Related papers: Immersions of spheres and algebraically constructi…

The main result of this article establishes the free analog of Grothendieck's Theorem on bijective polynomial mappings of $\mathbb{C}^g$. Namely, we show if $p$ is a polynomial mapping in $g$ freely non-commuting variables sending…

We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any $q > 0$ and…

Let g:X -> Y be a smooth (i.e. C^\infty differentiable) map between two smooth manifolds. In analogy with the case of complex polynomial functions, we say that y_0 in Y is a typical value of g if there exists an open neighbourhood U of y_0…

Let $f$ be a homogeneous polynomial of even degree $d$. We study the decompositions $f=\sum_{i=1}^r f_i^2$ where $\mathrm{deg} f_i=d/2$. The minimal number of summands $r$ is called the $2$-rank of $f$, so that the polynomials having…

The phenomena that cause a value of a polynomial function to be a bifurcation one are yet to be described when the fibers have dimension higher than $1$. In this note, the main result is the construction of a polynomial submersion function…

We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…

An elliptic curve may be immersed in ${\mathbf{P}}^{N-1}$ as a degree $N$ curve using level $N$ structure. In the case where $N$ is odd, there are well known classical results dating back to Bianchi and Klein. In this paper we study the…

We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.

S. Cappell and J. Shaneson constructed a pair of inequivalent embeddings of $(n-1)$-spheres in homotopy $(n+1)$-spheres for every square matrix of order $n$ with special properties (a Cappell-Shaneson matrix). A Cappell-Shaneson polynomial…

Let k be an algebraically closed field of characteristic zero. An element F from k(x_1,...,x_n) is called a closed rational function if the subfield k(F) is algebraically closed in the field k(x_1,...,x_n). We prove that a rational function…

If there exists a completely bounded projection of B(H) onto a von Neumann algebra M on H, then M is injective. If there exists a bounded projection and M is properly infinite, the same conclusion holds.

We consider the algebraic combinatorics of the set of injections from a $k$-element set to an $n$-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair $(S_k \times S_n,…

In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and…

Given arbitrary $r\ge1$, we construct an HK$_r$-integrable function which is not P$_1$-integrable. This is an extension of Musial et al.\ construction published recently in [Musial, P., Skvortsov, V., Tulone, F.: The HK$_r$-integral is not…

It is obtained that every constructive (in A.A.Markov's sense) function is summable, if it is defined almost everywhere in the interval $[0,1]$ and integrable in Riemann's sense.

A function is differentially algebraic (or simply D-algebraic) if there is a polynomial relationship between some of its derivatives and the indeterminate variable. Many functions in the sciences, such as Mathieu functions, the Weierstrass…

We consider A-hypergeometric functions associated to normal sets in the plane. We give a classification of all point configurations for which there exists a parameter vector such that the associated hypergeometric function is algebraic. In…

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact…

Let ${\mathcal S}(\mathfrak g)$ be the symmetric algebra of a reductive Lie algebra $\mathfrak g$ equipped with the standard Poisson structure. If ${\mathcal C}\subset\mathcal S(\mathfrak g)$ is a Poisson-commutative subalgebra, then ${\rm…

A real algebraic link in the 3-sphere is defined as the zero locus in the 3-sphere of a real algebraic function from $\mathbb{R}^4$ to $\mathbb{R}^2$. A real algebraic open book decomposition on the 3-sphere is by definition the Milnor…