Related papers: Generalized Loose Edge Factorization Theorems

We consider the problem of deterministically factoring a univariate polynomial over a finite field under the assumption of the Extended Riemann Hypothesis (ERH). This work builds upon the line of approach first explored by Gao in $2001$.…

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 define new generalized factorials in several variables over an arbitrary subset $\underline{S} \subseteq R^n,$ where $R$ is a Dedekind domain and $n$ is a positive integer. We then study the properties of the fixed divisor…

Following the works by Lin et al. (Circuits Syst. Signal Process. 20(6): 601-618, 2001) and Liu et al. (Circuits Syst. Signal Process. 30(3): 553-566, 2011), we investigate how to factorize a class of multivariate polynomial matrices. The…

A formula for factorizations of the full twist in the braid group $Br_{2m}$ depending on any four factorizations of the full twist in $Br_{m}$ is given. Applying this formula, a symplectic 4-manifold $X$ and two isotopic generic coverings…

Let $A = \mathbb{F}_p$ or $\mathbb{Z}_p$, and let $R = A[[x_1]][[x_2; \sigma_2, \delta_2]]\dots[[x_n;\sigma_n,\delta_n]]$, an iterated local skew power series ring over $A$. Under mild conditions, we show that (multiplicative) monomial…

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and…

In this paper, we study Grothendieck polynomials from a combinatorial viewpoint. We introduce the factorial Grothendieck polynomials, analogues of the factorial Schur functions and present some of their properties, and use them to produce a…

The Cholesky factorization of the moment matrix is considered for the generalized Charlier, generalized Meixner, and Gauss hypergeometric discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of…

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…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

We prove a Fej\'er-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on $\mathscr{W}\times\mathfrak{Y}$, where $\mathscr{W}$ is either the free semigroup $\langle x \rangle_g$ or the free product…

We construct real polarizable Hodge structures on the reduced leafwise cohomology of K\"ahler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's K\"ahlerian…

Generalized polynomials are mappings obtained from the conventional polynomials by the use of operations of addition, multiplication and taking the integer part. Extending the classical theorem of H. Weyl on equidistribution of polynomials,…

We consider the problem of interpolating functions partially defined over a distributive lattice, by means of lattice polynomial functions. Goodstein's theorem solves a particular instance of this interpolation problem on a distributive…

In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However,…

Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many…

In continuation to our recent work on noncommutative polynomial factorization, we consider the factorization problem for matrices of polynomials and show the following results. (1) Given as input a full rank $d\times d$ matrix $M$ whose…

We study actions by filtered automorphisms on classical generalized Weyl algebras (GWAs). In the case of a defining polynomial of degree two, we prove that the fixed ring under the action of a finite cyclic group of filtered automorphisms…

We discuss a consequence of Green and Tao's factorisation theorem for polynomial orbits on nilmanifolds, adjusted to the requirements of certain arithmetic applications. More precisely, we prove a generalisation of Theorem 16.4, Acta Arith.…