Related papers: Continuity of the Perron Root
The nonnegative inverse eigenvalue problem (NIEP) asks which lists of $n$ complex numbers (counting multiplicity) occur as the eigenvalues of some $n$-by-$n$ entry-wise nonnegative matrix. The NIEP has a long history and is a known hard…
Let $D$ be a square-free integer. Under certain conditions on $D$, we characterize non-constant arithmetic progressions of squares over quadratic extensions of $\mathbb{Q}(\sqrt{D})$.
Some easily verifiable sufficient conditions for the nonexistence of iterative roots for multifunctions on arbitrary nonempty sets are presented. Typically if the graph of the multifunction has a distinguished point with a relatively large…
Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have Perron eigenpair and Perron-Frobenius eigenpair. The Collatz method…
In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide…
We extend our previous work on Poisson-like formulas for subresultants in roots to the case of polynomials with multiple roots in both the univariate and multivariate case, and also explore some closed formulas in roots for univariate…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…
In this note, we characterize matrix functions that preserve the strong Perron-Frobenius property using the real Jordan canonical form of a real matrix.
When $A$ is a matrix with all eigenvalues in the disk $|z-1|<1$, the principal $p$th root of $A$ can be computed by Schr\"oder's method, among many other methods. In this paper we present a further study of Schr\"oder's method for the…
For real polynomials with (sparse) exponents in some fixed set, \[ \Psi(t)=x+y_1t^{k_1}+\ldots +y_L t^{k_L}, \] we analyse the types of root structures that might occur as the coefficients vary. We first establish a stratification of roots…
For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is a central variable. A pseudo-root of a polynomial p(t) is an element x in R, for which there exist polynomials q(t) and s(t) such that…
Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…
Extending the classical result that the roots of a polynomial with coefficients in $\mathbf{C}$ are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if $\mathcal{F} = \{f_{\lambda}…
Component-wise accurate algorithms for computing the principal square root of an M-matrix are designed in terms of triplet representations. A triplet representation of an M-matrix $A$ is the triple $(P, {\bf u},{\bf v})$, where the matrix…
This paper studies the stability of discrete-time polynomial dynamical systems on hypergraphs by utilizing the Perron-Frobenius theorem for nonnegative tensors with respect to the tensors Z-eigenvalues and Z-eigenvectors. Firstly, for a…
The Perron-Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of…
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 $n^{\text{th}}$…
The structured pseudospectra of a matrix A are sets of complex numbers that are eigenvalues of matrices X which are near to A and have the same entries as A at a fixed set of places. The sum of multiplicities of the eigenvalues of X inside…
Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result…