Related papers: Algorithm to Compute a Kharitonov-Type Sector Cont…

In our quest for the design, the analysis and the implementation of a subdivision algorithm for finding the complex roots of univariate polynomials given by oracles for their evaluation, we present sub-algorithms allowing substantial…

We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…

The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes…

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by…

We prove a Roth type theorem for polynomial corners in the finite field setting. Let $\phi_1$ and $\phi_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \subset \mathbb F_p \times \mathbb F_p$ with…

This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian…

In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…

We propose a symbolic-numeric algorithm to count the number of solutions of a polynomial system within a local region. More specifically, given a zero-dimensional system $f_1=\cdots=f_n=0$, with $f_i\in\mathbb{C}[x_1,\ldots,x_n]$, and a…

V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots $z\in\C$ of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described…

Highly efficient and even nearly optimal algorithms have been developed for the classical problem of univariate polynomial root-finding (see, e.g., \cite{P95}, \cite{P02}, \cite{MNP13}, and the bibliography therein), but this is still an…

We depart from our approximation of 2000 of all root radii of a polynomial, which has readily extended Sch{\"o}nhage's efficient algorithm of 1982 for a single root radius. We revisit this extension, advance it, based on our simple but…

Some near-optimal polynomial root-finders of 2024-25, based on subdivision iterations, approximate all complex roots of a polynomial or all roots in a fixed Region of Interest in the complex plane. The iterations can be applied to a black…

The polynomials with quaternion coefficients have two kind of roots: isolated and spherical. A spherical root generates a class of roots which contains only one complex number $z$ and its conjugate $\bar{z}$, and this class can be…

In many applications, the information about the number of eigenvalues inside a given region is required. In this paper, we propose a contour-integral based method for this purpose. The new method is motivated by two findings. There exist…

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$…

We describe here the package {\tt subdivision\\_solver} for the mathematical software {\tt SageMath}. It provides a solver on real numbers for square systems of large dense polynomials. By large polynomials we mean multivariate polynomials…

We present a variation of the modular algorithm for computing the Hermite Normal Form of an $\OK$-module presented by Cohen, where $\OK$ is the ring of integers of a number field K. The modular strategy was conjectured to run in polynomial…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.