🔢 mathematics

Optimal embedding dimension in the Nash--Tognoli theorem

This paper resolves a 1952 conjecture by Nash by proving that every smooth compact submanifold of $\R^n$ can be approximated via a small isotopy by the real locus of a nonsingular complex algebraic subset of $\C^n$ , with the additional result that if the codimension is at least two, these approximating sets can be chosen to share a predetermined biregular isomorphism type.

Original authors: Juliusz Banecki

Published 2026-05-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juliusz Banecki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Turning Smooth Shapes into "Mathematical Lego"

Imagine you have a smooth, perfect, curved object—like a polished marble statue or a sleek, curved piece of furniture. In the world of mathematics, this is called a smooth manifold. It is defined by its smooth curves and has no sharp edges or corners.

Now, imagine you want to build a copy of this object using only algebraic equations (think of these as the strict, rigid rules of "mathematical Lego"). These equations create shapes called algebraic sets. Usually, these shapes are a bit "crunchy"—they might have sharp corners, self-intersections, or singularities where the math breaks down.

For decades, mathematicians have asked: Can we take any smooth, curved object and find a set of strict algebraic equations that creates a shape almost identical to it?

This paper says yes, and it does so in the most efficient way possible.

The History: The "Extra Room" Problem

In the 1950s, a mathematician named John Nash proved that you could approximate these smooth shapes with algebraic ones, but he had to cheat a little. He had to move the object into a larger room (a higher-dimensional space) to make the math work. It was like trying to fold a complex origami crane; if the paper is too small, you can't do it, so you have to use a bigger sheet of paper.

Later, in the 1990s, Akbulut and King improved this. They showed you could get rid of some of the "extra" parts of the shape, but they still couldn't get rid of the need for the bigger room. They were stuck with the idea that to make the algebraic shape perfect, you had to add an extra dimension.

The Breakthrough: Fitting the Puzzle in the Same Box

Juliusz Banecki's paper solves this problem. He proves that you do not need a bigger room. You can take a smooth shape living in a specific space (say, 3D space) and find a perfect algebraic approximation that lives in that exact same space.

He calls this the "Optimal Embedding Dimension." It's like proving you can fold that complex origami crane perfectly on the original piece of paper without needing a larger sheet.

The Secret Sauce: "Shifting the Cracks"

How did he do it? The paper uses a clever technique called "singularities shifting."

Imagine your algebraic shape has some "cracks" or "kinks" (mathematicians call these singularities).

The Problem: You want the shape to be smooth everywhere, but the equations naturally create these kinks.
The Trick: Banecki uses a mathematical "sliding" technique. He takes the kinks and pushes them away from the smooth part of the shape.
The Double Slide: He doesn't just push them once. He pushes the kinks in one direction, then switches and pushes them in another direction, and finally pushes them all the way to "infinity" (off the board entirely).

By the time he is done, the smooth part of the shape is left perfectly clean, and all the "bad" math parts have been pushed so far away that they no longer interfere with the shape you are trying to build.

The "Complex" Twist

There is one more layer to this. The paper doesn't just look at the shape in our real world (Real numbers); it looks at the shape in a "complex" world (Complex numbers), which is like a super-charged version of reality with extra dimensions hidden inside.

Banecki proves that not only is the real shape smooth, but its "complex twin" is also perfectly smooth. This is a very strong condition. It's like saying that not only is your physical sculpture perfect, but if you were to look at it through a magical lens that reveals hidden dimensions, it would still look perfect.

What This Means for the "Copy"

The paper also shows something fascinating about the identity of the shape.

If you have a smooth shape that is topologically the same as a specific algebraic shape (like a donut shape), you can approximate it with an algebraic shape that is exactly that same type of donut.
You don't just get a "blob" that looks like a donut; you get a mathematically identical "donut" made of equations.

The One Caveat (The "One Dimension" Rule)

The author notes a small limitation. This magic trick works perfectly if the smooth shape is "thick" enough compared to the space it lives in. Specifically, if the space has at least two more dimensions than the shape itself (e.g., a 2D surface in 4D space), the trick works perfectly.

If the shape is just one dimension smaller than the space (like a 2D surface in 3D space), the paper doesn't prove it works yet, though the author suspects it might.

Summary

In short, Juliusz Banecki has proven that you can turn any smooth, curved object into a rigid, equation-based object without needing to move it to a bigger universe. He did this by mathematically "shoving" all the imperfections out of the way until only the perfect shape remained. This settles a question that mathematicians have been wrestling with for over 70 years.

Technical Summary: Optimal Embedding Dimension in the Nash–Tognoli Theorem

Problem Statement
The paper addresses the classical problem of approximating smooth compact submanifolds of $\mathbb{R}^n$ by real algebraic sets up to a small isotopy. While the Nash–Tognoli theorem established that any smooth compact connected manifold $M \subset \mathbb{R}^n$ (with $n > 2 \dim M$ ) can be approximated by a nonsingular connected component of a real algebraic set, it left open the question of whether the ambient dimension $n$ could be preserved without introducing extraneous connected components or increasing the dimension of the ambient space.

Specifically, the paper targets a conjecture posed by Nash in 1952: Can every smooth compact submanifold $M \subset \mathbb{R}^n$ be approximated by the real locus of a nonsingular complex algebraic subset of $\mathbb{C}^n$ defined over $\mathbb{R}$ ? Previous work by Akbulut and King (1992) achieved approximation by nonsingular algebraic sets but required an increase in ambient dimension (moving from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$ ) to ensure the resulting set was nonsingular. The goal of this paper is to resolve the approximation problem without increasing the ambient dimension and to ensure the entire complexification of the approximating set is smooth.

Methodology
The proof of the main result (Theorem 1.4) relies on a "singularities shifting" technique, originally developed by Akbulut and King, which is refined and extended in this work. The methodology proceeds through the following logical steps:

Polynomial Approximation: Starting with a smooth embedding $i: W \to \mathbb{R}^n$ of a nonsingular compact real algebraic set $W$ , the author first approximates $i$ by a polynomial map $j$ such that the complexification $j_C: W_C \to \mathbb{C}^n$ is a finite map (Lemma 3.1).
Singularities Shifting (Lemma 3.2): The core technical innovation involves manipulating the singular locus of the Zariski closure of the image. By constructing a specific polynomial perturbation (using Mergelyan's theorem and Stone-Weierstrass approximations), the author shifts the singularities of the complexified set away from the real locus. This is achieved by:
- Ensuring the projection $\pi_{1,2}$ (dropping the first two coordinates) is finite on the complexification.
- Constructing a polynomial function that separates the projection of the singular locus from the projection of the nonsingular real locus.
- Iteratively switching the roles of the first two variables ( $x_1$ and $x_2$ ) to shift singularities first in the $x_1$ direction, then the $x_2$ direction, and finally to infinity.
Removal of Singularities (Lemma 3.3 & Theorem 1.4): Once the singularities are separated from the real locus in the projection, the author employs a rational map (specifically a map involving a sum of squares of polynomials in the denominator) to "push" the singularities to infinity. This transforms the quasi-affine variety containing the real locus into a globally nonsingular complex algebraic set.

Key Contributions and Results

Theorem 1.3 (Main Result): The paper proves that every smooth compact submanifold $M \subset \mathbb{R}^n$ can be approximated by the real locus of a nonsingular complex algebraic subset of $\mathbb{C}^n$ defined over $\mathbb{R}$ . This settles the Nash conjecture regarding the optimal embedding dimension, confirming that no increase in ambient dimension is necessary.
Theorem 1.4 (Stronger Result): The paper establishes a stronger approximation theorem for embeddings of nonsingular compact real algebraic sets $W \subset \mathbb{R}^m$ $W \subset R^{m}$ into $\mathbb{R}^n$ $R^{n}$ (where $n \ge \dim W + 2$ $n \geq dim W + 2$ ). It shows that the embedding can be approximated by a regular map $j$ $j$ such that:
1. The image $j(W)$ is a nonsingular algebraic subset of $\mathbb{R}^n$ .
2. $j$ is a biregular isomorphism between $W$ and $j(W)$ .
3. The Zariski closure of $j(W)$ in $\mathbb{C}^n$ is a nonsingular complex algebraic set.
Corollary 1.6 & 1.7 (Isomorphism Types): For manifolds of codimension at least 2, the paper demonstrates that the approximating algebraic sets can be chosen to have a predetermined biregular isomorphism type. Specifically, if $M$ is diffeomorphic to a rational nonsingular real algebraic set, it can be approximated by rational nonsingular real algebraic sets. This provides a complete characterization of when rational approximation is possible in codimension $\ge 2$ .

Significance and Claims
The paper claims to resolve a long-standing open problem in real algebraic geometry that has seen no progress since the work of Akbulut and King in 1992. By removing the requirement to increase the ambient dimension (from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$ ), the result provides an "optimal" embedding dimension for the Nash–Tognoli theorem.

The author explicitly notes that the current approach does not cover the case where the codimension is 1 (i.e., $n = \dim W + 1$ ), though it is conjectured that the results hold in that case as well. The paper emphasizes that the result is false if one requires the Zariski closure in the projective space $\mathbb{CP}^n$ to be nonsingular, highlighting the specific nature of the affine nonsingularity achieved here.

The significance lies in the ability to approximate smooth manifolds by algebraic sets that are not only smooth in the real locus but possess a globally smooth complexification, thereby preserving the topological and algebraic structure of the original manifold without extraneous components or dimensional inflation.