Related papers: Roots multiplicity and square-free factorization o…

This work formalizes efficient Fast Fourier-based multiplication algorithms for polynomials in quotient rings such as $\mathbb{Z}_{m}[x]/\left<x^{n}-a\right>$, with $n$ a power of 2 and $m$ a non necessarily prime integer. We also present a…

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold…

Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. Assume that the factor algebra $\A=\mathbb{K}[x_1,...,x_m]/\I$…

We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the…

Matrix factorizations of a hypersurface yield a description of the asymptotic structure of minimal free resolutions over the hypersurface. We introduce a new concept of matrix factorizations for complete intersections that allows us to…

We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for…

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring $\mathbb{F}\langle x_1,x_2,\dots,x_n\rangle$ of polynomials over the field $\mathbb{F}$ and noncommuting variables $x_1,x_2,\ldots,x_n$.…

Let $f=f(x_1,\dots,x_m)$ be a multilinear polynomial over a field $F$. An $F$-algebra $A$ is said to be $f$-zpd ($f$-zero product determined) if every $m$-linear functional $\varphi\colon A^{m}\rightarrow F$ which preserves zeros of $f$ is…

A complex number $\alpha$ is said to satisfy the height reducing property if there is a finite set $F\subset \mathbb{Z}$ such that $\mathbb{Z}[\alpha]=F[\alpha]$, where $\mathbb{Z}$ is the ring of the rational integers. It is easy to see…

In this work several techniques to treat the partition function of the real scalar quartic quantum field theory on the Moyal plane is discussed. A factorisation approach requires the polytope volume for the diagonal subpolytope of symmetric…

The relationship between a polynomial's zeros and factors is well known. If a is a zero of f(x) then (x-a) is a factor of f(x). In this paper, we generalize this idea to polynomials of two variables and with real coefficients. We consider…

In this paper we factorize matrix polynomials into a complete set of spectral factors using a new design algorithm and we provide a complete set of block roots (solvents). The procedure is an extension of the (scalar) Horner method for the…

Let $R=K[x_{1},x_{2},\cdots, x_{m}]$ and $S=$ $K[y_{1},y_{2},\cdots, y_{m}]$ where $K$ is a field. %commutative ring with unity. In this paper, we propose a method showing how to obtain $3$-matrix factors for a given polynomial using either…

We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if $ f_1, \ldots, f_m $ are elements of the free associative algebra $…

Let $T=\alpha_0 I + \alpha_1 D + ...+\alpha_n D^n$, where $D$ is the differentiation operator and $\alpha_0\not= 0$, and let $f$ be a square-free polynomial with large minimum root separation. We prove that the roots of $Tf$ are close to…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative,…

Fractional polynomials are widely used for dose-response modelling, and recent Bayesian fractional polynomial work has renewed interest in this finite model class. We propose PMM-FP, a frequentist extension of Kunchenko's polynomial…

The remarkable properties of the real scalar quartic quantum field theory on the Moyal plane in combination with its similarity to the Kontsevich model make the model's partition function an interesting object to study. However, direct…

Frobenius companion matrices arise when we write an $n$-th order linear ordinary differential equation as a system of first order differential equations. These matrices and their transpose have very nice properties. By using the powers of…

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…