Related papers: On polynomial mappings from the plane to the plane

Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite…

We consider polynomial maps described by so-called "(multivariate) linearized polynomials". These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps…

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.

In this paper, we introduce some known map projections from a model of the Earth to a flat sheet of paper or map and derive the plotting equations for these projections. The first fundamental form and the Gaussian fundamental quantities are…

Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $\phi:\A\to \A$ that preserve zeros of $f$. Under certain technical…

This paper presents an alternative approach to simplify the proofs of some important results related to polynomial mappings in Computational Algebraic Geometry such as Polynomial Implicitization, Image Closure and some properties of the…

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…

We present a list of all polynomial dominanting maps of the complex plane with branched value curve isomorphic to the complex line, up to polynomial automorphisms.

We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The…

The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…

Let M be an oriented 3-manifold. For a generic f \in C^ \infty(M,R^3), there is a discrete set of swallowtail critical points. In that case, at any swallowtail point p there exists a well-oriented coordinate system centered at p, and a…

We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…

Two continuous maps $f, g : \mathbb{C}^2\to\mathbb{C}^2$ are said to be topologically equivalent if there exist homeomorphisms $\varphi,\psi:\mathbb{C}^2\to\mathbb{C}^2$ satisfying $\psi\circ f\circ\varphi = g$. It is known that there are…

This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

These results stem from a course on ring theory. Quantum planes are rings in two variables $x$ and $y$ such that $yx=qxy$ where $q$ is a nonzero constant. When $q=1$ a quantum plane is simply a commutative polynomial ring in two variables.…

For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with…

We construct a class of positive linear maps on matrix algebras. We find conditions when these maps are atomic, decomposable and completely positive. We obtain a large class of atomic positive linear maps. As applications in quantum…

We show that up to linear equivalence, there is only finitely many polynomial quadratic mappings $f:\Bbb C^2\to\Bbb C^2$ and $f:\Bbb R^2\to \Bbb R^2.$ We list all possibilities.