Using Real-Variable Techniques to Study Zeros of Complex-Valued Harmonic Functions

Dallin Robison; Eli Sampson; Jennifer Brooks; Kaden Parker; Katherine Perez; Mary Jenkins; Matthew G. Wilson; Samuel Liechty

REVIEW 3 major objections 5 minor 11 references

Using Real-Variable Techniques to Study Zeros of Complex-Valued Harmonic Functions

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For $p(z)=z^m+c(z^k+\bar z^k)-1$ with $\gcd(m,k)=1$, the zero count changes monotonically with $c$, rising from $m$ to $m+N$ when $k>0$ and falling from $M+N$ to $M$ when $k<0$, with $M$ and $N$ read from a short table depending on $m…

desk verdict Solid new zero-counting result for two harmonic trinomial families, but the proof is incomplete as printed—most cases are deferred to the reader. read the letter →

arxiv 2506.14966 v1 pith:WGFVA2RR submitted 2025-06-17 math.CV

Jennifer Brooks , Mary Jenkins , Samuel Liechty , Kaden Parker , Katherine Perez , Dallin Robison , Eli Sampson , Matthew G. Wilson This is my paper · ORCID

classification math.CV MSC 30C15

keywords complex-valuedharmonicfunctionszerospolynomialstrinomialsreal-variabletechniquesrootsofunityzerocountingmonotonecount

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

The paper proves that two one-parameter families of complex-valued harmonic functions have zero counts that change in a completely predictable way as the positive parameter $c$ increases. For $p(z)=z^m+c(z^k+\bar z^k)-1$ with $\gcd(m,k)=1$ and $m>|k|$, every zero lies on one of $2m$ rays, and on each ray the problem becomes a real-variable root count. The authors count the rays of each type using roots of unity and obtain Theorems 1 and 2: for $k>0$ the number of zeros increases monotonically from $m$ to $m+N$, and for $k<0$ it decreases monotonically from $M+N$ to $M$, where $N$ and $M$ are determined by $m \bmod 4$ and the parity of $k$. If these theorems are right, the zero count of these functions is fixed by the exponents and by whether $c$ lies below or above a threshold, with no further dependence on the coefficient. The written proof works through one representative residue class in each theorem and states that the other classes follow by the same counting.

What carries the argument

The carrying object is the ray-restricted real function $f_j(r)=(-1)^j r^m + 2c\cos(kj\pi/m)r^k -1$, whose positive zeros are exactly the moduli of the zeros of $p$ on the ray at angle $j\pi/m$. The argument splits into six cases according to the parity of $j$, the sign of $k$, and the sign of $\alpha=\cos(kj\pi/m)$; Lemmas 2 through 11 give 0, 1, or 2 positive roots for each case, with the two-root cases changing at an explicit threshold $c_0$. The remaining work is combinatorial: counting how many of the $2m$ values of $j$ fall into each case, using the fact that $\cos(kj\pi/m)=\operatorname{Re}(\omega^j)$ for $\omega=e^{ik\pi/m}$ and elementary counts of roots of unity with positive, negative, or zero real part.

What would settle it

In an omitted case, say $m\equiv 2 \pmod 4$ with $k>0$, enumerate the $2m$ values of $j$ and count how many even and odd $j$ have $\cos(kj\pi/m)>0$, $<0$, or $=0$; if the split differs from what the table requires, Theorem 1 fails. Alternatively, for small $m$ and $k$ in such a case, numerically count the zeros of $p(z)=z^m+c(z^k+\bar z^k)-1$ for $c$ below and above the threshold $c_0$ of Lemma 7 and check that the count is $m$ and $m+N$ respectively.

Watch

Extended reading notes

Core claim

The central discovery is a pair of exact zero-counting theorems for the harmonic functions $p(z)=z^m+c(z^k+\bar z^k)-1$. Theorem 1, for $k>0$ with $\gcd(m,k)=1$, says that as $c$ increases through the positive reals the number of zeros grows monotonically from $m$ to $m+N$, where $N=m$ for $m\equiv 0 \pmod 4$, $N=m-1$ or $m+1$ for $m\equiv 1 \pmod 4$ according as $k$ is odd or even, $N=m-2$ for $m\equiv 2 \pmod 4$, and $N=m+1$ or $m-1$ for $m\equiv 3 \pmod 4$ according as $k$ is odd or even. Theorem 2, for $k<0$, says the zero count decreases monotonically from $M+N$ to $M$, with paired values $(M,N)=(m+1,m-2)$ for $m\equiv 0 \pmod 4$; $(m-1,m+1)$ or $(m,m+1)$ for $m\equiv 1 \pmod 4$ according as $k$ is odd or even; $(m-1,m)$ for $m\equiv 2 \pmod 4$; and $(m+1,m-1)$ or $(m,m-1)$ for $m\equiv 3 \pmod 4$ according as $k$ is odd or even. The proof rests on two structural facts: the imaginary part $r^m\sin(m\theta)$ vanishes only on the rays $\theta=j\pi/m$, and on the $j$-th ray the modulus $r$ of a zero must satisfy $f_j(r)=(-1)^j r^m + 2c\cos(kj\pi/m)r^k -1=0$. The six possible forms of $f_j$ have one, two, or no positive real roots, with the two-root cases switching at an explicit threshold, and the root-of-unity count fixes how many rays of each form exist.

Load-bearing premise

The theorems assume that the counts of even- and odd-indexed rays with positive, negative, and zero values of $\cos(kj\pi/m)$ are exactly as tabulated for every residue class, but the paper writes out the proof for only one class in each theorem and says the others are left to the reader.

Editorial extensions

If this is right

For every admissible pair $(m,k)$ with $\gcd(m,k)=1$, the total number of zeros of $p$ is known exactly at small $c$ and at large $c$: $m$ and $m+N$ for $k>0$, and $M+N$ and $M$ for $k<0$.
The zero count is monotone in $c$, so roots are not created and destroyed repeatedly; each ray's contribution changes only at the threshold where the associated real function gains or loses two roots.
The threshold $c_0$ appearing in the two-root lemmas is explicit in terms of $m$, $k$, and $\alpha$, so the paper identifies not only the possible counts but the parameter value at which the transition happens.
The residue-class tables in Theorems 1 and 2 cover all $m$ with $m>|k|$ and $\gcd(m,k)=1$, so the result is complete in the parameter range considered, assuming the omitted cases are as stated.

Reading between the lines

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

The same ray-reduction is tied to having exactly one non-real term, so the general $a\ne b$ case is not accessible by these methods; the paper leaves it open how the maximum zero count there compares, and a numerical survey of small $a,b,m,k$ would be a direct test of whether a similar table exists.
The explicit threshold $c_0$ suggests the radii at which zeros appear or disappear could also be tracked, since the roots of $f_j$ vary continuously in $c$; this would give a quantitative prediction about where in the complex plane new zeros are born.
Because the paper's thresholds $c_0$ are explicit, one can numerically simulate the omitted residue classes, for example $m\equiv 2 \pmod 4$, and check the parity splits before relying on the full table; this is a direct computational verification of the unstated cases.

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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

Dear colleague,

The headline is that this is a genuine new result—exact zero-count transitions for the harmonic polynomials p(z)=z^m+c(z^k+\bar{z}^k)-1, with positive and negative k—obtained through a ray-reduction trick that is quite different from the earlier hypocycloid/argument-principle methods. The two families do not appear in the cited literature, and the real-variable counting on the 2m rays is a clean idea that will interest the zero-location community.

The Section 3 lemmas are correct. I checked the sign cases, the critical point calculations, and the threshold constants; they all line up. The one fully worked example in each theorem is consistent. The citations are appropriate and I don't see self-citation padding. No parameters were fitted; the proof is self-contained from standard calculus and roots-of-unity facts.

The soft spot is exactly what the reader flagged: the proofs of Theorems 1 and 2 are not complete. Section 4.3 proves one residue class and then says the rest are 'left to the reader.' Section 4.4 does the same. For a theorem whose entire content is a table over residue classes, that's not acceptable. Some of the unstated cases are not routine—for m≡0 mod 4 and k odd, rays with α=0 appear and need separate counting. An independent audit of the omitted cases, in the stress-test note, reproduces every stated M and N, so this is a completeness gap rather than a discovered error. But as printed, the central theorems are under-proved.

This is for specialists. If you work on zero location for harmonic polynomials, you'll want to know this result and this method. I'd bring it to reading group maybe, and I'd cite it if I needed a new family with exact counts.

Recommendation: send it to a serious referee. The referee should require the omitted case derivations, or at least a table giving the counts for every residue class, before acceptance. The paper's core seems right, but it isn't fully demonstrated yet.

Best,

Referee Report

3 major / 5 minor

Summary. The paper studies the zeros of the one-parameter families of complex-valued harmonic functions p(z)=z^m+c(z^k+\bar z^k)-1, where c>0, gcd(m,k)=1, m>|k|, and k is either positive (Theorem 1) or negative (Theorem 2). The main structural result, Lemma 1, shows that every zero lies on one of the 2m rays with angle jπ/m, which reduces the problem to counting positive real zeros of the real functions f_j(r)=(-1)^j r^m+2cα r^k-1 with α=cos(kjπ/m). Section 3 proves six lemmas describing how many positive real zeros f_j has according to the sign of k, the parity of j, and the sign of α; the thresholds in the two-zeros cases are given explicitly. Section 4 then counts, for each residue class of m modulo 4 and each parity of k, how many even and odd j yield α positive, negative, or zero, and uses those counts with the lemmas to assert exact minimum and maximum zero counts as c varies. Theorems 1 and 2 state monotone transitions from m to m+N (for positive k) and from M+N to M (for negative k), with N and M tabulated by m mod 4 and the parity of k. The paper's proofs, however, work out only one case in each theorem and leave the remaining five cases in each theorem to the reader.

Significance. If the missing cases are supplied, the paper would provide a genuinely elementary and exact zero-counting result for two families whose critical curves are not circles, complementing the argument-principle methods of Brilleslyper et al. and Brooks-Lee. The ray-reduction idea is clean, the real-variable lemmas are correct, and the worked examples are consistent with the stated theorems. The thresholds c0 in Lemmas 7 and 11 are explicit and parameter-free. The main weakness is that the published proofs of Theorems 1 and 2 are incomplete: only one case in each theorem is actually proved, and the exact tabulated values for all other cases rest on unverified parity-and-sign counts of roots of unity. The omitted counts appear to be correct in the cases I checked, but a complete proof is needed before the theorems can be accepted as proven.

major comments (3)

[Section 4.3, proof of Theorem 1] The proof explicitly states, 'A complete proof of the theorem would require many cases. We give the proof for one of the more complicated cases and leave the others to the reader.' The theorem has six cases depending on m mod 4 and the parity of k, but only the case m≡1 mod 4, k even is actually proved. The exact values of N in the theorem are computed from the asserted counts of even and odd j with Re(ω^j)>0, <0, or =0, and for the remaining five cases these counts are not established. These counts are load-bearing, not a routine repetition: for example, cases with m≡0 mod 4 and odd k have rays with α=0, whose contribution must be handled through Lemmas 2 and 3. As printed, the proof of Theorem 1 is therefore incomplete. The authors should either provide a complete case analysis or prove a general counting lemma that yields all six rows of the table.
[Section 4.4, proof of Theorem 2] The same incompleteness occurs in the proof of Theorem 2. Only the case m≡3 mod 4, k odd is worked out, and the other five cases are left to the reader. The statement 'we give the proof for one of the more complicated cases and leave the others to the reader' does not supply the required counting argument for the remaining residue classes. In particular, for m≡0 mod 4 and odd k, there are rays with cos(kjπ/m)=0 (e.g., m=12, k=5 gives j=6,18), so the claimed 'similar strategy' must also account for the fixed contributions of those rays via Lemmas 2 and 3. Since the M and N values in Theorem 2 depend on these unproved counts, the theorem is not established as written. A complete case analysis or a uniform counting lemma with full proof is required.
[Sections 4.3-4.4 and Lemmas 7, 11] The monotonicity claim is only implicit. The authors show that for small c the count is m (or M+N) and for large c it is m+N (or M), and that on each ray with a threshold the count changes by two at a single c0. Since the per-ray counts are monotone in c, the total count is nondecreasing (Theorem 1) or nonincreasing (Theorem 2), but this aggregation should be stated explicitly, including the behavior at the threshold values where a double zero occurs. This is a presentation issue rather than a substantive error, but it should be clarified in the revision.

minor comments (5)

[Section 4.3 and 4.4] In both proofs, the phrase 'j∈{0,2,...,2m−1}' should be 'j∈{0,1,...,2m−1}' to match Lemma 1 and the surrounding discussion.
[Theorem 2] In the bullet for 'm≡1 mod 4 and k is even', there is a missing comma: it should read 'M=m, N=m+1'.
[Lemma 3] There is a typo in the proof: 'postive real zeros' should be 'positive real zeros'.
[Examples 1 and 2] The displayed formulas should consistently use \bar z^k for the conjugate term; in the plain-text rendering both terms appear as z^k, which obscures the conjugate.
[References] Reference [9] to Sheil-Small is incomplete: it lists only a date and lacks a title, venue, or preprint identifier. Full bibliographic data should be supplied.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorems 1–2 are derived from directly proved real-variable lemmas and explicit root-of-unity counts, with no fitted parameters or load-bearing self-citation.

full rationale

The paper's central claims are exact zero-count transitions for p(z)=z^m+c(z^k+conj(z)^k)-1. The derivation chain is self-contained: Section 2 reduces the complex zero count to positive real zeros of f_j(r)=(-1)^j r^m+2c alpha r^k-1; Section 3 proves Lemmas 2–11 by elementary calculus (monotonicity, one critical point, signs at 0 and infinity); Section 4 counts the relevant indices j using Propositions 1–3, which are proved from roots-of-unity parity arguments. No parameter is fitted to data and no 'prediction' is reused as an input: c is the problem variable, and the thresholds c0 are explicitly computed in Lemmas 7 and 11 from the stated formula. The citations to Brilleslyper et al. [1] and Brooks–Lee [3] are contextual comparisons of proof strategy, not load-bearing premises; the authors explicitly note that those strategies are not viable here and prove everything needed for their own family. The only genuine weakness is a completeness gap, not circularity: Sections 4.3 and 4.4 state 'A complete proof of the theorem would require many cases. We give the proof for one of the more complicated cases and leave the others to the reader,' and the tabulated N and M values for the other residue classes are asserted rather than shown. That is an omitted-case issue, and the reader's independent audit suggests the counts reproduce, but incompleteness is not circularity: the omitted cases would be computed by the same already-proved lemmas and root-of-unity counts, not by assuming the theorem. Therefore the derivation does not reduce to its own inputs, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The only variable parameter is c, which is part of the family, not a fitted constant. No new entities or ad hoc analytic assumptions are introduced; the proof uses standard real analysis and roots of unity.

assumptions (2)

standard math Intermediate Value Theorem and Rolle's Theorem are used to count positive real zeros in Section 3.
Applied in Lemmas 4-11 to establish existence and uniqueness of roots of f_j.
standard math The set of q-th roots of unity is a cyclic group of order q; primitive root properties and parity of adjacent roots as in Propositions 1-3.
These facts drive the counting of even/odd j with cos(k j pi/m) positive, negative, or zero in Section 4.

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Cite this review

Pith. "Pith review of Using Real-Variable Techniques to Study Zeros of Complex-Valued Harmonic Functions." pith.science (2026). https://pith.science/paper/WGFVA2RR

@misc{pith2026250614966,
  author       = {Pith},
  title        = {Pith review of: Using Real-Variable Techniques to Study Zeros of Complex-Valued Harmonic Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WGFVA2RR}},
  note         = {Machine review of arXiv:2506.14966}
}

read the original abstract

We investigate the zeros of two one-parameter families of harmonic functions and describe how the number of zeros depends on the parameter. Our functions have the property that all zeros lie on certain rays in the complex plane and thus we are able to use real-variable techniques to count the zeros on each ray.

Figures

Figures reproduced from arXiv: 2506.14966 by the authors.

**Figure 1.** Inspired by the work of Brilleslyper et al., Brooks and Lee [3] investigated a related family of harmonic trinomials, each having a pole at the origin: (3) fc(z) = z n + c z k − 1 where n, k ∈ N with n > k and gcd(n, k) = 1. They showed that for this family, the number of zeros decreases from n + k to n − k as c increases through the positive reals. Not surprisingly, their proof strategy was the same as that in Bril… view at source ↗

**Figure 1.** Zeros of p(z) = z 5 + c(z 4 + z 4 ) − 1 Just as we added a single term to go from (1) to (2), we also investigate a family of harmonic functions related to the family (3). However, we write our functions in a way that makes their connection with (2) clearer: (4) p(z) = z m + c [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Zeros of p(z) = z 5 + c(z −4 + z −4 ) − 1 Our paper is structured as follows: In Section 2 we show that the zeros of (2) and (4) all lie on one of 2m rays in the complex plane, and we show how to reduce the question of counting zeros on each ray to a question of counting positive real zeros of a certain real-valued function associated with the ray. These real-valued functions take one of six forms, and in Section 3,… view at source ↗

Figures from the paper (1 more)

**Figure 3.** Figure 3: Zeros and rays of p(z) = z 5 + c(z 4 + z 4 ) − 1 By Lemma 1, we see that if z = reiθ is a zero of p, then Re p(z) = r m cos mjπ m + 2crk cos kjπ m − 1 = (−1)j r m + 2c cos kjπ m r k − 1 := fj (r). To simplify notation, let α = cos( kjπ m ). Because r is the…

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Zeros of a one-parameter family of harmonic trinomials

Michael Brilleslyper, Jennifer Brooks, Michael Dorff, Russell Howell, and Lis- beth Schaubroeck. Zeros of a one-parameter family of harmonic trinomials. Proc. Amer. Math. Soc. Ser. B , 7:82–90, 2020

work page 2020
[2]

Zeros of a family of complex-valued harmonic trinomials

Jennifer Brooks, Michael Dorff, Alexandra Hudson, Erin Pitts, Clay Whiffen, and Amy Woodall. Zeros of a family of complex-valued harmonic trinomials. Bull. Malays. Math. Sci. Soc. , 45(3):1079–1091, 2022

work page 2022
[3]

Zeros of a family of complex-valued har- monic functions with poles

Jennifer Brooks and Alexander Lee. Zeros of a family of complex-valued har- monic functions with poles. Computational Methods and Function Theory , 2024

work page 2024
[4]

Location of the zeros of certain complex-valued harmonic polynomials

Hunduma Legesse Geleta and Oluma Ararso Alemu. Location of the zeros of certain complex-valued harmonic polynomials. Journal of Mathematics , 2022(1):4886522, 2022

work page 2022
[5]

Geometry of trinomials

Aaron Melman. Geometry of trinomials. Pacific Journal of Mathematics , 259, 09 2012

work page 2012
[6]

Zero location for analytic and harmonic trinomials

Aaron Melman. Zero location for analytic and harmonic trinomials. Journal of Mathematical Analysis and Applications , 2024

work page 2024
[7]

Zeros of a family of complex harmonic polynomials

Samantha Sandberg. Zeros of a family of complex harmonic polynomials. Master’s thesis, Brigham Young University, 2021

work page 2021
[8]

On the zeros of polyanalytic polynomials

Olivier S` ete and Jan Zur. On the zeros of polyanalytic polynomials. Journal of Mathematical Analysis and Applications , page 128595, 2024

work page 2024

Show all 11 references

[9]

Sheil-Small, 02 1992

T. Sheil-Small, 02 1992

1992
[10]

The valence of harmonic polynomials

AS1443416 Wilmshurst. The valence of harmonic polynomials. Proceedings of the American Mathematical Society, pages 2077–2081, 1998

1998
[11]

Zeros of a two-parameter family of harmonic trinomials

David Work. Zeros of a two-parameter family of harmonic trinomials. Master’s thesis, Brigham Young University, 2021. ZEROS OF HARMONIC FUNCTIONS 13 Brigham Young University Mathematics Department, Provo UT 84602 USA Email address : jbrooks@mathematics.byu.edu Brigham Young Uni...

2021

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Zeros of a one-parameter family of harmonic trinomials

Michael Brilleslyper, Jennifer Brooks, Michael Dorff, Russell Howell, and Lis- beth Schaubroeck. Zeros of a one-parameter family of harmonic trinomials. Proc. Amer. Math. Soc. Ser. B , 7:82–90, 2020

work page 2020

[2] [2]

Zeros of a family of complex-valued harmonic trinomials

Jennifer Brooks, Michael Dorff, Alexandra Hudson, Erin Pitts, Clay Whiffen, and Amy Woodall. Zeros of a family of complex-valued harmonic trinomials. Bull. Malays. Math. Sci. Soc. , 45(3):1079–1091, 2022

work page 2022

[3] [3]

Zeros of a family of complex-valued har- monic functions with poles

Jennifer Brooks and Alexander Lee. Zeros of a family of complex-valued har- monic functions with poles. Computational Methods and Function Theory , 2024

work page 2024

[4] [4]

Location of the zeros of certain complex-valued harmonic polynomials

Hunduma Legesse Geleta and Oluma Ararso Alemu. Location of the zeros of certain complex-valued harmonic polynomials. Journal of Mathematics , 2022(1):4886522, 2022

work page 2022

[5] [5]

Geometry of trinomials

Aaron Melman. Geometry of trinomials. Pacific Journal of Mathematics , 259, 09 2012

work page 2012

[6] [6]

Zero location for analytic and harmonic trinomials

Aaron Melman. Zero location for analytic and harmonic trinomials. Journal of Mathematical Analysis and Applications , 2024

work page 2024

[7] [7]

Zeros of a family of complex harmonic polynomials

Samantha Sandberg. Zeros of a family of complex harmonic polynomials. Master’s thesis, Brigham Young University, 2021

work page 2021

[8] [8]

On the zeros of polyanalytic polynomials

Olivier S` ete and Jan Zur. On the zeros of polyanalytic polynomials. Journal of Mathematical Analysis and Applications , page 128595, 2024

work page 2024