arXiv subjects

Andrey Ryabichev

Publications and source records attributed to Andrey Ryabichev.

9 recordsLinked to original sources

arXiv2025

On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map onto the boundary of $(n+1)$-simplex. We show that for $h=\lfloor\frac{n+1}2\rfloor$, the function $\lambda(n,d)^h$ is almost linear in $d$ as $d\to\infty$ answering a question by O.Musin. All triangulations we obtain are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.

math.CO↗

arXiv2024

Chebyshev polynomials, their remarkable properties and connection with Catalan numbers

The main result of the article says that the formal power series equal to the ratio of two neighboring Chebyshev polynomials, after some renormalization, approximates the generating function of the Catalan numbers. We present a proof of this result using continued fractions. Also, for completeness of the presentation, we recall the basic properties of Chebyshev polynomials and formal power series that we use.

math.CO↗

arXiv2024

Roots of polynomials and tangents of circles

Given a real cubic function $f(x)$ with three roots, take an equilateral triangle $ABC$, the projections of which vertices are the roots of $f(x)$. There is a folklore fact that the vertical lines through the extrema of $f(x)$ are tangent to the inscribed circle of $ABC$. We generalise this fact to a regular $n$-gon and the corresponding degree $n$ polynomial.

math.CA↗

arXiv2023

Maps of manifolds of the same dimension with prescribed Thom-Boardman singularities

In this paper we extend Y.Eliashberg's $h$-principle to arbitrary generic smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity $Σ^I$ at each point. In dimension 3 we rephrase these conditions in terms of the Stiefel-Whitney classes and the cohomology classes of the given loci of folds, cusps and swallowtail points.

math.GT↗

arXiv2023

Short proof of the Kneser-Edmonds theorem on the degree of a map between closed surfaces

Suppose for closed surfaces $M,N$ there exists a continuous map $f:M\to N$ of geometric degree $d>0$. Then $χ(M)\le d\cdotχ(N)$. This inequality was first proved by Kneser in case of orientable surfaces and by Edmonds for arbitrary $M,N$. We give a new simple proof of this result. Our proof is completely elementary and does not use additional techniques (such as the factorisation theorem of Edmonds and the absolute degree theory of Hopf).

math.GT↗

arXiv2023

Maximal degree of a map of surfaces

Given closed possibly nonorientable surfaces $M,N$, we prove that if a map $f:M\to N$ has degree $d>0$, then $χ(M)\le d\cdotχ(N)$. We give all necessary comments on the definition and properties of geometric degree, which can be defined for any map. Our proof is based on the Kneser-Edmonds factorization theorem, simple natural proof of which is also presented.

math.GT↗

arXiv2023

Immersions of manifolds

This text is based on my talk at the popular science conference ``Dark geometry fest'' which was related to geometric methods and their applications, July 17, 2022. We will move towards the Smale-Hirsch theorem. To this end we will deal with the simplest partial case of this theorem -- the Whitney-Graustein theorem on regular curves in the plane. The text is written in an accessible and sometimes informal way, it is intended primarily for people who are interested in mathematics, but have not yet studied it deeply enough.

math.HO↗

arXiv2021

Cutting a triangle into equal triangles

We prove that almost every triangle can be dissected only into $n^2$ triangles which have to be equal one another. Moreover, such a dissection is unique for every $n$. It turns out that to solve this "simple" problem it is convenient to use some tools from linear algebra, calculus and measure theory.

math.MG↗

arXiv2019

Eliashberg's $h$-principle and generic maps of surfaces with prescribed singular locus

We extend Y.Eliashberg's $h$-principle to smooth maps of surfaces which are allowed to have cusp singularities, as well as folds. More precisely, we prove a necessary and sufficient condition for a given map of surfaces to be homotopic to one with given loci of folds and cusps. Then we use these results to obtain a necessary and sufficient condition for a subset of a surface $M$ to be realizable as the critical set of some generic smooth map from $M$ to a given surface $N$.

math.GT↗