Related papers: Roots multiplicity without companion matrices
In this paper some algorithms will be presented which can be used for the calculation of zeros of polynomials and eigenvalues of polynomial matrices with a multiplicity larger than one. The numerical values calculated with MATLAB are used…
We introduce a class of dynamical systems of algebraic origin, consisting of self-interacting irreducible polynomials over a field. A polynomial f is made to act on a polynomial g by mapping the roots of g. This action identifies a new…
Following a systematic analysis of existing results, we investigate when complete interlacing between the zeros of distinct polynomial sequences, $\{\mathcal{P}_n\}$ and $\{\mathcal{G}_n\}$ can be achieved by using a naturally arising extra…
We describe a simple approach to factorize non-commutative (nc) polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear…
We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph…
In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.
We give an algorithm for factoring quadratic polynomials over any UFD, Z in particular. We prove the correctness of this algorithm and give examples over Z and Z[i].
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 work presents a novel matrix-based method for constructing an approximation Hessian using only function evaluations. The method requires less computational power than interpolation-based methods and is easy to implement in matrix-based…
We introduce a new graph polynomial in two variables. This ``interlace'' polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant…
Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we…
Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose…
In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…
A pattern of interpolation nodes on the disk is studied, for which the interpolation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike…
Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\mathbb{F}_q$ of…
The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available…
A subalgebraic approximation algorithm is proposed to estimate from a set of time series the parameters of the observer representation of a discrete-time polynomial system without inputs which can generate an approximation of the observed…
We present an extension of state-feedback pole placement for quaternionic systems, based on companion forms and the Ackermann formula. For controllable single-input quaternionic LTI models, we define a companion polynomial that annihilates…
We analyze in the Landau gauge mixing of bosonic fields in gauge theories with exact and spontaneously broken symmetries, extending to this case the Lehmann-Symanzik-Zimmermann (LSZ) formalism of the asymptotic fields. Factorization of…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…