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arxiv: 2606.30792 · v1 · pith:YJRZCGHQnew · submitted 2026-06-29 · 🧮 math.CV

Carleman Approximation for certain sets with an isolated singularity

Harshith Alagandala , Sushil Gorai This is my paper

Pith reviewed 2026-07-01 01:38 UTC · model grok-4.3

classification 🧮 math.CV
keywords Carleman approximationpolynomial convexitytotally real subspacesisolated singularityLipschitz graphshyperbolic complex pointsseveral complex variables
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The pith

Local polynomial convexity at the origin suffices for Carleman approximation on unions of finitely many transverse totally real subspaces of maximal dimension.

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

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.

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.

What carries the argument

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

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

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

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.

read the original 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.

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.

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

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

pith-pipeline@v0.9.1-grok · 5633 in / 911 out tokens · 34084 ms · 2026-07-01T01:38:38.363338+00:00 · methodology

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Reference graph

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