (0.4518) is not 1/sqrt[5]{120}; that is actually (1/24)^{1/4}. The correct fifth root gives about 0.3839, which is below c3 ≈ 0.4264. With the printed value, φ ≤ max{c3,c5} = 0.4518, and the displayed maximum 1.4655 does not follow. This is a genuine gap in the written proof, but it is a one-line arithmetic fix; the intended argument is sound.

What is genuinely new: the explicit bound Γ(p) and the worst-case overestimation 1.4655n. The construction is standard—companion matrix scaled by τ, Cassini ovals, then enclosing each oval by a circle—but the tuning of the constants (2.15, 2, 3.15) to get within 2 percent of van der Sluis's lower bound is careful and not fitted to examples. The comparison with the Cauchy bound in Section 3.2 is also a nice extra. The paper is written honestly; the references are appropriate, and the only self-citation is to the prior 1.58n bound, which is entirely reasonable.

Other soft spots are minor. The notation τ as both a parameter and the max expression is slightly confusing. The τ=0 case is dispatched in one sentence; it should at least be stated that the quadratic factor can be handled directly. There are no numerical experiments, but for a worst-case theoretical bound that is fine; the van der Sluis benchmark is the right measure.

Overall, the core idea holds up after the c5 correction. The paper deserves a serious referee: it answers a question that has been open since 1970 in a nearly sharp way, with an explicit, computationally cheap bound. I would send it out, but I would require the author to correct the c5 computation and to double-check the few surrounding inequalities. For a reader in numerical analysis or polynomial root location, this is worth a close look.

Referee Report

1 major / 3 minor

Summary. The paper proposes an explicit, coefficient-based upper bound Γ(p) for the largest root modulus of a monic complex polynomial of degree n≥3. The construction uses a similarity-scaled Frobenius companion matrix, the Ostrowski-Brauer Cassini oval inclusion theorem, and circle enclosures of the resulting ovals to obtain a closed-form bound involving only the last few coefficients. The main result, Theorem 3.1, claims that Γ(p)/μ(p) ≤ 1.4655n, which lies within 5% of van der Sluis's optimality threshold for n≥11 and within 2% for n≥85. The paper also compares Γ(p) with the Cauchy bound and Fujiwara's bound, and proposes a min with Fujiwara's bound as a combined estimate.

Significance. If the stated theorem is correct, this is a strong result: it gives a simple, low-cost algebraic bound whose worst-case relative overestimation is close to the information-theoretic lower bound for absolute root bounds, improving on the classical Fujiwara bound's factor 2n and on earlier modifications from the literature. The proof is explicit and the constants 2.15, 2, and 3.15 are transparent design choices rather than fitted parameters. The paper's contribution is therefore significant for the theory of polynomial root bounds, provided the numerical issue identified below is repaired.

major comments (1)

[Section 3.1, proof of Theorem 3.1] The numerical value assigned to c5 is incorrect. The text defines c5 := 1/5√120 ≈ 0.4518. If this means 1/(120)^{1/5}, the value is approximately 0.3839; the number 0.4518 is instead (1/24)^{1/4}. With the printed value c5 = 0.4518, the line φ ≤ max{c3, c4, c5, ..., cn} = c3 ≈ 0.4264 is false, since c5 > c3. Substituting φ = 0.4518 into the four displayed terms of Γ(p)/(nμ(p)) gives approximately 1.389, 1.489, 1.445, and 1.477, so the maximum is about 1.489 rather than the claimed 1.4655. This is load-bearing because the theorem's constant depends on the maximum being attained at c3. Replacing 0.4518 by the correct 0.3839 makes c3 the maximum and reproduces the stated 1.4655n; thus the theorem is repairable, but the proof as printed does not establish it.

minor comments (3)

[Section 2, proof of Proposition 2.1, case 1] The displayed formula for sqrt(r_i(τ) r_j(τ)) assumes both rows i and j have the leading subdiagonal term t; for i=1 or j=1 the reduced row sum is |a0|/τ^{n-1} rather than t + |a_{i-1}|/τ^{n-i}. The subsequent estimate remains valid because τ ≥ |a0|^{1/n} implies r1/τ ≤ 1, but this should be stated explicitly to avoid an apparent minor gap.
[Section 3.1] The notation c5 := 1/5√120 is ambiguous in print; the intended expression should be written as 1/√[5]{120} or (120)^{-1/5}, and the decimal value should be corrected accordingly.
[Section 1.1] The statement that 42 of the 45 bounds in [9, Chap. 1] exceed the optimal threshold, with exceptions A5, A6 and B5, would benefit from a precise table or list reference, since that enumeration is not immediately verifiable from the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the root bound is an explicit parameter-free functional derived from standard external inclusion theorems and checked against an external optimality benchmark.

full rationale

The paper's derivation is self-contained in the relevant sense. Proposition 2.1 derives the bound Gamma(p) from the Ostrowski-Brauer Cassini oval inclusion theorem applied to a similarity-scaled companion matrix; this is an external standard result and is not obtained from the paper's own concluding bound. The constants 2.15 and 2 appearing in tau are explicit design choices in the proof, not fitted parameters, and no coefficient data are used to tune them. Theorem 3.1 then proves the relative overestimation estimate via Vieta's inequalities and elementary numerical bounds; the comparison threshold (2) is van der Sluis's external theorem. The only self-citation, Batra-Mignotte-Stefanescu [1], is used as background comparison for a modified Lagrange bound and is not load-bearing for Theorem 3.1. No equation in the paper defines its target quantity in terms of itself, and no fitted input is renamed as a prediction. The apparent arithmetic typo in the printed value of c5 in Section 3.1 is a correctness concern rather than a circularity, and it does not change the circularity assessment.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard results (companion matrices, Cassini oval inclusion, Vieta, van der Sluis lower bound) plus a few transparently chosen constants. No quantity is fitted to numerical data, and no new entities are postulated.

free parameters (3)

tau denominator for the cubic coefficient term = 2.15
Chosen by hand to control the row-sum factor (1+|a_{n-3}|/tau^3) <= 1+2.15 = 3.15. It is a design constant, not fitted to root data.
tau denominator for the quartic coefficient term = 2
Chosen so that the fourth-root contribution is c4 = (1/48)^(1/4) approximately 0.3799, below the dominant c3 approximately 0.4264. Explicit in equation (3).
row-sum caps in equation (4) = 2, 3, and 3.15
Derived from the denominators above. These caps determine the compact Gamma(p) formula and the final worst-case constant 1.4655.

assumptions (4)

standard math Companion matrix eigenvalues coincide with polynomial roots.
Used in Section 2 to transfer polynomial root bounding to matrix eigenvalue bounding via C(t) = S C_F S^{-1}.
standard math Ostrowski-Brauer Cassini oval inclusion theorem for eigenvalues of a matrix.
Invoked in Section 2 from reference [17]. It guarantees that eigenvalues of C(t) lie in the union of the sets O_{i,j}(t) defined by reduced row sums.
standard math Vieta coefficient bounds |a_{n-k}| <= mu^k C(n,k) <= mu^k n^k/k!.
Used in Section 3.1 to bound phi = tau/(n mu) and to compute the worst-case constant 1.4655.
standard math van der Sluis lower bound that the Cauchy bound minimizes worst-case relative overestimation among absolute root bounds.
Used in Section 1.1 as the optimality benchmark: limsup rho(p)/mu(p) = 1/(2^(1/n)-1). It is an external theorem, not derived in this paper.

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Cite this review

Pith. "Pith review of Bounds for the maximum modulus of polynomial roots with nearly optimal worst-case overestimation." pith.science (2026). https://pith.science/paper/TOLT5NQA

@misc{pith2026241116385,
  author       = {Pith},
  title        = {Pith review of: Bounds for the maximum modulus of polynomial roots with nearly optimal worst-case overestimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TOLT5NQA}},
  note         = {Machine review of arXiv:2411.16385}
}

read the original abstract

Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum root-modulus we establish a simple, nearly optimal result.

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [5]

On eﬃcient computation and asymptotic sharpness of Kala ntari’s bounds for zeros of polynomials

Yi Jin. On eﬃcient computation and asymptotic sharpness of Kala ntari’s bounds for zeros of polynomials. Math. Comp., 75(256):1905–1912, 2006

work page 1905

[2] [1]

Improvements of Lagrange’s bound for polynomial roots

Prashant Batra, Maurice Mignotte, and Doru S ¸tef˘ anescu. Improvements of Lagrange’s bound for polynomial roots. J. Symbolic Comput. , 82:19– 25, 2017

work page 2017

[3] [2]

The theory of equations: With an introduction to the theory of binary alge braic forms

William Snow Burnside and Arthur William Panton. The theory of equations: With an introduction to the theory of binary alge braic forms. Dover Publications, Inc., New York, 1960. 2 volumes

work page 1960

[4] [3]

¨Uber die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung

Fujiwara, M. ¨Uber die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung. Tˆ ohoku Math. Journal, 10:167– 171, 1916

work page 1916

[5] [4]

On the inﬁnite families of zero bounds of complex polynomials

Shun-Pin Hsu and Feng-Chi Cheng. On the inﬁnite families of zero bounds of complex polynomials. J. Comput. Appl. Math. , 256:219–229, 2014

work page 2014

[6] [6]

An inﬁnite family of bounds on zeros of analytic functions and relationship to Smale’s bound

Bahman Kalantari. An inﬁnite family of bounds on zeros of analytic functions and relationship to Smale’s bound. Math. Comp., 74(250):841– 852, 2005

work page 2005

[7] [7]

Bounds for the zeros of polynomials from compression matrix inequalities

Fuad Kittaneh, Mohammad Odeh, and Khalid Shebrawi. Bounds for the zeros of polynomials from compression matrix inequalities. Filomat, 34(3):1035–1051, 2020

work page 2020

[8] [8]

Marden, M. M. The Geometry of Polynomials . AMS, Providence, Rhode Island, second edition, 1966

work page 1966