Related papers: Factorizations of tropical and sign polynomials
To directed graphs with unique sink and source we associate a noncommutative associative alsgebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when…
We develop a fast algorithm for computing the bound of an Ore polynomial over a skew field, under mild conditions. As an application, we state a criterion for deciding whether a bounded Ore polynomial is irreducible, and we discuss a…
H. Lenstra has pointed out that a cubic polynomial of the form (x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of {0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor divides one summand but not the…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism…
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…
In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real ("motion polynomials"), we only require the absence of real…
We consider systems of strict multivariate polynomial inequalities over the reals. All polynomial coefficients are parameters ranging over the reals, where for each coefficient we prescribe its sign. We are interested in the existence of…
We identify a recursive structure among factorizations of polynomial values into two integer factors. Polynomials for which this recursive structure characterizes all non-trivial representations of integer factorizations of the polynomial…
We compute the factorization homology of a polynomial algebra over a compact and closed manifold with trivialized tangent bundle up to weak equivalence in a new way. This calculation is based on the model of a graph complex and an explicit…
We generalize the classical lifting and recombination scheme for rational and absolute factorization of bivariate polynomials to the case of a critical fiber. We explore different strategies for recombinations of the analytic factors,…
Motion polynomials are a specific type of polynomial over a Clifford algebra that can conveniently describe rational motions. There exists an algorithm for the factorization of motion polynomials that works in generic cases. It hinges on…
The number of positive solutions of a system of two polynomials in two variables defined in the field of real numbers with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions.…
We use character polynomials to obtain a positive combinatorial interpretation of the multiplicity of the sign representation in irreducible polynomial representations of $GL_n(\mathbb{C})$ indexed by two-column and hook partitions. Our…
We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…
Tropical algebra emerges in many fields of mathematics such as algebraic geometry, mathematical physics and combinatorial optimization. In part, its importance is related to the fact that it makes various parameters of mathematical objects…
The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The…
In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials
We provide an analogue of Wedderburn's factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring polynomials of degree $n$ with $n$ conjugacy classes of…
This paper deals with function field analogues of famous theorems of Laudau which counted the number of integers which have $t$ prime factors and R. Hall which researched the distribution of divisors of integers in residue classes.\;We…