Related papers: Subtropical Real Root Finding
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the…
In this paper, we provide a new method to find all zeros of polynomials with quaternionic coefficients located on only one side of the powers of the variable (these polynomials are called simple polynomials). This method is much more…
This paper presents new six solutions for sixth degree polynomial equation in general forms basing on new theorems, where the possibility to calculate the six roots of any sixth degree equation nearly simultaneously. The proposed roots for…
Nonlinear systems of polynomial equations arise naturally in many applied settings, for example loglinear models on contingency tables and Gaussian graphical models. The solution sets to these systems over the reals are often positive…
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…
By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper…
We present an algorithm for isolating the roots of an arbitrary complex polynomial $p$ that also works for polynomials with multiple roots provided that the number $k$ of distinct roots is given as part of the input. It outputs $k$ pairwise…
The LC method described in this work seeks to approximate the roots of polynomial equations in one variable. This book allows you to explore the LC method, which uses geometric structures of Lines L and Circumferences C in the plane of…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with…
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…
This paper investigates the stratification of the discriminant hypersurface associated with a univariate polynomial via the number of its distinct complex roots. We introduce two novel approaches different from the one based on…
Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view on this problem leads to efficient preprocessing techniques, applying polyhedral…
We propose an approach to constructing iterative methods for finding polynomial roots simultaneously. One feature of this approach is using the fundamental theorem of symmetric polynomials. Within this framework, we reconstruct many of the…
We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these…
We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…
Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of…