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arxiv: 2605.22928 · v1 · pith:IMR54VD2new · submitted 2026-05-21 · 🧮 math.AG

Optimal embedding dimension in the Nash--Tognoli theorem

Juliusz Banecki This is my paper

Pith reviewed 2026-05-25 05:26 UTC · model grok-4.3

classification 🧮 math.AG
keywords Nash conjecturereal algebraic setsisotopy approximationembedding dimensionnonsingular varietiesreal locismooth submanifoldsalgebraic realization
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The pith

Every smooth compact submanifold of R^n can be isotoped to the real locus of a nonsingular real algebraic variety in C^n.

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

The paper proves that any smooth compact submanifold embedded in Euclidean space R^n admits a small isotopy making it coincide with the real points of a nonsingular complex algebraic set defined over R inside C^n. This settles one version of Nash's 1952 conjecture on algebraic realization of manifolds. The approximation works in the original ambient dimension without any increase. When the codimension is at least two, all such approximating sets can moreover be taken to have one fixed biregular isomorphism type chosen in advance.

Core claim

Every smooth compact submanifold of R^n can be approximated up to a small isotopy by the real locus of a nonsingular complex algebraic subset of C^n defined over R.

What carries the argument

Small isotopy approximation of a smooth compact submanifold by the real locus of a nonsingular real algebraic variety inside the same ambient dimension n.

If this is right

  • When the codimension is at least two, the approximating algebraic sets can all be chosen from a single predetermined biregular isomorphism class.
  • Real algebraic geometry applies directly to smooth manifolds inside their original Euclidean embedding space.
  • The 1952 Nash conjecture holds without any increase in embedding dimension.
  • Algebraic models of manifolds become available in the minimal dimension where the manifold sits.

Where Pith is reading between the lines

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

  • The uniform algebraic type result for high codimension may simplify classification or deformation problems that mix topology and algebra.
  • The same-dimension result suggests that many differential-topological constructions on embedded manifolds can be replaced by algebraic ones without changing the ambient space.
  • It remains open whether analogous statements hold for noncompact manifolds or for approximations with additional constraints such as prescribed singularities.

Load-bearing premise

The given submanifold is smooth, compact, and already embedded in R^n, so the algebraic approximation does not require raising the ambient dimension.

What would settle it

Exhibit one smooth compact submanifold M inside some R^n together with a neighborhood in the space of embeddings such that no embedding in that neighborhood is the real locus of a nonsingular real algebraic set in C^n.

read the original abstract

We prove that every smooth compact submanifold of $\R^n$ can be approximated up to a small isotopy by the real locus of a nonsingular complex algebraic subset of $\C^n$ defined over $\R$. This settles a version of a conjecture posed in 1952 by Nash. Moreover, we show that if the codimension of the manifold being approximated is at least two, then the approximating real algebraic sets can be chosen to all have the same predetermined biregular isomorphism type.

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 / 2 minor

Summary. The manuscript proves that every smooth compact submanifold of R^n can be approximated up to a small isotopy by the real locus of a nonsingular complex algebraic subset of C^n defined over R. This settles a version of Nash's 1952 conjecture. For manifolds of codimension at least 2, the approximating algebraic sets may be chosen with a fixed predetermined biregular isomorphism type.

Significance. If the result holds, it establishes the optimal (original) embedding dimension for the Nash-Tognoli theorem, improving on prior work that required stabilization in higher dimensions. The direct construction inside C^n, without invoking higher-dimensional embeddings, and the uniform biregular type in codimension >=2 are notable strengths for applications in real algebraic geometry.

minor comments (2)
  1. The abstract states the main theorem clearly, but the manuscript should include explicit references to the sequence of polynomial approximations and isotopy adjustments used in the construction (as described in the skeptic's note) to allow verification of the direct embedding in C^n.
  2. Notation for the real locus and the biregular isomorphism type should be defined at first use in the introduction for readers unfamiliar with the Nash-Tognoli literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of our result and for highlighting its significance in settling a version of Nash's conjecture at the optimal embedding dimension. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript advances a direct proof of an approximation theorem for smooth compact submanifolds of R^n by real loci of nonsingular complex algebraic sets in C^n, using sequences of polynomial approximations and isotopy adjustments that remain within the original ambient dimension. No load-bearing step reduces to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain; the argument is self-contained against the external Nash conjecture and does not invoke prior results by the same author as uniqueness theorems or ansatzes. The codimension >=2 case fixes biregular type independently of the main construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the result relies on standard background from differential topology and real algebraic geometry with no new free parameters or invented entities mentioned.

axioms (1)
  • standard math Standard properties of smooth manifolds, isotopies, and nonsingular algebraic varieties over the reals
    The theorem statement invokes these as background assumptions from the field.

pith-pipeline@v0.9.0 · 5594 in / 1132 out tokens · 23024 ms · 2026-05-25T05:26:48.317923+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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