Related papers: On finite polynomial mappings

We prove that if E a subset of an n-dimensional manifold, then every continuous R^n-valued map on E that is zero-free on the interior of E can be approximated in the fine topology, and hence, in particular, in the uniform topology, by a…

We study Gauss maps of order $k$, associated to a projective variety $X$ embedded in projective space via a line bundle $L.$ We show that if $X$ is a smooth, complete complex variety and $L$ is a $k$-jet spanned line bundle on $X$, with…

In the paper "Geometry of polynomial mapping at infinity via intersection homology" the second and third authors associated to a given polynomial mapping $F : \C^2 \to \C^2$ with nonvanishing jacobian a variety whose homology or…

We show that the Zariski closure of the set of hypersurfaces of degree $M$ in ${\mathbb P}^{M}$, where $M\geq 5$, which are either not factorial or not birationally superrigid, is of codimension at least $\binom{M-3}{2}+1$ in the parameter…

Let G be a semi-simple algebraic group over a finitely generated field K of characteristic zero, and let \Gamma < G(K) be a finitely generated Zariski-dense subgroup. In this note we prove that the set of K-generic elements of \Gamma (whose…

Let $X$ be a variety defined over an algebraically closed field $k$ of characteristic $0$, let $N\in\mathbb{N}$, let $g:X\dashrightarrow X$ be a dominant rational self-map, and let $A:\mathbb{A}^N\to \mathbb{A}^N$ be a linear transformation…

For given polynomial map $F:\C^2\to\C^2$ with nonvanishing jacobian we associate a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set $Z\subset \mathbb{A}^{n-2}\subset \mathbb{A}^{n}$, we construct an…

Let $X$ be a complete variety of dimension $n$ over an algebraically closed field $\mathbf{K}$. Let $V_\bullet$ be a graded linear series associated to a line bundle $L$ on $X$, that is, a collection $\{V_m\}_{m\in\mathbb{N}}$ of vector…

We say that a metrizable space $M$ is a Krasinkiewicz space if any map from a metrizable compactum $X$ into $M$ can be approximated by Krasinkiewicz maps (a map $g\colon X\to M$ is Krasinkiewicz provided every continuum in $X$ is either…

Assume that there exists a smooth map between two closed manifolds $M^m\to N^k$ with only finitely many cone-like singular points, where $2\leq k\leq m\leq 2k-1$. If $(m,k)\not\in\{(2,2), (4,3), (5,3), (8,5), (16,9)\}$, then $M^m$ admits a…

The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial $f$ in the matrix algebra $M_n(K)$ is a vector space for every $n \in {\mathbb N}$. We prove this conjecture for the case where $f$ has degree…

Let F(X) be the set of finite nonempty subsets of a set X. We have found the necessary and sufficient conditions under which for a given function f:F(X)-->R there is an ultrametric on X such that f(A)=diam A for every A\in F(X). For finite…

We show that for a polynomial mapping F = (f_1,...,f_m): C^n \to C^m the Lojasiewicz exponent at infinity of F is attained on the set {z \in C^n : f_1(z)...f_m(z) = 0}

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a…

Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on…

A "tensor space" is a vector space equipped with a finite collection of multi-linear forms. In previous work, we showed that (for each signature) there exists a universal homogeneous tensor space, which is unique up to isomorphism. Here we…

Let $X$ be a (real or complex) infinite dimensional linear space. We establish conditions on a homogeneous polynomial $P$ on $X$ so that, if $W$ is any finite dimensional subspace of $X$ on which $P$ vanishes, then $P$ vanishes on an…

Suppose L is any finite algebraic extension of either the ordinary rational numbers or the p-adic rational numbers. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms…

We study the postcritically-finite (PCF) maps in the moduli space of complex polynomials $\mathrm{MP}_d$. For a certain class of rational curves $C$ in $\mathrm{MP}_d$, we characterize the condition that $C$ contains infinitely many PCF…