arxiv: 2604.18977 · v1 · submitted 2026-04-21 · 🧮 math.DG · math.SP

Recognition: unknown

The Steklov spectrum of convex polygonal domains II: investigating spectral determination

Carlos Villegas-Blas, Carolyn Gordon, Emily B. Dryden, Javier Moreno, Julie Rowlett

Authors on Pith no claims yet

Pith reviewed 2026-05-10 02:14 UTC · model grok-4.3

classification 🧮 math.DG math.SP

keywords Steklov spectrumspectral determinationconvex polygonstrianglesquadrilateralscharacteristic polynomialpolygonal domains

0 comments

The pith

The Steklov spectrum determines almost all triangles uniquely within the class of triangles and distinguishes triangles and quadrilaterals from smooth domains.

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

The paper investigates how much the Steklov spectrum reveals about the shape of convex polygons. It establishes that almost all triangles are uniquely identified by their Steklov spectrum among all triangles, with additional specific results based on angle types. For special quadrilaterals like rectangles and parallelograms, the spectrum determines the shape up to a few possibilities. The work also shows that triangles and quadrilaterals have spectra that cannot match those of smoothly bounded domains, indicating that corners are detectable in the spectrum.

Core claim

We prove that almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles; further results depending on the types of angles in the triangles are given. We examine three special classes of convex quadrilaterals—rectangles, parallelograms, and kites—and obtain results ranging from unique spectral determination to determination up to three possibilities. For regular n-gons, we prove spectral determination within certain classes. Triangles and quadrilaterals are spectrally distinguished from smoothly bounded domains, and matching such a spectrum imposes restrictions on edge lengths for higher n-gons.

What carries the argument

The characteristic polynomial, which encodes the Steklov spectrum for convex polygons.

Load-bearing premise

The characteristic polynomial from prior works accurately encodes the Steklov spectrum for all convex polygons under consideration.

What would settle it

A counterexample consisting of two non-congruent triangles with identical Steklov spectra would disprove the unique determination for almost all triangles.

Figures

Figures reproduced from arXiv: 2604.18977 by Carlos Villegas-Blas, Carolyn Gordon, Emily B. Dryden, Javier Moreno, Julie Rowlett.

**Figure 1.** Figure 1: A triangle with angles and edges labeled as in [18]. Every curvilinear n-gon Ω has an associated characteristic polynomial PΩ, first introduced in [20, Equation (2.20)] (see also [18]). This will be our main tool for obtaining inverse spectral results. Definition 2.2. Let Ω be a curvilinear n-gon with data ℓ and α as in Notational Conventions 2.1. Define c(αj ) = cos π 2 2αj and s(αj ) = sin π 2 2αj … view at source ↗

**Figure 2.** Figure 2: Since the sum of the distances from the vertex α1 to the other two vertices is fixed and equal to ℓ1 + ℓ2, the vertex α1 must lie on the ellipse with foci at the vertices α2 and α3 as depicted here. If we restrict to the class of isosceles triangles, the characteristic polynomial suffices to uniquely determine each triangle. Theorem 3.3. Within the class of isosceles triangles, each element is uniquely det… view at source ↗

**Figure 3.** Figure 3: The angle of T(v) at vertex v is α1 and it determines T(v) uniquely. Proposition 3.8. Suppose T satisfies either of the following conditions: • T is an acute triangle with no angle in { π 3 , π 5 , π 7 }. • T is a right triangle. Then any triangle that has the same characteristic polynomial as T is congruent to T. Proof. Under either of these assumptions, T is not obtuse. So, any non-obtuse triangle that h… view at source ↗

**Figure 4.** Figure 4: A convex kite is a quadrilateral that has two pairs of adjacent sides that are of equal lengths. Consequently, splitting the kite along the dashed segment, one obtains two congruent triangles. We will see that for almost all kites K, the characteristic polynomial of K determines K up to at most three possibilities within the set of all kites, and a substantial collection of kites are uniquely determined. T… view at source ↗

read the original abstract

The extent to which the geometry of an object is determined by some associated spectral data is a longstanding problem. We investigate this problem in the context of the Steklov spectrum, focusing on convex polygons. We prove that almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles; further results depending on the types of angles in the triangles are given. We examine three special classes of convex quadrilaterals--rectangles, parallelograms, and kites--and obtain results ranging from unique spectral determination to determination up to three possibilities. For regular $n$-gons, we are again able to prove spectral determination within certain classes of polygons. More generally, we investigate the extent to which the Steklov spectrum distinguishes convex polygons from simply-connected domains with smooth boundary; that is, does the Steklov spectrum detect corners? We prove that triangles and quadrilaterals are spectrally distinguished from such smoothly bounded domains; moreover, we show that having the same Steklov spectrum as such a domain imposes substantial restrictions on the edge lengths of higher-order $n$-gons. Throughout, our main tool is the characteristic polynomial developed in works by Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher.

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

This paper applies an existing characteristic polynomial to prove that Steklov spectra determine almost all triangles uniquely among triangles and distinguish triangles and quadrilaterals from smooth domains, with partial results for special quadrilaterals.

read the letter

The core contribution is a set of concrete uniqueness theorems for the Steklov spectrum on convex polygons. Using the characteristic polynomial from Krymski, Levitin, Parnovski, Polterovich, and Sher, the authors show that almost all triangles are spectrally rigid within the class of triangles, with additional statements that depend on whether angles are acute or obtuse. For rectangles, parallelograms, and kites they get determination up to a small finite number of possibilities. They also prove that no triangle or quadrilateral can share its Steklov spectrum with a smooth simply-connected domain, and they give length restrictions for higher n-gons that match a smooth spectrum. These are new applications of the tool rather than a new derivation of it. The work is straightforward: it takes the polynomial, analyzes its roots and coefficients under the geometric constraints of the polygons, and extracts the stated conclusions. That approach is clean and stays within the literature the authors cite. The main limitation is that the entire argument rests on the polynomial being an exact and complete encoding of the Steklov eigenvalues for every convex polygon under consideration. The paper does not re-derive the polynomial or provide independent numerical checks for the angle ranges or multiple-eigenvalue cases that arise here, so any gaps in the earlier papers transfer directly. For regular n-gons the results are again conditional on the same tool. Readers working on inverse Steklov problems or spectral geometry of polygons will find the explicit statements useful. The paper is not a broad advance but a targeted extension that is worth refereeing because the claims are precise and the method is reproducible from the cited source. I would send it to review with the expectation that referees check the polynomial's applicability to the new angle configurations.

Referee Report

3 major / 2 minor

Summary. The manuscript investigates spectral determination via the Steklov spectrum for convex polygonal domains. It proves that almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles, with additional results conditioned on angle types. For three special classes of convex quadrilaterals (rectangles, parallelograms, and kites), it obtains results ranging from unique determination to determination up to three possibilities. Spectral determination is also shown for regular n-gons within certain classes. More generally, the paper proves that triangles and quadrilaterals are spectrally distinguished from simply-connected domains with smooth boundaries and derives substantial restrictions on edge lengths for higher-order n-gons sharing a Steklov spectrum with such domains. All results are obtained by analyzing roots or coefficients of the characteristic polynomial developed in prior works by Krymski, Levitin, Parnovski, Polterovich, and Sher.

Significance. If the central tool encodes the Steklov spectrum exactly for the polygons under consideration, the results would advance the inverse Steklov problem by supplying explicit uniqueness theorems inside the polygonal class and concrete distinctions between polygons and smooth domains. The algebraic approach via the characteristic polynomial is a methodological strength when the encoding is complete and free of unexamined edge cases.

major comments (3)

[triangle uniqueness section] The proofs of uniqueness for almost all triangles (abstract and the section applying the characteristic polynomial to triangles) rest entirely on the assumption that the characteristic polynomial from Krymski et al. encodes the full Steklov spectrum without multiplicity issues or failures at obtuse angles; no independent derivation, numerical cross-check, or explicit verification for these cases appears in the manuscript, making this a load-bearing gap for the central claim.
[special quadrilaterals section] In the analysis of rectangles, parallelograms, and kites (the section on special quadrilaterals), the determination results up to three possibilities are derived from root analysis of the same polynomial; the manuscript does not address whether the polynomial remains exact when eigenvalues coincide or when right angles are present, which directly affects the claimed distinction count.
[smooth-boundary distinction section] The distinction between triangles/quadrilaterals and smooth domains, as well as the edge-length restrictions for higher n-gons (the section on smooth-boundary distinction), inherits any unstated restrictions of the characteristic polynomial at corners; without a self-contained check that the polynomial captures the Dirichlet-to-Neumann map for all convex polygons considered, the distinction claims cannot be fully evaluated.

minor comments (2)

[introduction] The introduction could include a brief self-contained statement of the characteristic polynomial (including its degree and dependence on angles and side lengths) rather than relying solely on citations to prior works.
[notation and preliminaries] Notation for the roots and coefficients of the polynomial is used consistently but would benefit from an explicit table summarizing which geometric quantities they determine for each polygon class.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below. The central tool is the characteristic polynomial established in the cited prior work, and we will make partial revisions to add explicit references, summaries of applicability, and clarifications to strengthen the presentation without changing the core proofs.

read point-by-point responses

Referee: The proofs of uniqueness for almost all triangles (abstract and the section applying the characteristic polynomial to triangles) rest entirely on the assumption that the characteristic polynomial from Krymski et al. encodes the full Steklov spectrum without multiplicity issues or failures at obtuse angles; no independent derivation, numerical cross-check, or explicit verification for these cases appears in the manuscript, making this a load-bearing gap for the central claim.

Authors: The characteristic polynomial is rigorously derived in Krymski et al. to encode the complete Steklov spectrum, including multiplicities via root analysis, for all convex polygonal domains. The derivation is general and applies to obtuse angles without exception. Our triangle uniqueness results follow by direct application of this established result. To address the concern, we will add a brief summary paragraph in the triangle section (or introduction) citing the specific theorems from the prior work and confirming applicability to obtuse cases and multiplicities. This is a partial revision to improve clarity. revision: partial
Referee: In the analysis of rectangles, parallelograms, and kites (the section on special quadrilaterals), the determination results up to three possibilities are derived from root analysis of the same polynomial; the manuscript does not address whether the polynomial remains exact when eigenvalues coincide or when right angles are present, which directly affects the claimed distinction count.

Authors: Rectangles, parallelograms, and kites are convex quadrilaterals covered by the general theory in the cited work, which proves the polynomial encodes the spectrum exactly, with eigenvalue coincidences handled by multiple roots and right angles included in the convex case. The root analysis in our section already accounts for these. We will revise the special quadrilaterals section with a short clarifying paragraph referencing the prior results on these features to justify the distinction counts explicitly. revision: partial
Referee: The distinction between triangles/quadrilaterals and smooth domains, as well as the edge-length restrictions for higher n-gons (the section on smooth-boundary distinction), inherits any unstated restrictions of the characteristic polynomial at corners; without a self-contained check that the polynomial captures the Dirichlet-to-Neumann map for all convex polygons considered, the distinction claims cannot be fully evaluated.

Authors: The prior work establishes that the characteristic polynomial fully captures the Dirichlet-to-Neumann map for every convex polygon, with no additional restrictions at corners. Thus the distinctions for triangles/quadrilaterals and edge-length restrictions for higher n-gons are valid. We will add an explicit statement and citation at the start of the smooth-boundary distinction section to make this foundation clear within the manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity detected; results derived from external characteristic polynomial

full rationale

The paper's core claims (unique determination of almost all triangles, distinctions for quadrilaterals and regular n-gons, and corner detection versus smooth domains) are obtained by algebraic analysis of roots and coefficients of the characteristic polynomial introduced in prior independent works by Krymski, Levitin, Parnovski, Polterovich, and Sher. No derivation step reduces to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain within this manuscript. The cited polynomial is treated as an established external tool whose validity is independent of the present results, making the extension to new polygon classes a standard non-circular mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the characteristic polynomial from cited prior papers as the central computational tool; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The characteristic polynomial developed in prior works accurately represents the Steklov spectrum for convex polygons.
Invoked as the main tool throughout the described results.

pith-pipeline@v0.9.0 · 5548 in / 1143 out tokens · 37951 ms · 2026-05-10T02:14:42.463365+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 23 canonical work pages

[1]

M. S. Agranovich,On a mixed Poincar´ e-Steklov type spectral problem in a Lipschitz domain, Russ. J. Math. Phys.13 (2006), no. 3, 239–244, DOI 10.1134/S1061920806030010. MR2262827

work page doi:10.1134/s1061920806030010 2006
[2]

non-smooth

M. ˇS. Birman and M. Z. Solomjak,The principal term of the spectral asymptotics for “non-smooth” elliptic problems, Funkcional. Anal. i Priloˇ zen.4(1970), no. 4, 1–13 (Russian). MR0278126

1970
[3]

Colbois, A

B. Colbois, A. Girouard, C. Gordon, and D. Sher,Some recent developments on the Steklov eigenvalue problem, Rev. Mat. Complut.37(2024), no. 1, 1–161, DOI 10.1007/s13163-023-00480-3

work page doi:10.1007/s13163-023-00480-3 2024
[4]

E. B. Dryden, C. Gordon, J. Moreno, J. Rowlett, and C. Villegas-Blas,The Steklov spectrum of convex polygonal domains I: spectral finiteness, J. Geom. Anal.35(2025), no. 3, Paper No. 91, 38, DOI 10.1007/s12220-025-01922-8. MR4861159

work page doi:10.1007/s12220-025-01922-8 2025
[5]

Durso,On the inverse spectral problem for polygonal domains, ProQuest LLC, Ann Arbor, MI, 1988

C. Durso,On the inverse spectral problem for polygonal domains, ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.)– Massachusetts Institute of Technology. MR2941198

1988
[6]

Differential Geom.122 (2022), no

Alberto Enciso and Javier G´ omez-Serrano,Spectral determination of semi-regular polygons, J. Differential Geom.122 (2022), no. 3, 399–419, DOI 10.4310/jdg/1675712993. MR4544558

work page doi:10.4310/jdg/1675712993 2022
[7]

Girouard, J

A. Girouard, J. Lagac´ e, I. Polterovich, and A. Savo,The Steklov spectrum of cuboids, Mathematika65(2019), no. 2, 272–310, DOI 10.1112/s0025579318000414

work page doi:10.1112/s0025579318000414 2019
[8]

Girouard, L

A. Girouard, L. Parnovski, I. Polterovich, and D Sher,The Steklov spectrum of surfaces: asymptotics and invariants, Math. Proc. Cambridge Philos. Soc.157(2014), no. 3, 379–389, DOI 10.1017/S030500411400036X

work page doi:10.1017/s030500411400036x 2014
[9]

Girouard and I

A. Girouard and I. Polterovich,Spectral geometry of the Steklov problem (survey article), J. Spectr. Theory7(2017), no. 2, 321–359, DOI 10.4171/JST/164. MR3662010

work page doi:10.4171/jst/164 2017
[10]

,On the Hersch-Payne-Schiffer estimates for the eigenvalues of the Steklov problem, Funktsional. Anal. i Prilozhen. 44(2010), no. 2, 33–47, DOI 10.1007/s10688-010-0014-1 (Russian, with Russian summary); English transl., Funct. Anal. Appl.44(2010), no. 2, 106–117. MR2681956

work page doi:10.1007/s10688-010-0014-1 2010
[11]

Gordon, P

C. Gordon, P. Herbrich, and D. Webb,Steklov and Robin isospectral manifolds, J. Spectr. Theory11(2021), no. 1, 39–61, DOI 10.4171/jst/335. MR4233205

work page doi:10.4171/jst/335 2021
[12]

Gordon, D

C. Gordon, D. Webb, and S. Wolpert,Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math. 110(1992), no. 1, 1–22, DOI 10.1007/BF01231320. MR1181812

work page doi:10.1007/bf01231320 1992
[13]

Grieser and S

D. Grieser and S. Maronna,Hearing the shape of a triangle, Notices Amer. Math. Soc.60(2013), no. 11, 1440–1447, DOI 10.1090/noti1063. MR3154630 26

work page doi:10.1090/noti1063 2013
[14]

Hezari and S

H. Hezari and S. Zelditch,One can hear the shape of ellipses of small eccentricity, Ann. of Math. (2)196(2022), no. 3, 1083–1134, DOI 10.4007/annals.2022.196.3.4. MR4502596

work page doi:10.4007/annals.2022.196.3.4 2022
[15]

Hezari, Z

H. Hezari, Z. Lu, and J. Rowlett,The Neumann isospectral problem for trapezoids, Ann. Henri Poincar´ e18(2017), no. 12, 3759–3792, DOI 10.1007/s00023-017-0617-7. MR3723340

work page doi:10.1007/s00023-017-0617-7 2017
[16]

,The Dirichlet isospectral problem for trapezoids, J. Math. Phys.62(2021), no. 5, Paper No. 051511, 13, DOI 10.1063/5.0036384. MR4262854

work page doi:10.1063/5.0036384 2021
[17]

Karpukhin, J

M. Karpukhin, J. Lagac´ e, and I. Polterovich,Weyl’s law for the Steklov problem on surfaces with rough boundary, Arch. Ration. Mech. Anal.247(2023), no. 5, Paper No. 77, 20, DOI 10.1007/s00205-023-01912-6. MR4629464

work page doi:10.1007/s00205-023-01912-6 2023
[18]

Krymski, M

S. Krymski, M. Levitin, L. Parnovski, I. Polterovich, and David A. Sher,Inverse Steklov spectral problem for curvilinear polygons, Int. Math. Res. Not. IMRN1(2021), 1–37, DOI 10.1093/imrn/rnaa200. MR4198492

work page doi:10.1093/imrn/rnaa200 2021
[19]

Kuznetsov, T

N. Kuznetsov, T. Kulczycki, M. Kwa´ snicki, A. Nazarov, S. Poborchi, I. Polterovich, and B. Siudeja,The legacy of Vladimir Andreevich Steklov, Notices Amer. Math. Soc.61(2014), no. 1, 9–22, DOI 10.1090/noti1073. MR3137253

work page doi:10.1090/noti1073 2014
[20]

Levitin, L

M. Levitin, L. Parnovski, I. Polterovich, and D. A. Sher,Sloshing, Steklov and corners: asymptotics of Steklov eigenvalues for curvilinear polygons, Proc. Lond. Math. Soc. (3)125(2022), no. 3, 359–487, DOI 10.1112/plms.12461. MR4480880

work page doi:10.1112/plms.12461 2022
[21]

Lu and J

Z. Lu and J. Rowlett,One can hear the corners of a drum, Bull. Lond. Math. Soc.48(2016), no. 1, 85–93, DOI 10.1112/blms/bdv094. MR3455751

work page doi:10.1112/blms/bdv094 2016
[22]

,The sound of symmetry, Amer. Math. Monthly122(2015), no. 9, 815–835, DOI 10.4169/amer.math.monthly.122.9.815. MR3418203

work page doi:10.4169/amer.math.monthly.122.9.815 2015
[23]

Mancino, July 2025

D. Mancino, July 2025. Private communication

2025
[24]

Nilsson, J

E. Nilsson, J. Rowlett, and F. Rydell,The isospectral problem for flat tori from three perspectives, Bull. Amer. Math. Soc. (N.S.)60(2023), no. 1, 39–83, DOI 10.1090/bull/1770. MR4520776

work page doi:10.1090/bull/1770 2023
[25]

Nursultanov, J

M. Nursultanov, J. Rowlett, and D. Sher,The heat kernel on curvilinear polygonal domains in surfaces, Ann. Math. Qu´ e. 49(2025), no. 1, 1–61, DOI 10.1007/s40316-024-00237-4 (English, with English and French summaries). MR4894857

work page doi:10.1007/s40316-024-00237-4 2025
[26]

G. V. Rozenblum,Weyl asymptotics for Poincar´ e-Steklov eigenvalues in a domain with Lipschitz boundary, J. Spectr. Theory13(2023), no. 3, 755–803, DOI 10.4171/jst/477. MR4670344 Emily Dryden, Department of Mathematics, Bucknell University, Lewisburg, PA 17837 USA URL:http://www.unix.bucknell.edu/~ed012/ Email address:emily.dryden@bucknell.edu Carolyn Gor...

work page doi:10.4171/jst/477 2023