The average number of unit-circle zeros in Stanley’s reciprocal polynomial family
Henry Zweiman
September 9, 2026
Abstract
For a finite set of positive integers, put . We determine the asymptotic average number of unit-circle zeros when is chosen uniformly subject to and reciprocity of . The average is , with zeros counted with multiplicity. More generally, the mean unit-circle zero measure, normalized by the degree, converges to times uniform angular measure. This answers the average-count question posed by Stanley. The proof parametrizes the family by independent signs, preserves its zero-set law while randomizing a fixed boundary term, and applies the general real universality theorem of Nguyen and Vu. A Gaussian density computation gives the bulk limit, and the Erdős–Turán discrepancy theorem controls the endpoints.
1. Introduction and statement
For a nonempty finite set define
If , this is a monic integer polynomial of degree and constant coefficient one. Let
There are such sets. Throughout the paper is uniform on , and every zero is counted with its algebraic multiplicity. Write for the number of zeros of on the unit circle.
Stanley introduced these polynomials in connection with enumerative identities and cyclotomic sets [5]. His question [6, Addendum 2] asks for an analytic estimate of the number of unit-circle zeros of
or, equivalently, for the average of . The same unit-circle phenomenon is noted in the aside on page 9 of [5].
Theorem 1.
For every continuous function on the unit circle,
In particular,
Corollary 2.
If denotes the number of unit-circle zeros of , then
For every fixed arc of angular length , the average number of unit-circle zeros of on is
The constant is classical for independent Gaussian cosine polynomials; see the discussion in [1]. General Gaussian zero-density formulas are developed in [2]. The coefficients of (1), however, are differences of constrained indicators and are not independent. We establish the exact reduction needed for this family, including its fixed endpoint coefficients. The probabilistic input is the existing general universality theorem [3, Theorem 2.6], whose hypotheses we verify below. We do not assert a new universality theorem. Nor does the average-count result classify those for which all zeros are roots of unity, the broader problem in [5].
2. Independent signs and exact preservation of zeros
Put and . Let be the indicator of for , and put . Reciprocity of (1) is equivalent to
Indeed, multiplication of this identity by gives the reciprocity identity, and the implication reverses because is nonzero. Consequently
The pairs besides may be chosen independently, proving .
Set . Thus , , and . Under uniform choice of , the variables
are independent Rademacher variables, each taking and with probability . Pairing the terms indexed by gives
For completeness, substituting first gives
Using in this last expression proves (5), including its sign.
Make the change of variable . The signs are again independent Rademacher variables. Define, on ,
There are no missing endpoint zeros: and is odd and nonzero. Complex conjugation pairs the zeros in the two open semicircles. The maps and nonvanishing factors in (7) preserve multiplicity. Hence is twice the number of zeros of in .
Lemma 3.
Let be a random analytic function with and equal in law, and let be deterministic and analytic on the same domain. Assume is almost surely not identically zero. If is an independent Rademacher variable, then and have the same zero-multiset distribution.
Proof.
The functions and have the same distribution of zeros, since and have the same law and multiplication by does not change a zero or its multiplicity. The two choices of therefore give identical zero-multiset laws.
To work with entire functions, define
This is a real trigonometric polynomial of degree . By Lemma 3, its zeros in have the same law as the transformed zeros of in the upper semicircle. Let be obtained by replacing all signs in (8) by independent standard real Gaussians . On this interval its zeros are those of
3. Universality on compact subintervals
For a real analytic function , write when has compact support in a domain of analyticity. Multiplicities are included.
Proposition 4.
For every there is such that
Proof.
We verify Conditions C1 and C2 of [3] for Theorem 2.6, using its case . Fix such that the support of lies in the interior of . Rescale by and use the real center set
There are random terms. Their deterministic basis functions are
These are entire and real on the real axis. The random coefficients in both ensembles have mean zero, variance one, and uniformly bounded moments. Thus C1 holds with and matching moments. Take and in C2. The remaining parameters are those prescribed by Theorem 2.6; the estimates below hold for every fixed .
Local zero counts.
Both ensembles are trigonometric polynomials of degree at most . Their coefficient of is or , respectively, and is nonzero surely or almost surely. For , multiplying either polynomial by produces an algebraic polynomial of degree at most . On any disk of radius in the plane the exponential map is injective for large . The number of zeros in a unit disk in the plane is therefore at most , including multiplicity. In particular for sufficiently large . The truncated moment in C2(1) is then zero.
Size, delocalization, and derivatives.
On complex distance two neighborhoods of the basis functions and their first two derivatives are bounded by a constant independent of . Indeed, the rescaled frequencies are at most and the imaginary parts of their arguments are bounded. For real in a compact subinterval of , the geometric sum gives
To see uniformity, the oscillatory part is half the real part of ; its denominator is bounded away from zero on this subinterval. For , the identity
shows that the same lower bound applies to the sum of squared moduli. Also is bounded below here. It follows, uniformly for , that
This proves C2(4). The sums of squared first derivatives, and of the squared suprema of second derivatives on unit disks, are . Together with (12), this proves C2(5), even without its allowed factor . Its mean-dependent term is zero.
For the Rademacher ensemble the function itself is bounded by on the relevant disks. For the Gaussian ensemble the event that every coefficient has absolute value at most has complement of probability at most ; on that event the function is bounded by . For any fixed , these bounds imply C2(3), whose size threshold is and exceptional probability is .
of size , with all deterministic coefficients equal to one. Choose a fixed parameter . In an interval of length centered at , the lemma supplies a deterministic point such that, for the appropriate Rademacher or Gaussian variables ,
The interval-length hypothesis holds for all sufficiently large . The lemma explicitly permits arbitrary index sets of the given size; it does not require consecutive frequencies. Condition on the distinguished coefficient . At , division of the function in (8) by leaves the sum in (13) plus . The uniformity in makes the estimate independent of this conditioning. Since , eventually
Thus the probability that the rescaled function at is smaller than is . Moreover . This is precisely the nearby-point condition C2(2), for each ensemble.
All constants can be chosen uniformly over . Theorem 2.6 of [3] therefore compares smooth real-zero statistics on intervals of radius in the variable with error . Its tests have derivatives through order six bounded by one; by linearity the same conclusion holds with any fixed derivative bound. Take a smooth partition of unity in with mesh , pieces supported in intervals of radius less than , and uniformly bounded derivatives. Multiply the pieces by . Only pieces occur, their derivatives through order six stay bounded independently of , and their centers lie in for large . Summing the local estimates proves (10).
4. The Gaussian density
Proposition 5.
Uniformly on , the mean real-zero density of is
Consequently, for every ,
Proof.
Write , and let
On the chosen interval and . For , summation by parts applied to the bounded partial sums of gives
The constants are uniform because is bounded below. Using (11) and yields
In particular for large . The Gaussian real-zero formula [2, Theorem 3.1] gives
Equivalently, this follows by conditioning on in the Kac–Rice formula: the conditional derivative variance is . The covariance determinant is positive, so the formula applies on the interval and multiple Gaussian zeros have probability zero. Substituting (16) gives (14). On the support of , the zeros of and agree. Integrating their common density and applying Proposition 4 proves (15).
5. Endpoint control and completion of the proof
We need control near only for the original integer-polynomial ensemble. A deterministic estimate supplies it.
Lemma 6.
Let be an arc of angular length . For every ,
The same upper bound holds when only unit-circle zeros are counted.
Proof.
Every coefficient of lies in , and its leading and constant coefficients are one. Therefore . The Erdős–Turán discrepancy inequality in the form [4, Theorem 2] bounds the angular discrepancy by for a monic degree- polynomial with constant coefficient one, where
Here , giving (17). Restricting to unit-circle zeros can only decrease the count. Endpoint conventions for arcs may be handled by enlarging the arc and taking a limit; multiplicities are included in the discrepancy theorem.
Let be times the expected zero-counting measure of on ; we consider . By the exact zero-law reduction, its total mass is at most . Proposition 5 says that its integral against each smooth compactly supported test converges to the integral against
For , the two intervals and correspond to two angular arcs, each of length , near and in the upper semicircle. Lemma 6 and the zero-law identity imply
with absolute implied constants for large . By first inserting a smooth interior cutoff, then approximating a continuous test uniformly on the interior, and finally letting decrease to zero in (18), we obtain, for every continuous on ,
All approximations are uniform in because the total masses are bounded.
For a continuous on the unit circle put
Conjugation and the change of variables in (7) identify the left side of (2) with . Also
Multiplicities add under products, so . For arcs, approximate their indicators from above and below by continuous functions. The limiting angular measure has no atoms, and Theorem 1 then gives the asserted limit.
6. Exact checks and further questions
The accompanying script uses rational arithmetic to verify the reciprocity parametrization and to count unit-circle roots without a numerical tolerance. For each reciprocal degree- polynomial it forms the integer polynomial defined by
The recurrence , , represents . Squarefree factorization of , followed by Sturm counts in weighted by the factor multiplicities, gives half the unit-circle count. There are no roots at , because . For , this checks all polynomials and reproduces
These are consistency checks against the initial data in [6]; the asymptotic proof does not depend on finite computation. In converting counts to fractions, the degree is , one more than .
The theorem determines an average, with no claimed concentration rate or second-order expansion. Natural next questions are fluctuations of , a quantitative error term uniform up to the endpoint regions, and the interaction between the noncyclotomic unit-circle zeros and irreducibility. The number of entirely cyclotomic members of is a separate arithmetic enumeration problem. The reductions above identify a tractable random-function representation without settling that classification.
References
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[2] A. Edelman and E. Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. 32 (1995), 1–37. arXiv:math/9501224.
[3] O. Nguyen and V. Vu, Roots of random functions: A framework for local universality, Amer. J. Math. 144 (2022), 1–74. arXiv:1711.03615.
[4] K. Soundararajan, Equidistribution of zeros of polynomials, Amer. Math. Monthly 126 (2019), 226–236. doi:10.1080/00029890.2019.1546078.
[5] R. P. Stanley, Some enumerative applications of cyclotomic polynomials, Enumer. Comb. Appl. 6:1 (2026), Article S2R4. doi:10.54550/ECA2026V6S1R4.
[6] R. Stanley, Polynomials with many zeros of absolute value 1, MathOverflow question 461829, January 8, 2024; Addendum 2, January 29, 2024. mathoverflow.net/questions/461829.