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

We investigate monogenicity and prime splitting in extensions generated by roots of iterated quadratic polynomials. Let $f(x)\in\mathbb{Z}[x]$ be an irreducible, monic, quadratic polynomial, and write $f^n(x)$ for the $n^{\text{th}}$…

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $\alpha_1,\alpha_2,\alpha_3$ are nonzero elements from $F$ such that $\alpha_1+\alpha_2+\alpha_3=0$,…

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…

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain…

In this thesis we study when a homogeneous polynomial $f$ decomposes or "splits" additively. Up to base change this means that it is possible to write $f = g + h$ where $g$ and $h$ are polynomials in independent sets of variables. This…

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…

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…

Multiplicative relations between the roots of a polynomial in $\mathbb{Q}[x]$ have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other…

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

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…

We investigate rank revealing factorizations of $m \times n$ polynomial matrices $P(\lambda)$ into products of three, $P(\lambda) = L(\lambda) E(\lambda) R(\lambda)$, or two, $P(\lambda) = L(\lambda) R(\lambda)$, polynomial matrices. Among…

For a positive real $\alpha$, we can consider the additive submonoid $M$ of the real line that is generated by the nonnegative powers of $\alpha$. When $\alpha$ is transcendental, $M$ is a unique factorization monoid. However, when $\alpha$…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

This paper studies so-called "null polynomials modulo m", i.e., polynomials with integer coefficients that satisfy f(x)=0 (mod m) for any integer x. The study on null polynomials is helpful to reduce congruences of higher degrees modulo m…

Multiresolution Matrix Factorization (MMF) was recently introduced as an alternative to the dominant low-rank paradigm in order to capture structure in matrices at multiple different scales. Using ideas from multiresolution analysis (MRA),…

We present a few factorizations of polynomials over finite fields. These factorizations are related to traces, compositions of polynomials and binomial coefficients. As a corollary we obtain a description of all irreducible polynomials…

Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a…

Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full…

Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate…