Related papers: The complexity and geometry of numerically solving…
In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton's method. In this paper we reconsider his algorithm in…
The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of $n$ complex polynomials in $n$ unknowns in time polynomial, on the…
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…
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.…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
Smale's alpha-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale…
We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real solutions to such systems, particularly…
This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate…
In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii) Geometry of Polynomials Level Sets, and iii) Shape…
We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…
We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound…
We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring.…
Cyclotomic polynomials play an important role in several areas of mathematics and their study has a very long history, which goes back at least to Gauss (1801). In particular, the properties of their coefficients have been intensively…