Related papers: Complex Roots of Quaternion Polynomials
We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that…
Mixed trigonometric-polynomials (MTPs) are functions of the form $f(x,\sin{x}, \cos{x})$ with $f\in\mathbb{Q}[x_1,x_2,x_3]$. In this paper, an algorithm ``isolating" all the real roots of an MTP is provided and implemented. It automatically…
A fundamental problem in the theory of linearized and projective polynomials over finite fields is to characterize the number of roots in the coefficient field directly from the coefficients. We prove results of this type, of a recursive…
A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…
This paper presents new formulary solutions for quantic polynomial equations in general forms, where we present five solutions for any fifth degree polynomial equation with real coefficients, and thereby having the possibility to calculate…
The roots of a complex polynomial depend continuously on the coefficients; that is, an infinitesimal perturbation of the coefficients results in an infinitesimal perturbation of the roots. A short, straightforward proof of this is possible…
We introduce a new approach to isolate the real roots of a square-free polynomial $F=\sum_{i=0}^n A_i x^i$ with real coefficients. It is assumed that each coefficient of $F$ can be approximated to any specified error bound. The presented…
In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree $n \ge 2$ and height bounded by $H \ge 2$. The polynomial is…
Consider a random polynomial $G_n(z)=\xi_nz^n+...+\xi_1z+\xi_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\log(1+\log(1+|\xi_0|))$ has a slowly varying tail. Then the distribution of the complex roots of…
The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region…
We consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using (variants of) the continued fraction algorithm (CF). We introduce a novel way to compute a lower bound on the positive real…
We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or…
We present a quite efficient method to calculate the roots of Bernstein-Sato polynomial for a defining polynomial $f$ of a projective hypersurface $Z\subset{\mathbb P}^{n-1}$ of degree $d$ having only weighted homogeneous isolated…
Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring…
This paper derives numerical bounds for and implements the splitting circle method for finding roots of a univariate polynomial in the presence of fixed precision.
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 $n^{\text{th}}$…
We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion.
Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated "on the left" and "on the right". If the interpolation problem involves interpolation conditions…
This paper is concerned with the distribution in the complex plane of the roots of a polynomial sequence $\{W_n(x)\}_{n\ge0}$ given by a recursion $W_n(x)=aW_{n-1}(x)+(bx+c)W_{n-2}(x)$, with $W_0(x)=1$ and $W_1(x)=t(x-r)$, where $a>0$,…
Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.