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

A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the…

The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the…

We investigate the automorphism groups of the algebraic curves \[ \mathcal{C}_d : y^d = \varphi_d(x), \] where $\varphi_d(x)$ denotes the Chebyshev polynomial of degree $d$, defined over a field $k$ with $p:=\operatorname{char}(k) \nmid…

We determine the generic complete eigenstructures for $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More precisely, we show that the set of $n \times n$ complex symmetric matrix polynomials of odd…

In this paper, we will show that if two meromorphic mappings $f$ and $g$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ have the same inverse images for $(2n+2)$ moving hyperplanes $\{a_i\}_{i=1}^{2n+2}$ with multiplicities counted to level…

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…

Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) =…

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components, and $\mathcal{C}(N)$ be the complex of curves of $N$. Suppose that $g + n \leq 3$ or $g + n \geq 5$. If $\lambda : \mathcal{C}(N) \rightarrow…

The order type of a point set in $R^d$ maps each $(d{+}1)$-tuple of points to its orientation (e.g., clockwise or counterclockwise in $R^2$). Two point sets $X$ and $Y$ have the same order type if there exists a mapping $f$ from $X$ to $Y$…

We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a…

A Thurston map is a branched covering map $f\colon S^2\to S^2$ that is postcritically finite. Mating of polynomials, introduced by Douady and Hubbard, is a method to geometrically combine the Julia sets of two polynomials (and their…

This paper presents two algorithms. In their simplest form, the first algorithm decides the existence of a pointed homotopy between given simplicial maps f, g from X to Y and the second computes the group $[\Sigma X,Y]^*$ of pointed…

We consider drawings of graphs in the plane in which vertices are assigned distinct points in the plane and edges are drawn as simple curves connecting the vertices and such that the edges intersect only at their common endpoints. There is…

We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…

A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial…

We show that if two tensor algebras of topological graphs are algebraically isomorphic, then the graphs are locally conjugate. Conversely, if the base space is at most one dimensional and the edge space is compact, then locally conjugate…

We introduce a new notion, called quasi-holomorphic maps. These are real smooth maps equipped with a structure that imitates the singularities and singularity stratifications of holomorphic maps on the source and target manifolds, although…

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…

We present explicit constructions of centrally symmetric 2-neighborly d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20 k^2 2^k.…