Number and location of pre-images under harmonic mappings in the plane

Jan Zur; Olivier S\`ete

REVIEW 2 major objections 4 minor 38 references

Number and location of pre-images under harmonic mappings in the plane

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that pre-image counts of harmonic mappings are determined by winding numbers of caustics, and that harmonic polynomials of degree $n$ achieve every zero count from $n$ to $n^2$.

desk verdict Solid extension of the caustic-winding formula to non-degenerate harmonic mappings, but two fixable gaps—one in the A∞ finiteness argument, one in the perturbation lemma—make the main theorem conditional as written. read the letter →

arxiv 1908.08759 v2 pith:NIM2LAZW submitted 2019-08-23 math.CV

Olivier Sète , Jan Zur This is my paper · ORCID

classification math.CV MSC 30C5531A0555M25

keywords harmonicmappingspre-imagescausticsargumentprinciplevalencepolynomialswindingnumberPoincaréindex

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 establishes an exact counting formula for the solutions of $f(z)=\eta$, where $f$ is a harmonic mapping in the plane and $\eta$ is any value that is not a caustic point. The formula expresses the number of pre-images as a pole contribution plus twice the total winding number of the caustic curves around $\eta$, with the factor two coming from the fact that every critical arc borders one sense-preserving and one sense-reversing region. A sympathetic reader should care because this turns a purely analytic counting problem into a geometric one: the image count can be read off from a plot of the caustics, and changes only when $\eta$ crosses a caustic arc. As an application, the paper proves that for every degree $n\ge 2$ and every integer $k$ between $n$ and $n^2$, some harmonic polynomial of degree $n$ has exactly $k$ zeros.

What carries the argument

The load-bearing object is the critical curve: a closed parametrization $\gamma$ of a component of the critical set $C=\{z:J_f(z)=0\}$, oriented by the second complex dilatation through $\omega(\gamma(t))=e^{it}$, whose image $f\circ\gamma$ is a caustic. The identity $N_\eta(f)=P(f)+2\sum_{\gamma\in\mathrm{crit}} n(f\circ\gamma;\eta)$ carries the whole argument, converting pre-image counting into a winding-number sum; the factor $2$ is forced because every critical arc borders a sense-preserving region on one side and a sense-reversing region on the other. Around a fold caustic, the paper supplements the global formula with a local normal-form analysis and uses harmonic Newton iteration to show that the two pre-images that appear or disappear lie near $z_0\pm i\sqrt{t b_1/a_1}$.

What would settle it

For the map $f(z)=z^2+1/z+\overline{z}+1+2\log|z|$ from Example 3.10, compute the winding numbers of the critical curves about a point $\eta$ in each caustic tile and compare $N_\eta(f)=4+2\sum n(f\circ\gamma;\eta)$ with the actual number of pre-images obtained by solving. A mismatch in any tile, or an $\eta\notin f(C)$ with infinitely many pre-images accumulating at the pole, would falsify Theorem 3.4.

Watch

Extended reading notes

Core claim

The central claim, Theorem 3.4, is that for a non-degenerate harmonic mapping $f$ — defined on the Riemann sphere except for finitely many poles, with a bounded critical set and with local expansions at the poles whose leading coefficients are unequal in modulus — the identity $N_\eta(f)=P(f)+2\sum_{\gamma\in\mathrm{crit}} n(f\circ\gamma;\eta)$ holds for every $\eta\notin f(C)$. Here $\mathrm{crit}$ is the finite collection of closed curves obtained by parametrizing the components of the critical set $C=\{z:J_f(z)=0\}$ through $\omega(\gamma(t))=e^{it}$, and $P(f)$ is the sum of the absolute Poincaré indices of all poles. The proof partitions the sphere into the connected components of the complement of $C$, applies the argument principle on each component, and sums; each critical arc contributes twice because it separates a sense-preserving from a sense-reversing region. The paper then derives structural consequences: the number of pre-images is constant on caustic tiles, crossing a single caustic arc changes it by two, and for harmonic polynomials $f=p+\overline{q}$ with $\deg p=n>\deg q=m$, every zero count $k=n,n+1,\ldots,V_{n,m}$ occurs; with $m=n-1$, this gives every $k=n,n+1,\ldots,n^2$.

Load-bearing premise

The counting formula assumes that in the unbounded component of the complement of the critical set, the equation $f(z)=\eta$ has only finitely many solutions; the proof's inversion step $z\mapsto 1/z$ does not close off the possibility that zeros accumulate at the pole of the inverted map at $0$.

Editorial extensions

If this is right

The number of pre-images is constant on each connected component of $C\setminus f(C)$, and crossing a single caustic arc changes the count by exactly two.
For sufficiently large $|\eta|$, all pre-images lie near the poles, and the total number of pre-images equals $P(f)$.
Every intermediate zero count between the minimum $n$ and the maximum $V_{n,m}$ is attained by some harmonic polynomial $p+\overline{q}$ with $\deg p=n$ and $\deg q=m$; in particular, all $k=n,n+1,\ldots,n^2$ occur for $m=n-1$.
A zero count $k$ of parity opposite to $n$ forces the polynomial to have a singular zero, while the same-parity counts can be realized by non-singular polynomials.
The formula is a geometric algorithm: from the caustic plot, winding numbers of the critical curves around $\eta$ give the image count without solving $f(z)=\eta$.

Reading between the lines

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

The same argument-principle mechanism should extend to harmonic mappings on bounded domains and compact Riemann surfaces, since the local contributions are identical; the paper lists this as an outlook rather than a theorem.
For gravitational-lensing models of the form $r(z)-\overline{z}$, the identity refines the odd-number-of-images theorem: image multiplicity is governed by caustic winding numbers, so caustic topology alone should determine which multiplicities are possible.
Because the formula reduces valence to a winding-number sum, the open problem of computing $V_{n,m}$ becomes a computational topology question: find the caustic tile where the winding sum is maximal.
The local fold analysis suggests an explicit numerical recipe for tracking image pairs in lensing applications: start harmonic Newton iteration at $z_0\pm i\sqrt{t b_1/a_1}$ to capture the two pre-images that bifurcate from a fold.

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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 main counting formula—N_eta(f) = P(f) + 2 sum n(f∘γ;η)—is right, and it deserves a serious referee. But the proof has a genuine gap in the A∞ part, and the Section 5 application leans on an unverified hypothesis. Both are fixable, so treat this as conditional.

What's actually new: Sète and Zur take the known rational-function result [21] and push it to all non-degenerate harmonic mappings, with clean argument-principle machinery. The caustic-tile interpretation and the ±2 rule are nice and useful. Theorem 4.2, which effectively locates the two pre-images near a fold and gives the direction in which they appear/disappear, is a good complement to Lyzzaik's local theory. Corollary 5.6—every k between n and n² occurs for some degree-n harmonic polynomial—is a real new result, generalizing Bleher et al.

The soft spots. In the proof of Theorem 3.3, to apply the argument principle in A∞, the paper needs finitely many zeros of f−η in the unbounded component. The justification says otherwise fη(1/z) would have infinitely many non-singular zeros accumulating at 0, contradicting isolation. That doesn't work: fη(1/z) generally has a pole at 0, and isolated zeros may accumulate at a pole. The statement is probably true under the non-degeneracy hypothesis: at a pole at infinity, |a_n|≠|b_n| forces |z^n f(z)| to stay bounded away from zero, so zeros can't get arbitrarily large. But the paper doesn't give that argument, and Theorem 3.4 depends on this step. So the central formula is conditional as written.

The second issue is in Theorem 5.4. The proof invokes Lemma 5.3, which requires the maximum-valence polynomial to have only non-singular zeros. That's not established, and the claim that a cubic perturbation resolves all multiple caustic arcs without introducing new ones is asserted rather than proved. These are application-level gaps, not a fatal flaw, but they need work.

The citation pattern looks normal; the two authors cite their own Newton-iteration paper and the rational-function paper, both appropriately.

Bottom line: this is a solid, useful paper with a likely correct main theorem and a new corollary. It needs a serious referee, not a desk reject. If the authors close the A∞ gap and tighten Section 5, I'd be happy to see it accepted.

Best,

Referee Report

2 major / 4 minor

Summary. The paper derives a formula for the number of pre-images of a non-caustic point under a non-degenerate harmonic mapping in terms of the number of poles and the winding numbers of the critical curves. The main result, Theorem 3.4, states that N_eta(f) = P(f) + 2 * sum over critical curves of n(f composed with gamma; eta), and this is used to show that the pre-image count changes by +/-2 when crossing a caustic arc and to give a geometric method for determining valence. The authors also provide a local description of pre-images near fold caustics (Theorem 4.2) and apply the global formula to harmonic polynomials, proving in Theorem 5.4 that for every n > m >= 1 and every k between n and V_{n,m} there is a harmonic polynomial p(z)+q(z), with deg p = n and deg q = m, having exactly k zeros; Corollary 5.6 then gives the range k = n, n+1, ..., n^2 when m = n-1.

Significance. If the main assertions are correct, the paper gives an elegant geometric algorithm for computing the valence of non-degenerate harmonic maps and settles the intermediate-valence question for harmonic polynomials in the full degree range, generalizing the earlier result of Bleher et al. for rational harmonic functions. The argument-principle approach is natural, and the paper contains several well-chosen worked examples and figures. The central formula is derived from standard principles rather than from the authors' own results, and I see no circularity in the main counting argument. The presentation is generally clear, but the proof contains two repairable gaps that need attention before the paper is fully convincing.

major comments (2)

[Section 3.1, proof of Theorem 3.3] The proof's claim that f_eta has only finitely many zeros in A_infinity is not established. The argument is that infinitely many zeros in {|z| >= R} would give infinitely many non-singular zeros of f_eta(1/z) in {|z| <= 1/R}, contradicting isolation of non-singular zeros. This ignores that f_eta(1/z) generally has a pole at 0 when f has a pole at infinity; zeros of a function may accumulate at a pole, so isolation of each zero does not give a contradiction. The statement is very likely true under the non-degeneracy condition |a_n| != |b_n| in Definition 3.1(2), because that condition implies |f(z) - eta| >= c |z|^n for sufficiently large |z|, which bounds all zeros; the proof should supply this or an equivalent argument.
[Section 5, Theorem 5.4 and Lemma 5.3] The perturbation step in the proof of Theorem 5.4 has two unverified hypotheses. First, Lemma 5.3 is applied to a maximum-valence polynomial f, but Lemma 5.3 requires all zeros of f to be non-singular; the authors do not show that an extremal polynomial can be chosen with this property. Second, Lemma 5.2 only separates the images of two prescribed critical points, while the proof of Theorem 5.4 asserts that a degree-3 perturbation can resolve all multiple caustic arcs simultaneously 'such that no other multiple caustic arcs occur'; this global assertion is not proved. The subsequent claim that there is a path phi from eta_n to 0 intersecting the caustics only in single folds also depends on these unresolved points. These gaps are load-bearing for Corollary 5.6 and need to be repaired.

minor comments (4)

[Example 3.11] The displayed formula for the harmonic polynomial is missing the conjugation on the q-part; as written, f(z) = z^n + (z-1)^n + i z^n - i(z-1)^n is an analytic polynomial of degree n and cannot have n^2 zeros. The intended expression should involve the conjugate of the second group of terms.
[Section 2.1, after (2.7)] The statement 'By construction f is sense-preserving to the left of gamma, and sense-reversing to the right of gamma' is used later in the proof of Theorem 3.3, but it is not justified at this point; a short explanation of the orientation convention would improve readability.
[Section 4, Theorem 4.2] The proof relies on [33, Lem. 5.1, Thm. 5.2] and 'their proofs' to obtain convergence of the harmonic Newton iteration in the two disks around z_+ and z_-; stating the hypotheses of those results explicitly would make the argument more self-contained.
[Section 5, Lemma 5.2] In the proof of Lemma 5.2, the Hermite interpolation polynomial of degree 3 satisfying p(z_1)=epsilon, p(z_2)=-epsilon, p'(z_1)=p'(z_2)=0 exists uniquely when z_1 != z_2; the authors should mention that epsilon is taken sufficiently small so that later perturbation arguments remain valid.

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citations appear, but the central counting formula and zero-count application are not circular.

full rationale

The derivation chain is not circular. Theorem 3.4's formula N_eta(f) = P(f) + 2 * sum over crit of n(f o gamma; eta) is obtained by applying the argument principle (Theorem 2.5) to the components of C\C, with zero indices supplied by Proposition 2.7 and pole indices by Proposition 2.10; the formula is not an input to those propositions. The geometric consequences in Theorems 3.6 and 3.7 are corollaries of the same argument, not restatements of the assumptions. The paper's self-citations do not carry the central claim: [21] is cited only as a source of a similar partition for rational harmonic mappings, [24] concerns the index of a special singular zero, and [33] supplies independent convergence results for the harmonic Newton iteration used in Theorem 4.2. None of these citations defines the counting formula or forces the valence conclusion by construction. There are genuine proof gaps, but they are correctness risks rather than circularity: the finiteness of zeros of f - eta in A_infinity is justified by an inversion argument that overlooks the pole of f_eta(1/z) at 0, and the perturbation argument in Lemma 5.3 assumes the maximal-valence polynomial has only non-singular zeros, which is not established. These gaps do not make any theorem equivalent to its inputs by definition, so the paper receives only a low score for the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central formula depends on standard complex analysis and on external theorems for local structure of harmonic mappings and Newton convergence. No fitted parameters and no new physical or mathematical entities are introduced.

assumptions (5)

standard math Argument principle for continuous functions with finitely many exceptional points (Theorem 2.5, after [3], [34]).
Used to balance winding on boundaries of components A against pole indices and zero indices; this is the engine of Theorem 3.4.
standard math Poincare index formulas for poles and non-singular zeros of harmonic mappings (Propositions 2.7 and 2.10, after [35]).
Gives P(f) and the sign conventions in the argument-principle balance.
domain assumption Local standard forms and valence statements for light harmonic mappings (Lyzzaik [27, Thm. 5.1], Neumann [28, Thm. 6.7]).
Used in Section 4 to classify fold and cusp local behavior and to cap local valence at 2 or 3.
domain assumption Convergence of harmonic Newton iteration (Sete and Zur [33, Lem. 5.1, Thm. 5.2]).
Proves existence and approximate location of the two pre-images near a fold in Theorem 4.2.
ad hoc to paper After a generic degree-3 perturbation, a maximum-valence harmonic polynomial can be taken with no multiple caustic arcs; an admissible path from eta_n to 0 crossing only folds exists.
Assumed in the proof of Theorem 5.4; not fully proved, and Lemma 5.3 is invoked without verifying that all zeros of the maximum-valence polynomial are non-singular.

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Pith. "Pith review of Number and location of pre-images under harmonic mappings in the plane." pith.science (2026). https://pith.science/paper/NIM2LAZW

@misc{pith2026190808759,
  author       = {Pith},
  title        = {Pith review of: Number and location of pre-images under harmonic mappings in the plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIM2LAZW}},
  note         = {Machine review of arXiv:1908.08759}
}

abstract

We derive a formula for the number of pre-images under a non-degenerate harmonic mapping $f$, using the argument principle. This formula reveals a connection between the pre-images and the caustics. Our results allow to deduce the number of pre-images under $f$ geometrically for every non-caustic point. We approximately locate the pre-images of points near the caustics. Moreover, we apply our results to prove that for every $k = n, n+1, \ldots, n^2$ there exists a harmonic polynomial of degree $n$ with $k$ zeros.

Figures

Figures reproduced from arXiv: 1908.08759 by the authors.

**Figure 1.** Number of pre-images of η under f(z) = z − z 2/(z 3 − 0.6 3) for an η in the respective regions; see also Example 3.9 below. The black lines mark the caustics (critical values) of f. The number of pre-images of η in the outer tile corresponds to the number of poles of f (including ∞). f(z) = r(z)−z have been studied intensively [17, 7, 25, 26, 22], since these are of interest when modeling the phenomenon of gravitat… view at source ↗

**Figure 2.** Left: Aε (shaded) and oriented critical arcs near a zero z0 of ω 0 . Right: Deformation of γj in the proof of Theorem 3.3. The +/− signs indicate regions where f is sense-preserving/sense-reversing. let z1, . . . , zk be the poles of f in A, and define P(f; A) = Pk j=1|ind(f; zj )|. Then, for η ∈ C such that f − η is non-zero on ∂A, the number Nη(f; A) of pre-images of η under f in A is Nη(f; A) = P(f; A) +Xn j=1 n(… view at source ↗

**Figure 3.** Then, the change of the winding number n(f ◦ γ; η2) − n(f ◦ γ; η1) can be determined with the next result. Proposition 3.8 ([31, Prop. 3.4.4]). Let γ be a smooth closed curve and η /∈ trace(γ). Let further R be a ray from η to ∞ in direction e iϕ, such that R is not a tangent at any point on γ. Then R intersects γ at finitely many points γ(t1), . . . , γ(tk) and we have for the winding number of γ about η n(γ; η) = … view at source ↗

Figures from the paper (5 more)

**Figure 3.** Figure 3: Intersection index and caustics in the η-plane. Let η1, η2 be in two adjacent caustic tiles separated by a single caustic arc. We call two sets adjacent, if they share a common boundary arc. Consider the ray R from η1 to ∞ through η2, and let it intersect the caustic b…

**Figure 4.** Figure 4: Critical curves (left) and caustics (right) of the functions in Exam [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: Behavior at a fold (top) and cusp (bottom); cf. [21, Figs. 4, 7]. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: Caustics of f(z) = 1 3 z 3 + 1 2 z 2 ; see Example 4.4 [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]

**Figure 7.** Figure 7: Phase plots of fηj (z) = 1 3 z 3 + 1 2 z 2 − ηj (see [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

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Pith tools

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

[1] [21]

Liesen and J

J. Liesen and J. Zur, How constant shifts aﬀect the zeros of certain rational har- monic functions, Comput. Methods Funct. Theory, 18 (2018), pp. 583–607

work page 2018

[2] [1]

L. V. Ahlfors, Lectures on quasiconformal mappings, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966

work page 1966

[3] [2]

J. H. An and N. W. Evans, The Chang–Refsdal lens revisited, Monthly Notices Roy. Astronom. Soc., 369 (2006), pp. 317–334

work page 2006

[4] [3]

M. B. Balk, Polyanalytic Functions, vol. 63 of Mathematical Research, Akademie- Verlag, Berlin, 1991. 25

work page 1991

[5] [4]

A. F. Beardon, Complex analysis. The argument principle in analysis and topology, John Wiley & Sons, Ltd., Chichester, 1979

work page 1979

[6] [5]

Bénéteau and N

C. Bénéteau and N. Hudson, A survey on the maximal number of solutions of equations related to gravitational lensing, inComplexanalysisanddynamicalsystems, Trends Math., Birkhäuser/Springer, Cham, 2018, pp. 23–38

work page 2018

[7] [6]

Bergweiler and A

W. Bergweiler and A. Eremenko, On the number of solutions of a transcenden- tal equation arising in the theory of gravitational lensing, Comput. Methods Funct. Theory, 10 (2010), pp. 303–324

work page 2010

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P. M. Bleher, Y. Homma, L. L. Ji, and R. K. W. Roeder, Counting zeros of harmonic rational functions and its application to gravitational lensing, Int. Math. Res. Not. IMRN, (2014), pp. 2245–2264

work page 2014