Related papers: An Alternative Approach to Polynomial Mappings

In this work we provide a novel approach for computing the coefficients of the characteristic polynomial of a square matrix. We demonstrate that each coefficient can be efficiently represented by a set of circle graphs. Thus, one can employ…

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…

Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on…

This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…

This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.

We present a new algorithm for reconstructing an exact algebraic number from its approximate value using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on…

We consider several distinct characterizations of finite implication algebras. One of these leads to a new characterization of Boolean polymatroids.

This paper considers generalizations of open mappings, closed mappings, pseudo-open mappings, and quotient mappings from topological spaces to generalized topological spaces. Characterizations of these classes of mappings are obtained and…

Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper…

The different notions of matings of pairs of equal degree polynomials are introduced and are related to each other as well as known results on matings. The possible obstructions to matings are identified and related. Moreover the relations…

Holonomy R-matrices parametrized by finite-dimensional representations are constructed for quantized universal enveloping algebras of simple Lie algebras at roots of 1.

A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel…

In this paper, we introduce the concept of complementary edge ideals of graphs and study their algebraic properties and invariants.

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.

This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…

A new technique for approximating the entire solution set for a nonlinear system of relations (nonlinear equations, inequalities, etc. involving algebraic, smooth, or even continuous functions) is presented. The technique is to first plot…

In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial…

We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this…

In this note we prove that certain polynomial mappings $P_\mathfrak{g}^k(\mathbf{x}) \in \mathbf{Z}[\mathbf{x}]$ in $n$-variables obtained from simple complex Lie algebras $\mathfrak{g}$ of arbitrary rank $n \ge 1$, are exceptional.

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…