The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present-day computing. We briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms. We end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost.

The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial q(x) = qnx n +...+q0 with root separation ρ, coefficients |qn | ≥ 1 and |qi | ≤ 2 τ, it needs coefficient approximations to O(n(log(1/ρ)+τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ)+τ) 2) bit operations.

Abstract--We substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p(x). Our new algorithms save both Boolean and arithmetic sequential time, versus the previous best algorithms of SchSnhage [1], Pan [2], and Neff and Reif [3]. In parallel (NC) implementation, we dramatically decrease the number of processors, versus the parallel algorithm of Neff [4], which was the only NC algorithm known for this problem so far. Specifically, under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p(x) to the unit disc {x: Ix [ < 1}, our algorithms (which promise to be practically effective) approximate all the zeros of p(x) within the absolute error bound 2-b, by using order of n arithmetic operations and order of (b + n)n 2 Boolean (bitwise) operations (in both cases up to within polylogarithmic factors). The algorithms allow their optimal (work preserving) NC parallelization, so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or (b + n)n 2 Boolean processors. All the cited bounds on the computational complexity are within polylogarithmic factors from the optimum (in terms of n and b) under both arithmetic and Boolean models of computation (in the Boolean case, under the additional (realistic) assumption that n = O(b)).

Abstract. We show that if a polynomial filled Julia set has empty interior, then it is computable. 1.

The known record complexity estimates for approximating polynomial zeros rely on geometric constructions on the complex plane, which achieve initial approximation to the zeros and/or their clusters as well as their isolation from each other, and on the subsequent fast analytic refinement of the initial approximations. We modify Weyl's classical geometric construction for approximating all the n polynomial zeros in order to more rapidly achieve their strong isolation. For approximating the isolated zeros or clusters of zeros, we propose a new extension of Newton's iteration to yield quadratic global convergence (right from the start), under substantially weaker requirements to their initial isolation than one needs in the known algorithms.

A new formula for the approximation of root of polynomials with complex coe#cients is presented. For each simple root there exists a neighborhood such that given any input within this neighborhood, the formula generates a convergent sequence, computed via elementary operations on the input and the corresponding derivative values. Each element of the sequence is defined in terms of the quotient of two determinants, computable via a recursive formula. Convergence is proved by deriving an explicit error estimate. For special polynomials explicit neighborhoods and error estimates are derived that depend only on the initial error. In particular, the latter applies to the approximation of root of numbers. The proof of convergence utilizes a family of iteration functions, called the Basic Family; a nontrivial determinantal generalization of Taylor's theorem; a lower bound on determinants; Gerschgorin's theorem and Hadamard's inequality; as well as several new key results. The convergence resu...

1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of

. After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in G ottingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflosung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought. 1. Logic and Foundations of Mathematics in G ...

. Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a - p(a)Dm-2 (a)/Dm-1 (a)}, where Dm (a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each Dm (a) is also representable as a Toeplitz determinant. Except possibly for the locus of points equidistant to two distinct roots, given any input a, the Basic Sequence converges to a root of p. The roots of p partition the Euclidean plane into Voronoi regions. Under some regularity assumption (e.g. simplicity of the roots), for almost all inputs within the Voronoi polygon of a root, the corresponding Basic Sequence converges to that root. The discovery of the Basic Sequence, its error estimates, and several of its properties are consequences of our previous analysis of a fundamental family of iteration functions {Bm(z)}, called the Basic Family. Given any fixed m # 2 and an appropriat...

Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and non-fractal images created using the mathematical convergence properties of iteration functions. ” An individual image is called a “polynomiograph.” The word polynomiography is a combination of the word “polynomial ” and the suffix “-graphy. ” It is meant to convey the idea that it represents a certain graph of polynomials, but not in the usual sense of graphing, say a parabola for a quadratic polynomial. Polynomiographs are obtained using algorithms requiring the manipulation of thousands of pixels on a computer monitor. Depending upon the degree of the underlying polynomial, it is possible to obtain beautiful images on a laptop computer in less time than a TV commercial. Polynomials form a fundamental class of mathematical objects with diverse applications; they arise in devising algorithms for such mundane task as multiplying two numbers, much faster than the ordinary way we have all learned to do this task (FFT). According to the Fundamental Theorem of Algebra, a polynomial of degree n, with real or complex coefficients, has n zeros (roots) which may or may not be distinct. The task of approximation of the zeros of polynomials is a problem that was known to Sumerians (third millennium B.C.). This problem