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Dragging the roots of a polynomial to the unit circle

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Pith's one-line read The paper proves two sharp thresholds—one Fourier, one discriminant—control when a self-inversive polynomial's roots land on the unit circle and interlace the roots of unity.

desk verdict The interlace-number framework is genuinely new and mostly careful, but the headline unboundedness result rests on a concrete root-set error in Prop. 7.8 and needs major repair before this is publishable. read the letter →

arxiv 1908.03208 v3 pith:3FI7LJZT submitted 2019-08-08 math.CO math.CV

Arnaldo Mandel , Sinai Robins This is my paper · ORCID
classification math.COmath.CV MSC 12D1026C1030C1511C0811L0314P1052B99
keywords self-inversivepolynomialpalindromicrootsinterlacingofunitypolyhedralfandiscriminantfiniteFouriertransform
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 asks when a self-inversive polynomial, one whose coefficients are symmetric under reversal so its roots come in reciprocal-conjugate pairs, has all roots on the unit circle, and when those roots fall one in each angular sector between consecutive roots of unity. The authors attach to each trimmed such polynomial $p$ a one-parameter family $p_\alpha(x)=\alpha(x^n+1)+p(x)$, with $\alpha$ dragging the roots toward the circle, and define two sharp thresholds: the interlace number $\operatorname{il}(p)$ and the circle number $\operatorname{cn}(p)$. The main claim is that $\operatorname{il}(p)$ is exactly half the largest negative value of $p$ evaluated at the $n$th roots of unity, so one finite Fourier transform of the coefficients settles interlacing, and that $\operatorname{cn}(p)$ is the largest real root of a discriminant built from $p_\alpha$ divided by its common factor with $x^n+1$. The two thresholds always satisfy $\operatorname{cn}(p)\le\operatorname{il}(p)$, and the paper classifies the exact polynomials where equality holds, while showing the ratio can grow without bound as the degree grows.

What carries the argument

The central object is the dragged family $p_\alpha(x)=\alpha(x^n+1)+p(x)$, whose roots move continuously toward the unit circle as $\alpha\to\infty$. The argument is carried by three devices: the discrete Fourier transform of the coefficient vector, since evaluating $p$ at $\omega\in U_n$ is exactly that transform and Theorem 3.7 converts the sign of those evaluations into angle-interlacing with the roots of unity; the Möbius root correspondence, which maps the unit circle minus a point to the real line and turns circle rootedness of a self-inversive polynomial into real rootedness of an associated real polynomial, so that interlacing theorems on the line apply; and the discriminant of the normalized family, whose largest real root locates the first double root and hence the circle number. The interlace certs, the roots of unity attaining the maximum in the interlace formula, index the cones of a complete polyhedral fan in the space of trim palindromic polynomials.

What would settle it

Compute $p_\alpha$ for the degree-4 palindromic polynomial $p(x)=\sqrt{2}(x^3+x)+2x^2$ at a value $\beta$ equal to its circle number, or just above it, and check directly whether $p_\beta$ together with $x^4+1$ admit a common angle interlacing set; the proposition as printed predicts the interlacing holds for every $\alpha\ge\beta$, so finding one $\alpha$ in $(\operatorname{cn}(p),\operatorname{il}(p))$ whose roots fail to share a common angle interlace with $x^4+1$ refutes the universal form of Proposition 6.6.

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Extended reading notes

Core claim

For a trim self-inversive polynomial $p$ of darga $n$ (darga is the sum of the smallest and largest indices of nonzero coefficients), the paper defines $p_\alpha=\alpha(x^n+1)+p(x)$ and proves two exact threshold formulas. The interlace number, the least $\alpha$ beyond which $p_\alpha$ strictly angle-interlaces the $n$th roots of unity, equals $\tfrac12\max\{-p(\omega):\omega\in U_n\}$ (Theorem 4.2); because a polynomial that angle-interlaces a set of $n$ circle points is automatically circle-rooted, this gives a cheap sufficient condition. The circle number, the least $\alpha$ beyond which $p_\alpha$ is circle-rooted for every larger parameter, equals the largest real root of $\operatorname{Disc}\bigl(p_\alpha(x)/\gcd(p(x),x^n+1)\bigr)$ (Theorem 6.9). Interlacing always implies circle rootedness, so $\operatorname{cn}(p)\le\operatorname{il}(p)$, and the paper calls $p$ exact when equality holds; exact polynomials have an interlace cert, a root of unity attaining the maximum, that is a double root at the threshold. The same toolkit reinterprets known coefficient criteria as upper bounds on $\operatorname{il}$, organizes real palindromic polynomials of fixed darga into a complete polyhedral fan according to which root of unity witnesses the interlace number, and shows the quotient $\operatorname{il}(p)/\operatorname{cn}(p)$ can be arbitrarily large in high degree.

Load-bearing premise

The load-bearing premise is that circle-rootedness of $p_\alpha$ for all $\alpha\ge\beta$ can be tested by a single common interlace between $p_\beta$ and $x^n+1$ via the Möbius map and Proposition 2.4; the printed version of this equivalence in Proposition 6.6 makes a stronger for-every-$\alpha$ statement that the proof only establishes at $\beta$ and that already fails for degree-4 examples in Section 8.3.

Editorial extensions

If this is right

  • For any trim self-inversive polynomial with integer coefficients, the interlace number is an algebraic integer, and verifying interlacing with roots of unity reduces to checking $n$ Fourier evaluations rather than factoring or root-finding.
  • Known circle-rootedness criteria in the literature become upper bounds on $\operatorname{il}(p)$, so any family satisfying such a bound is automatically circle-rooted and interlaces $U_n$.
  • The circle number is semi-algebraic in the coefficients and computable from a single discriminant; the parity simplification in Theorem 6.13 cuts the determinant size roughly in half.
  • Polynomials whose interlace cert is $1$ or $-1$ are exact, and exactness is decidable by checking whether the threshold polynomial has a double root.
  • In high degree the ratio $\operatorname{il}/\operatorname{cn}$ is unbounded, so no coefficient-only condition can approximate the circle number uniformly.

Reading between the lines

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

  • A testable extension is that the same minimax Fourier formula might certify interlacing for matrix-valued polynomials or for roots constrained to a finite union of circles, since the threshold is read from finitely many evaluations.
  • The unbounded quotient suggests that for large degree the practical route to circle-rootedness certificates should go through interlacing-certified subfamilies rather than through coefficient-only conditions.
  • The fan-of-certs classification suggests a probabilistic corollary: for random palindromic polynomials, the cone containing the polynomial, and hence which root of unity is the cert, could be studied from the geometry of the interlace simplex.
  • The exact-polynomial condition, equality of the two thresholds with a double root at the threshold, may provide an algebraic certificate of tight interlacing analogous to real-rooted interlacing pairs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper introduces two threshold invariants for a trim self-inversive polynomial p of darga n: the interlace number il(p), the least parameter such that p_α = α(x^n+1)+p strictly angle-interlaces the nth roots of unity, and the circle number cn(p), the least parameter such that p_α is circle-rooted for all larger α. The main results are the Interlace Formula (Theorems 4.2 and 4.3), expressing il(p) as half the maximum of −p on the relevant roots of unity; a discriminant characterization of cn(p) (Theorems 6.7 and 6.9); a polyhedral-fan classification of palindromic polynomials by their interlacing certificates (Section 5); and a study of exact polynomials where il = cn (Section 7). The paper also claims that the quotient il/cn is unbounded as the degree grows (Theorem 7.7) and includes small-degree computations plus several families of examples.

Significance. If the main theorems are correct, the framework is a genuinely useful organizing device: the Interlace Formula reduces interlacing with roots of unity to a finite Fourier transform, the discriminant formula gives an explicit algebraic algorithm for the circle number, and the fan of interlace certificates provides a clean geometric classification. The interlace formula and the small-darga computations are carefully argued and appear reproducible, with no fitted parameters. The discriminant characterization and the exact-polynomial criteria are attractive and likely to be cited. However, the paper's headline unboundedness claim depends on Proposition 7.8, whose proof contains a concrete false statement about the roots of x^n+1 and Q_n^2; as written, Theorem 7.7 is not established. In addition, Proposition 6.6 is stated in a stronger form than its proof supports and is contradicted by a degree-4 example, and Theorem 6.13 has sign errors in the displayed candidate values. These issues require substantive repair before the paper can be recommended for publication.

major comments (3)
  1. [§7, Proposition 7.8] The proof of Proposition 7.8 contains an incorrect description of the roots involved. With n=4m, θ=π/(2n) and ω=e^{iθ}, one has (ω^{2r})^n = e^{irπ} = (−1)^r, so the roots of x^n+1 are the ω^{2r} with r odd, not the full sequence ω^2, ω^4, ..., ω^{2n}. Moreover, F_{4j+1} has roots at x = ω^{±2(4j+1)}, whose exponents are congruent to 2 and 4m−2−8j modulo n, not the listed even exponents ω^2, ω^6, ..., ω^{2n−2}. For n=4, Q_4^2 has only the two roots e^{±iπ/4}, each double, so P_4 + x^4 + 1 does not angle-interlace x^4+1; consequently the appeal to Corollary 6.8 to obtain cn(P_4)=1 is unsupported. Because the lower bound on il(P_n) is the engine of Theorem 7.7, the unboundedness result is not established by this argument and needs a corrected construction or a genuinely different upper bound for cn(P_n).
  2. [§6, Proposition 6.6] The proposition is stated in a stronger form than the proof establishes. The proof uses Proposition 2.4 and root correspondence to show that p_α is circle-rooted for all α≥β is equivalent to S_ω(p_β) being real rooted and having a common interlace with S_ω(x^n+1), i.e. to a common angle interlace between p_β and x^n+1 at the single parameter β. The printed statement instead requires p_α and x^n+1 to have a common angle interlace for every α≥β. That stronger assertion is false for the degree-4 example p(x)=2x^2 from Section 8.3: at β=cn(p)=1, p_β=(x^2+1)^2 has double roots ±i, which cannot have a common angle interlace with x^4+1. The statement should be weakened to the β-level condition actually used, and the dependence of Corollary 6.8 and the circle-number computation on this assertion should be re-examined.
  3. [§6, Theorem 6.13] The displayed formulas for r1 and r2 contain sign errors. As printed, r1 = p(−1)/2 and r2 = p(−1)/2 (n even) or p′(−1)/n (n odd), but the proof identifies the candidate parameters as −p(1)/2, −p(−1)/2, and −p′(−1)/n, respectively. The computations in Section 8.3 use the negative forms (for example r1=−b−1 and r2=b−1 for darga 4), so the theorem as stated cannot be applied directly. The statement must be corrected; otherwise the algorithmic content of the circle-number computation is misstated.
minor comments (3)
  1. [§9.5, Eq. (28)] The displayed formula i l(B_n) = 2^{n−1} cos^n(π/n) − 1 appears to have a sign error: for n=3 it gives −1/2, whereas direct computation from the Interlace Formula gives 3/2. The proof text also omits the constant contribution p(ω) = (1+ω)^n − 2 when applying the Interlace Formula.
  2. [References] Reference [29] lists arXiv:1908.00839 in the citation line but ends with the identifier arXiv:1902.04231; please reconcile the announced and actual identifiers.
  3. [Throughout] There are several small typos and spacing artifacts, including 'wich' for 'which' in the introduction and 'Erhart' for 'Ehrhart' in Problem 9; a careful copyedit would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the interlace and circle numbers are derived from the coefficients via independent theorems, not fitted or self-referentially defined.

full rationale

The central quantities il(p) and cn(p) are defined directly as thresholds from the parametric family p_alpha(x)=alpha(x^n+1)+p(x), independently of each other and of the formulas later derived for them. Theorem 4.2 obtains the finite Fourier formula il(p)=1/2 max{-p(omega)} from the sign-interlacing criterion of Theorem 3.7, whose proof is given in the paper; no coefficient is fitted, and the formula is not assumed in the definition of il. Theorem 6.7 and Theorem 6.9 derive the double-root and discriminant characterizations of cn via the Cayley map and Proposition 2.4, both developed algebraically inside the paper, with the gcd(p, x^n+1) correction handled explicitly. Proposition 3.4 only uses the geometric fact that strict angle-interlacing forces circle-rootedness, so the inequality cn <= il is not an input. The Fan of Interlace Certs repackages the already-proved Interlace Formula but does not redefine il by the fan. The only self-citations are [14] and [29], used for gcd-polynomial examples and an asymptotic refinement; these are separate published results and are not premises of the main theorems. The acknowledged computational restatements of Lakatos-Losonczi and Kwon bounds are translations of external theorems into the framework, not circular predictions. Possible mathematical issues, such as the root-set description in Proposition 7.8, would be correctness defects rather than circularity. Overall the derivation chain is self-contained for its central claims.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters: the definitions involve no fitted constants. The paper relies on standard results (fundamental theorem of algebra, continuity of roots, root-of-unity sums, discriminant subresultants, real-algebraic quantifier elimination, Gamma asymptotics), all stated as background. New objects (interlace cert, circle cert, FOIC) are explicit mathematical definitions, not unexplained postulates, so the invented-entities ledger is empty.

assumptions (6)
  • standard math Fundamental theorem of algebra and continuity of polynomial roots as functions of coefficients.
    Used throughout, e.g., in Propositions 2.3, 2.4 and the root-correspondence arguments in Section 6.
  • standard math Properties of roots of unity and discrete Fourier transforms, including sum identities over U_n.
    Basis for the Interlace Formula in Theorems 4.2 and 4.3 and for the FOIC relations in Proposition 5.4.
  • standard math Discriminant and subresultant theory for detecting double roots and counting real roots.
    Used in the double-root formula (Theorem 6.7), Theorem 6.9, and the subresultant discussion in Section 2.
  • standard math Tarski-Seidenberg quantifier elimination, so the graph of 'largest real root' and the set of polynomials with given gcd are semi-algebraic.
    Invoked in the sketch of Corollary 6.10 and in the semi-algebraic claims of the introduction.
  • standard math Gamma function asymptotics and elementary product estimates, as in Lemma 7.9.
    Used in Lemma 7.10 to estimate D_n and hence the lower bound on BE(n) in Proposition 7.8.
  • standard math Möbius transformation (Cayley map) root correspondence and its preservation of interlacing.
    Stated and proved as Proposition 6.3, then used as a load-bearing tool in Section 6 for the circle number.

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Pith. "Pith review of Dragging the roots of a polynomial to the unit circle." pith.science (2026). https://pith.science/paper/3FI7LJZT

@misc{pith2026190803208,
  author       = {Pith},
  title        = {Pith review of: Dragging the roots of a polynomial to the unit circle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FI7LJZT}},
  note         = {Machine review of arXiv:1908.03208}
}
abstract

Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family of polynomials $p_\alpha$ to each such polynomial $p$, and define $\mathscr{cn}(p)$, $\mathscr{il}(p)$ to be the sharp threshold values of $\alpha$ that guarantee that, for all larger values of the parameter, $p_\alpha$ has, respectively, all roots in the unit circle and all roots interlacing the roots of unity of the same degree. Interlacing implies circle rootedness, hence $\mathscr{il}(p)\geq\mathscr{cn}(p)$, and this inequality is often used for showing circle rootedness. Both $\mathscr{cn}(p)$ and $\mathscr{il}(p)$ turn out to be semi-algebraic functions of the coefficients of $p$, and some useful bounds are also presented, entailing several known results about roots in the circle. The study of $\mathscr{il}(p)$ leads to a rich classification of real self-inversive polynomials of each degree, organizing them into a complete polyhedral fan. We have a close look at the class of polynomials for which $\mathscr{il}(p)=\mathscr{cn}(p)$, whereas in general the quotient $\frac{\mathscr{il}(p)}{\mathscr{cn}(p)}$ is shown to be unbounded as the degree grows. Several examples and open questions are presented.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.