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.

� � � We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n +1 complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context. Further complexity theory is developed. 1. Introduction. 1A. Bezout’s Theorem Revisited. Let f: � n+1 → � n be a system of homogeneous polynomials f =(f1,...,fn), deg fi = di, i=1,...,n. The linear space of such f is denoted by H (d) where d = (d1,...,dn). Consider the

this paper is to show that the problem of finding approximately a zero of a polynomial system of equations can be solved in polynomial time, on the average. The number of arithmetic operations is bounded by cN

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There are a variety of needs for the discretization of a manifold—statistical sampling, quadrature rules, starting points for Newton’s method, computer-aided design, interpolation schemes, finite element tessellations—to name but a few. So let us assume we are given a d-dimensional manifold A in the Euclidean space R d′ and wish to determine, say, 5000 points that “represent A”. How can we go about this if A is described by some geometric property or by some parametrization of the unit cube U d: = [0, 1] d in R d? Naturally, we must be guided by the particular application in mind. For an historical perspective as well as a brief motivational journey, let’s look at the simple case when A is the interval [−1, 1] ⊂ R. One obvious choice for N points that discretize A is the set of equally spaced points xk,N = −1 +

Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S0. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1,..., fp and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f1,..., fp.

this paper asserts:

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)).

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo [10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in [10] yields a method for symbolically solving a zerodimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zerodimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined affine degree of the equation system. Replacing the affine degree of the equation system by a suitably defined real degree of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straightline program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.

We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V = 1=r s ; where s ? 0 and r is Euclidean distance. With Ed(s; N) denoting the minimal energy for such N-point arrangements we obtain bounds (valid for all N) for E d (s; N) in the cases when 0 ! s ! d and 2 d ! s. For s = d, we determine the precise asymptotic behavior of Ed(d; N) as N !1. As a corollary, lower bounds are given for the separation of any pair of points in an N-point minimal energy configuration, when s d 2. For the unit sphere in R 3 (d = 2), we present two conjectures concerning the asymptotic expansion of E 2 (s; N) that relate to the zeta function iL(s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of iL (s) when 0 ! s ! 2 (the divergent case). 1 Introduction and ...