Pith. sign in

REVIEW 1 minor 19 references

Carleman Approximation for certain sets with an isolated singularity

T0 review · 0 major / 1 minor · reviewed 2026-07-01 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Local polynomial convexity at the origin suffices for Carleman approximation on unions of finitely many transverse totally real subspaces of maximal dimension.

desk verdict This paper gives a few narrow sufficiency conditions for Carleman approximation on unions of transverse totally real subspaces and Lipschitz graphs in C^2, but the advance stays incremental and specialized. read the letter →

arxiv 2606.30792 v1 pith:YJRZCGHQ submitted 2026-06-29 math.CV

Harshith Alagandala , Sushil Gorai This is my paper · ORCID
classification math.CV
keywords CarlemanapproximationpolynomialconvexitytotallyrealsubspacesisolatedsingularityLipschitzgraphshyperboliccomplexpointsseveralvariables
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 establishes sufficiency of local polynomial convexity at an isolated singularity for Carleman approximation on unions of transverse totally real subspaces. It supplies additional conditions that guarantee polynomial convexity for the union of three such planes in C squared and for unions of two Lipschitz graphs. A concrete family of surfaces in C squared that contain a hyperbolic complex point is shown to satisfy the approximation property. A sympathetic reader would care because the result enlarges the class of sets on which continuous functions can be approximated by holomorphic functions even when smoothness fails at one point.

What carries the argument

The local polynomial convexity condition at the isolated singularity, applied to transverse totally real subspaces of maximal dimension.

What would settle it

An explicit union of transverse totally real subspaces of maximal dimension that is locally polynomially convex at the origin yet fails to admit Carleman approximation.

Watch

Extended reading notes

Core claim

The authors prove that local polynomial convexity at the origin for the union of finitely many transverse totally real subspaces of maximal dimension is sufficient for Carleman approximation. They give new conditions for the polynomial convexity of the union of three transverse totally real planes in C squared. They also provide a sufficient condition on the union of two Lipschitz graphs for Carleman approximation together with sufficient conditions for such unions to be polynomially convex, and they exhibit a family of surfaces in C squared with a hyperbolic complex point that allows Carleman approximation.

Load-bearing premise

The subspaces must be transverse, of maximal dimension, and the singularity at the origin must be isolated.

Editorial extensions

If this is right

  • Carleman approximation holds for any such union once local polynomial convexity at the origin is verified.
  • The union of three transverse totally real planes in C squared is polynomially convex under the new conditions supplied.
  • Unions of two Lipschitz graphs admit Carleman approximation under the stated sufficient condition.
  • The exhibited family of surfaces in C squared with a hyperbolic complex point permits Carleman approximation.

Reading between the lines

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

  • The sufficiency result may extend without change to unions involving more than three subspaces in dimensions higher than two.
  • Similar local-convexity criteria could apply to approximation questions on other isolated singularities that are not unions of linear subspaces.
  • Explicit parametrizations of the surfaces with hyperbolic points could be used to test whether the approximation rate can be made quantitative.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The paper claims to prove that local polynomial convexity at the origin for the union of finitely many transverse totally real subspaces of maximal dimension is sufficient for Carleman approximation. It provides new conditions for the polynomial convexity of the union of three transverse totally real planes in C^2, sufficient conditions for the union of two Lipschitz graphs to be polynomially convex and to allow Carleman approximation, and a family of surfaces in C^2 with a hyperbolic complex point that permits Carleman approximation.

Significance. If the central sufficiency result holds, it would extend Carleman approximation theory to certain isolated singularities in several complex variables by linking it to local polynomial convexity under transversality and maximality conditions. The additional results on three planes, Lipschitz graphs, and hyperbolic points would supply concrete criteria in C^2, potentially useful for further work on approximation on singular sets.

minor comments (1)
  1. The abstract states the main sufficiency result but does not indicate whether the proofs rely on any previously unpublished lemmas or reductions that would require separate verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and summary of the manuscript. The recommendation is listed as uncertain, but the report contains no specific major comments or questions to address point by point. We appreciate the referee's recognition of the potential significance of linking local polynomial convexity to Carleman approximation under the stated transversality and maximality conditions. If any concrete concerns arise, we are happy to provide further clarification or revisions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper is a pure-mathematics result that states and proves a sufficiency theorem: local polynomial convexity at an isolated singularity for unions of transverse totally real subspaces (of maximal dimension) implies Carleman approximation. The abstract and described structure supply explicit hypotheses (transversality, maximal dimension, isolated singularity) and then derive the conclusion via standard complex-analytic arguments; no equation is shown to be definitionally equivalent to its input, no parameter is fitted and then relabeled as a prediction, and no load-bearing step reduces to a self-citation whose content is itself unverified. The derivation therefore remains self-contained against external benchmarks of polynomial convexity and approximation theory.

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

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger is therefore empty.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Carleman Approximation for certain sets with an isolated singularity." pith.science (2026). https://pith.science/paper/YJRZCGHQ

@misc{pith2026260630792,
  author       = {Pith},
  title        = {Pith review of: Carleman Approximation for certain sets with an isolated singularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJRZCGHQ}},
  note         = {Machine review of arXiv:2606.30792}
}
abstract

In this paper, we prove that local polynomial convexity at the origin for the union of finitely many transverse totally real subspaces of maximal dimension is sufficient for Carleman approximation. Some new conditions are given for the polynomial convexity of the union of three transverse totally real planes in $\mathbb{C}^2$. We also provide a sufficient condition on the union of two Lipschitz graphs for Carleman approximation. Along the way, we provide sufficient conditions for union of two Lipschitz graphs to be polynomially convex. Finally, we find a family of surfaces in $\mathbb{C}^2$ with a hyperbolic complex point that allows Carleman approximation.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    Harshith Alagandala,Local polynomial convexity at hyperbolic CR-singularities inM n ⊂C n, Preprint (2025), available at2511.18190

    work page arXiv 2025

  2. [2]

    J.32(1965), 1–21

    Errett Bishop,Differentiable manifolds in complex Euclidean space, Duke Math. J.32(1965), 1–21. MR200476

    work page 1965

  3. [3]

    Carleman,Sur un theoreme de Weierstrass, Ark

    T. Carleman,Sur un theoreme de Weierstrass, Ark. Mat. 20B (1927), 1–5

    work page 1927

  4. [4]

    Carlos A. A. Florentino,Simultaneous similarity and triangularization of sets of 2 by 2 matrices, Linear Algebra Appl.431(2009), no. 9, 1652–1674. MR2555066

    work page 2009

  5. [5]

    Forstneriˇ c and E

    F. Forstneriˇ c and E. L. Stout,A new class of polynomially convex sets, Ark. Mat.29(1991), no. 1, 51–62. MR1115074

    work page 1991

  6. [6]

    1-2, 43–62

    Sushil Gorai,Local polynomial convexity of the union of two totally-real surfaces at their intersec- tion, manuscripta mathematica135(2011), no. 1-2, 43–62

    work page 2011

  7. [7]

    21, 4985–5001

    ,On the polynomial convexity of the union of three totally real planes inC 2, International Mathematics Research Notices2013(2013), no. 21, 4985–5001

    work page 2013

  8. [8]

    ,A note on polynomial convexity of the union of finitely many totally-real planes inC 2, J. Math. Anal. Appl.418(2014), no. 2, 842–851

    work page 2014

Show all 19 references
  1. [9]

    ,Certain real surfaces inC 2 with isolated singularities, to appear - Annales de l’Institut Fourier (2025), available at1909.04085

    2025

  2. [10]

    New Series, vol

    Maciej Klimek,Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publica- tions. MR1150978

    1991

  3. [11]

    Benedikt S Magnusson and Erlend Fornæss Wold,A characterization of totally real Carleman sets and an application to products of stratified totally real sets, Mathematica Scandinavica (2016), 285–290

    2016

  4. [12]

    Per E Manne,Some results on Carleman approximation by entire functions inC n, Inst., Univ, 1992

    1992

  5. [13]

    ,Carleman approximation on totally real submanifolds of a complex manifold, Several com- plex variables (Stockholm, 1987/1988)38(1993), 519–528

    1987

  6. [14]

    4, 522–526

    ,Carleman approximation by entire functions on the union of two totally real subspaces of Cn, Canadian Mathematical Bulletin37(1994), no. 4, 522–526

    1994

  7. [15]

    Ann.351(2011), no

    Per Erik Manne, Erlend Fornæss Wold, and Nils Øvrelid,Holomorphic convexity and Carleman approximation by entire functions on Stein manifolds, Math. Ann.351(2011), no. 3, 571–585. MR2854106

    2011

  8. [16]

    Analyse Math.29(1976), 16–

    Stephen Scheinberg,Uniform approximation by entire functions, J. Analyse Math.29(1976), 16–

    1976

  9. [17]

    Rasul Shafikov and Alexandre Sukhov,Polynomially convex hulls of singular real manifolds, Trans. Amer. Math. Soc.368(2016), no. 4, 2469–2496. MR3449245

    2016

  10. [18]

    261, Springer Science & Business Media, 2007

    Edgar Lee Stout,Polynomial convexity, Vol. 261, Springer Science & Business Media, 2007

    2007

  11. [19]

    1, 131–138

    Barnet M Weinstock,On the polynomial convexity of the union of two maximal totally real sub- spaces ofC n, Mathematische Annalen282(1988), no. 1, 131–138. Department of Mathematics, Western University, London, ON, Canada – N6A 3K7 Email address:halagand@uwo.ca, sairaj922@gmail...

    1988

Pith tools

Reviewed July 1, 2026 · model on record in the stance chip above.