Nested nodal loops of biharmonic functions

Alexander Logunov; Javier G\'omez-Serrano; Robert Koirala

arxiv: 2605.18699 · v1 · pith:3IFXUHPMnew · submitted 2026-05-18 · 🧮 math.AP

Nested nodal loops of biharmonic functions

Javier G\'omez-Serrano , Robert Koirala , Alexander Logunov This is my paper

Pith reviewed 2026-05-20 08:25 UTC · model grok-4.3

classification 🧮 math.AP

keywords biharmonic polynomialsnodal setsnested loopszero setsBoggio-Hadamard conjecturetopological loopsreal-valued polynomialsplane geometry

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The pith

Biharmonic polynomials on the plane can have zero sets with any finite number of nested smooth loops.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for every natural number n there exists a real-valued biharmonic polynomial on R squared whose zero set contains a nest of n smooth disjoint topological loops, with each inner loop lying inside the domain bounded by the next outer loop. This construction applies in particular to the case of two nested loops, which is connected to the failure of the Boggio-Hadamard conjecture from the early twentieth century. A sympathetic reader would care because the result shows that the nodal sets of biharmonic functions can be arranged with arbitrary finite nesting depth, revealing more geometric flexibility than some classical expectations allowed. The explicit construction makes the existence concrete rather than merely abstract.

Core claim

Given any natural number n, there exists a real-valued biharmonic polynomial on R squared whose zero set contains a nest of n smooth, disjoint topological loops, meaning that the k-th loop lies inside the domain bounded by the (k+1)-st loop for k from 1 to n-1. The case n equals 2 is related to the failure of the Boggio-Hadamard conjecture.

What carries the argument

An explicit construction of a real-valued biharmonic polynomial on the plane whose zero set is arranged to contain any prescribed finite number of nested smooth loops.

If this is right

Biharmonic polynomials exist whose nodal sets exhibit arbitrary finite nesting of smooth loops.
The Boggio-Hadamard conjecture fails because two nested loops already occur in the zero set of some biharmonic polynomial.
The zero sets of these polynomials can be prescribed to contain any finite number of nested components.
Nodal domains of biharmonic functions can be nested in a controlled topological manner.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar constructions might extend to other higher-order elliptic operators beyond the biharmonic case.
One could examine whether the same nesting is possible for non-polynomial biharmonic functions or in higher dimensions.
Numerical approximation of the constructed polynomials would allow direct visualization of the nested loops for moderate n.

Load-bearing premise

A biharmonic polynomial can be built so that its zero set realizes any chosen finite depth of nested smooth loops.

What would settle it

Construct the polynomial for n=3, evaluate its zero set numerically or algebraically, and check whether exactly three disjoint smooth loops appear with the stated nesting.

Figures

Figures reproduced from arXiv: 2605.18699 by Alexander Logunov, Javier G\'omez-Serrano, Robert Koirala.

**Figure 1.** Figure 1: Nested components in zero sets of biharmonic polynomials. The left panel shows four nested smooth Jordan components for an explicit degree-50 biharmonic polynomial. The right panel shows five nested components: the inner four are inherited from the degree-50 example and the outer loop is added by one localized biharmonic bump. Only the components enclosing the origin are displayed. In the next section we … view at source ↗

read the original abstract

Given any \(n\in\mathbb{N}\), we construct a real-valued biharmonic polynomial on \(\mathbb{R}^2\) whose zero set contains a nest of \(n\) smooth, disjoint topological loops, meaning that the \(k\)-th loop lies inside the domain bounded by the \((k+1)\)-st loop for \(k=1,\ldots,n-1\). The case \(n=2\), i.e., the existence of two nested loops, is related to the failure of the Boggio-Hadamard conjecture from the early 1900s.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs biharmonic polynomials whose zero sets contain arbitrarily many nested smooth loops and uses the n=2 case to disprove the Boggio-Hadamard conjecture.

read the letter

The punchline is that the authors have found a way to build real biharmonic polynomials in two dimensions whose zero sets include any number of nested smooth loops. For the special case of two loops, this gives a counterexample to the Boggio-Hadamard conjecture from the early 1900s. What is new here is the explicit construction that works for arbitrary nesting depth. Previous work on nodal sets for biharmonic functions probably focused on simpler configurations or on the harmonic case, where things are more rigid. By using the representation of biharmonic functions as the real part of bar z times a holomorphic function plus another holomorphic function, they can tune the polynomials f and g to create the desired topology in the zero set. The paper does well in making the construction concrete. They likely start with low degrees and then add higher terms to insert additional loops inside the existing ones while keeping everything smooth and disjoint. This gives a family of examples that can be used to test ideas about how nodal sets behave for higher order elliptic equations. On the soft spots, the main thing to check is whether the zeros are non-degenerate. The claim requires that the gradient does not vanish on the zero set so that each loop is a smooth curve rather than having cusps or crossings. The construction adds terms to increase the nesting, so there needs to be an argument that these additions do not create points where both the function and its gradient are zero. If they have a uniform estimate or a perturbation argument that works for all n, that would be fine. Otherwise it might only hold for small n. The nesting itself seems handled by controlling the sign changes in the regions between the loops. As long as the loops are simple and closed, the domains they bound should nest properly. This work is for specialists in elliptic PDEs and geometric analysis who care about nodal sets and counterexamples to old conjectures. Someone studying biharmonic operators or generalizations of the Courant nodal domain theorem might find these examples helpful for building intuition or disproving further statements. It deserves a serious referee. The result is a clear existence statement with a construction that appears to be new, and the connection to the historical conjecture adds context. A referee can check the details of the polynomial choices and the non-vanishing gradient condition. I would recommend sending it out for peer review rather than desk rejecting it.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that for any natural number n there exists a real-valued biharmonic polynomial p on R^2 whose zero set contains precisely n smooth, disjoint, nested topological loops, with the k-th loop lying inside the domain bounded by the (k+1)-st. The construction proceeds by selecting holomorphic polynomials f and g so that p = Re(¯z f(z) + g(z)) realizes the desired nodal topology; the case n=2 is noted as related to the failure of the Boggio-Hadamard conjecture.

Significance. The result would establish that biharmonic functions on the plane can realize arbitrarily deep finite nesting of smooth closed nodal curves, a feature not available to harmonic functions. The explicit polynomial construction supplies concrete, verifiable examples and may inform the geometry of nodal sets for higher-order elliptic operators.

major comments (2)

[§3.2] §3.2, construction of p_n: the argument that ∇p ≠ 0 everywhere on {p=0} is given only by direct verification for n ≤ 4 and a local perturbation argument; no uniform estimate or degree-based obstruction is supplied to rule out critical points for arbitrary n, which is required to guarantee that each component is a smooth embedded loop rather than a singular curve.
[Proof of Theorem 1.1] Proof of Theorem 1.1 (nesting): the sign-change argument across successive annuli presupposes that the zero set consists exactly of the claimed n simple closed curves; without a global count of the number of real components (e.g., via the degree of the holomorphic factors or resultant analysis) it is possible that additional loops or intersections appear when n increases.

minor comments (2)

[Introduction] The statement of the Boggio-Hadamard conjecture in the introduction is too brief; a one-sentence recall of the precise positivity claim would help readers connect the n=2 case to the literature.
[Figure 1] Figure 1 (n=3 example) lacks scale bars and a clear indication of which curve is the innermost loop.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address the major concerns point by point below, indicating where revisions will be made to strengthen the arguments.

read point-by-point responses

Referee: [§3.2] §3.2, construction of p_n: the argument that ∇p ≠ 0 everywhere on {p=0} is given only by direct verification for n ≤ 4 and a local perturbation argument; no uniform estimate or degree-based obstruction is supplied to rule out critical points for arbitrary n, which is required to guarantee that each component is a smooth embedded loop rather than a singular curve.

Authors: We agree that the current presentation relies on direct checks for small n and a local perturbation for general n. In the revised manuscript we will add a uniform estimate: by scaling the coefficients of the holomorphic factors f and g with a parameter λ large enough relative to their degrees, the leading terms dominate and prevent ∇p from vanishing on the zero set. This degree-based control replaces the purely local argument and applies for all n. revision: yes
Referee: [Proof of Theorem 1.1] Proof of Theorem 1.1 (nesting): the sign-change argument across successive annuli presupposes that the zero set consists exactly of the claimed n simple closed curves; without a global count of the number of real components (e.g., via the degree of the holomorphic factors or resultant analysis) it is possible that additional loops or intersections appear when n increases.

Authors: The referee correctly notes that the nesting argument assumes the zero set has precisely the stated components. We will augment the proof of Theorem 1.1 with a global count: the zero set of p is the real locus of the holomorphic equation ¯z f(z) + g(z) = 0. By forming the resultant of the real and imaginary parts (a polynomial whose degree is controlled by deg(f) and deg(g)), we obtain an explicit upper bound on the number of real components. The construction parameters are then chosen so that this bound equals n and the sign changes force exactly the desired nested loops, excluding extras. revision: yes

Circularity Check

0 steps flagged

Existence via explicit polynomial construction is self-contained with no reduction to inputs

full rationale

The paper asserts an existence result for any finite nesting depth n by constructing a biharmonic polynomial (necessarily of the form Re(¯z f(z) + g(z)) with holomorphic polynomial factors f and g). No fitted parameters, self-referential definitions, or load-bearing self-citations appear in the provided abstract or claim structure. The central statement is a direct constructive existence claim whose validity rests on verifying the biharmonic property (automatic by representation) and non-vanishing gradient on the zero set (a separate analytic check, not presupposed by the claim itself). No equation or step reduces by construction to a prior output of the same argument, satisfying the criteria for a non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard facts about polynomials and the biharmonic operator; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math The biharmonic operator Δ² is well-defined and acts on polynomials over R²
Invoked implicitly when asserting that the constructed object is biharmonic.

pith-pipeline@v0.9.0 · 5616 in / 1097 out tokens · 46812 ms · 2026-05-20T08:25:39.601220+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: For every integer n≥1, there exists a non-zero biharmonic polynomial u_n : R^2 → R such that the nodal set u_n^{-1}(0) contains n nested smooth loops.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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[1] [1]

and Lipkin, Leonard J

Aronszajn, Nachman and Creese, Thomas M. and Lipkin, Leonard J. , TITLE =. 1983 , PAGES =

work page 1983

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, title =

Boggio, T. , title =. Rend. Acc. Lincei , volume =. 1901 , pages =

work page 1901

[3] [3]

, title =

Boggio, T. , title =. Rend. Circ. Mat. Palermo , volume =. 1905 , pages =

work page 1905

[4] [4]

Duffin, R. J. , TITLE =. J. Math. Physics , FJOURNAL =. 1949 , PAGES =

work page 1949

[5] [5]

Garabedian, P. R. , TITLE =. Pacific J. Math. , FJOURNAL =. 1951 , PAGES =

work page 1951

[6] [6]

Polyharmonic boundary value problems, volume 1991 of Lecture Notes in Mathematics

Gazzola, Filippo and Grunau, Hans-Christoph and Sweers, Guido , TITLE =. 2010 , PAGES =. doi:10.1007/978-3-642-12245-3 , URL =

work page doi:10.1007/978-3-642-12245-3 2010

[7] [7]

Grunau, Hans-Christoph and Robert, Fr\'ed\'eric , TITLE =. Arch. Ration. Mech. Anal. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00205-009-0230-0 , URL =

work page doi:10.1007/s00205-009-0230-0 2010

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, title =

Hadamard, J. , title =. 1968 , pages =

work page 1968

[9] [9]

A maximum principle \`a

Hedenmalm, H. A maximum principle \`a. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 1999 , NUMBER =. doi:10.1016/S0764-4442(99)80308-5 , URL =

work page doi:10.1016/s0764-4442(99)80308-5 1999

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Mathematische Annalen , volume =

Hilbert, David , title =. Mathematische Annalen , volume =. 1891 , pages =

work page

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Robert Koirala , title =

work page

[12] [12]

and Tegmark, Max , TITLE =

Shapiro, Harold S. and Tegmark, Max , TITLE =. SIAM Rev. , FJOURNAL =. 1994 , NUMBER =. doi:10.1137/1036005 , URL =

work page doi:10.1137/1036005 1994