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Juliusz Banecki

Publications and source records attributed to Juliusz Banecki.

13 recordsLinked to original sources
arXiv2026

Optimal embedding dimension in the Nash--Tognoli theorem

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.

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arXiv2026

Invariants of real affine varieties based on their complexifications

We introduce a new family of invariants of real algebraic sets defined in terms of the topology of their complexifications and compute some of these invariants for spheres. This allows us to completely classify topological isomorphism classes of algebraic vector bundles over products of two spheres. We also obtain new results concerning both the existence and nonexistence of regular maps from products of spheres into spheres. Additionally, we show that the newly defined invariants provide obstructions to weak algebraic approximation of smooth submanifolds of real algebraic sets, disproving a conjecture of Kucharz and Kurdyka.

math.AG↗
arXiv2025

Uniform rationality of products with the plane

Let $X$ be a nonsingular rational variety. We prove that $X\times \mathbb{C}^2$ is uniformly rational. It follows that nonsingular stably rational varieties are stably uniformly rational.

math.AG↗
arXiv2025

Homotopy properties of regular mappings into real retract rational varieties

We study homotopy properties of regular mappings from spheres into a real retract rational variety $Y$. We show that the homotopy classes which are represented by such mappings form subgroups of the homotopy groups of $Y$, and that the groups are independent of the choice of the basepoint on $Y$ as long as $Y$ is connected. We also construct regular representatives of all the Whitehead products in all the homotopy groups of $Y$.

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arXiv2024

Relative Stone-Weierstrass theorem for mappings between varieties

We introduce a class of real algebraic varieties characterised by a simple rationality condition, which exhibit strong properties regarding approximation of continuous and smooth mappings by regular ones. They form a natural counterpart to the classes of malleable and quasi-malleable varieties. The approximation property studied here is stronger than those considered before and allows us to deduce non-trivial facts about extensions of regular mappings between varieties.

math.AG↗
arXiv2024

Approximation of maps between real algebraic varieties

A nonsingular real algebraic variety Y is said to have the approximation property if for every real algebraic variety X the following holds: if f:X-->Y is a C^inf map that is homotopic to a regular map, then f can be approximated in the C^inf topology by regular maps. In this paper, we characterize the varieties Y with the approximation property. We also characterize the varieties Y with the approximation property combined with a suitable interpolation condition. Some of our results have variants concerning the regular approximation of continuous maps defined on (possibly singular) real algebraic varieties.

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arXiv2024

Separately regular and separately Nash functions

We strengthen certain known results saying that separately regular functions are rational and separately Nash functions are semialgebraic. The approach presented here unifies and highlights the similarities between the two problems.

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arXiv2024

Algebraic homotopy classes

We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic to a regular one. Furthermore we prove that algebraic homotopy classes of spheres form a subgroup of the homotopy group, and that a similar result holds also for cohomotopy groups of arbitrary varieties.

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arXiv2023

Extensions of k-regulous functions from two-dimensional varieties

We prove that a $k$-regulous function defined on a two-dimensional non-singular affine variety can be extended to an ambient variety. Additionally we derive some results concerning sums of squares of $k$-regulous functions; in particular we show that every positive semi-definite regular function on a non-singular affine variety can be written as a sum of squares of locally Lipschitz regulous functions.

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arXiv2021

Distortion in the group of circle homeomorphisms

Let $G$ be the group ${\rm PAff}_+({\bf S}^1)$ of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in $G$ is given.

math.DS↗
arXiv2020

Sums of even powers of k-regulous functions

We provide an example of a nonnegative $k$-regulous function on $\mathbb{R}^n$ for $k\geq 1$ and $n \geq 2$ which cannot be written as a sum of squares of $k$-regulous functions. We then obtain lower bounds for Pythagoras numbers $p_{2d}(\mathcal{R}^k(\mathbb{R}^n))$ of $k$-regulous functions on $\mathbb{R}^n$ for $k\geq 1$ and $n\geq 2$. We also prove that the second Pythagoras number of the ring of $0$-regulous functions $\mathcal{R}^0(X)$ on an irreducible $0$-regulous affine variety $X$ is finite and bounded from above by $2^{\dim X}$.

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