Related papers: The polyhedral type of a polynomial map on the pla…

Two proper polynomial maps $f_1, f_2 \colon \mathbb{C}^2 \longrightarrow \mathbb{C}^2$ are said to be \emph{equivalent} if there exist $\Phi_1, \Phi_2 \in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=\Phi_2 \circ f_1 \circ \Phi_1$. We…

Let $f,g:X \to Y$ be continuous mappings. We say that $f$ is topologically equivalent to $g$ if there exist homeomorphisms $\Phi : X\to X$ and $\Psi: Y\to Y$ such that $\Psi\circ f\circ \Phi=g.$ Let $X,Y$ be complex smooth irreducible…

Two proper polynomial maps $f_1, \,f_2 \colon \mC^n \lr \mC^n$ are said to be \emph{equivalent} if there exist $\Phi_1,\, \Phi_2 \in \textrm{Aut}(\mC^n)$ such that $f_2=\Phi_2 \circ f_1 \circ \Phi_1$. In this article we investigate proper…

Let $X, Y$ be smooth algebraic varieties of the same dimension. Let $f, g : X \to Y$ be finite polynomial mappings. We say that $f, g$ are equivalent if there exists a regular automorphism $\Phi \in Aut(X)$ such that $f = g\circ \Phi$. Of…

The following numerical control over the topological equivalence is proved: two complex polynomials in $n\not= 3$ variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of…

We describe the topology of a general polynomial mapping $f:\Bbb C^2\to\Bbb C^2.$

The study of the topology of polynomial maps originates from classical questions in affine geometry, such as the Jacobian Conjecture, as well as from works of Whitney, Thom, and Mather in the 1950-70s on diffeomorphism types of smooth maps.…

In this paper we study the topology of three different kinds of spaces associated to polynomial knots of degree at most $d$, for $d\geq2$. We denote these spaces by $\mathcal{O}_d$, $\mathcal{P}_d$ and $\mathcal{Q}_d$. For $d\geq3$, we show…

We describe the topology of a general polynomial mapping $F=(f, g):X\to\Bbb C^2$, where $X$ is a complex plane or a complex sphere.

For every polynomial f of degree n with no double roots, there is an associated family C(f) of harmonic algebraic curves, fibred over the circle, with at most n-1 singular fibres. We study the combinatorial topology of C(f) in the generic…

In a nutshell, we show that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent. Given two polytops $A\subseteq B$ and a number $k$, the Nested Polytope Problem (NPP) asks, if there exists a polytope…

We define two classes of topological infinite degree covering maps modeled on two families of transcendental holomorphic maps. The first, which we call exponential maps of type $(p,q)$, are branched covers and is modeled on transcendental…

In this paper, we consider polynomial correspondences $f (x, y)$ in $\mathbb{C}[x, y]$ of degree $d \ge 2$ in both the variables and obtain necessary and sufficient conditions in order that the equation $f (x, y) = 0$ can be expressed as…

We classify quadratic polynomial mappings from $\mathbb{C}^3$ to $\mathbb{C}^2$ up to affine equivalence and topological equivalence. This is a part of a larger project, we have already classified mappings from $\mathbb{C}^2$ to…

In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…

For given natural numbers $d_1,d_2$ let $\Omega_2(d_1,d_2)$ be the set off all polynomial mappings $F=(f,g):\mathbb{C}^2\to\mathbb{C}^2$ such that deg $f\le d_1$, deg $g\le d_2$. We say that the mapping $F$ is topologically stable in…

In this article we analyze the global diffeomorphism property of polynomial maps $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials…

We describe an effective method for computing the topological degree of continuous functions $R:S^2 \to S^2$, where $S^2$ is the Riemann sphere. Our approach generalizes the degree formula for rational functions of complex polynomials,…

We construct, for every even dimensional sphere $S^n$, $n >1$, and every odd integer $k$, a homogeneous polynomial map $f: S^{n}\to S^{n}$ of Brouwer degree $k$ and algebraic degree $2|k|-1$.

We show that, up to linear equivalence, there are only finitely many polynomial quadratic mappings $F:\mathbb{C}^2\to\mathbb{C}^n$ and $F:\mathbb{R}^2\to\mathbb{R}^n$. We list all possibilities.