Polynomial Real Root Isolation Research Articles

On the maximum computing time of the bisection method for real root isolation

The bisection method for polynomial real root isolation was introduced by Collins and Akritas in 1976. In 1981 Mignotte introduced the polynomials Aa,n(x)=xn−2(ax−1)2, a an integer, a≥2 and n≥3. First we prove that if a is odd then the computing time of the bisection method when applied to Aa,n dominates n5(log⁡d)2 where d is the maximum norm of Aa,n. Then we prove that if A is any polynomial of degree n with maximum norm d then the computing time of the bisection method, with a minor improvement regarding homothetic transformations, is dominated by n5(log⁡d)2. It follows that the maximum computing time of the bisection method is codominant with n5(log⁡d)2.

On the computing time of the continued fractions method

The maximum computing time of the continued fractions method for polynomial real root isolation is at least quintic in the degree of the input polynomial. This computing time is realized for an infinite sequence of polynomials of increasing degrees, each having the same coefficients. The recursion trees for those polynomials do not depend on the use of root bounds in the continued fractions method. The trees are completely described. The height of each tree is more than half the degree. When the degree exceeds one hundred, more than one third of the nodes along the longest path are associated with primitive polynomials whose low-order and high-order coefficients are large negative integers. The length of the forty-five percent highest order coefficients and of the ten percent lowest order coefficients is at least linear in the degree of the input polynomial multiplied by the level of the node. Hence the time required to compute one node from the previous node using classical methods is at least proportional to the cube of the degree of the input polynomial multiplied by the level of the node. The intervals that the continued fractions method returns are characterized using a matrix factorization algorithm.

Computing the asymptotes for a real plane algebraic curve

The purpose of this paper is to present an algorithm for computing all the asymptotes of a real plane algebraic curve. By this algorithm, all the asymptotes of a real plane algebraic curve may be represented via polynomial real root isolation.

Determination of the tangents for a real plane algebraic curve

The purpose of this paper is to present an algorithm for computing the tangents of a real plane algebraic curve. By this algorithm, all the slopes of the tangents to a real plane algebraic curve at a particular point may be accurately represented via polynomial real root isolation.

New bounds for the Descartes method

We give a new bound for the number of recursive subdivisions in the Descartes method for polynomial real root isolation. Our proof uses Ostrowski’s theory of normal power series from 1950 which has so far been overlooked in the literature. We combine Ostrowski’s results with a theorem of Davenport from 1985 to obtain our bound. We also characterize normality of cubic polynomials by explicit conditions on their roots and derive a generalization of one of Ostrowski’s theorems.

New bounds for the Descartes method

We give a new bound for the number of recursive subdivisions in the Descartes method for polynomial real root isolation. Our proof uses Ostrowski's theory of normal power series from 1950 which has so far been overlooked in the literature. We combine Ostrowski's results with a theorem of Davenport from 1985 to obtain our bound. We also characterize normality of cubic polynomials by explicit conditions on their roots and derive a generalization of one of Ostrowski's theorems. The poster is based on a paper that is to appear in the Journal of Symbolic Computation [1]. In addition to the results of the paper the poster presents facsimiles of pertinent mathematical works in French, German, and English that span a period of 400 years. We use color-coding to relate the historical results to our theory.

Trees and jumps and real roots

When the nodes of a tree are visited in depth-first order there are occasional jumps from a deeper level of the tree to a higher level. On the set of all full binary trees with a given number of nodes there is about 1 jump for every 2 internal nodes, and the average jump distance is about 2 levels. These averages are close to averages for trees that arise in polynomial real root isolation.

Interval Arithmetic in Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition requires many very time consuming operations, including resultant computation, polynomial factorization, algebraic polynomial gcd computation and polynomial real root isolation. We show how the time for algebraic polynomial real root isolation can be greatly reduced by using interval arithmetic instead of exact computation. This substantially reduces the overall time for cylindrical algebraic decomposition.

A short note on a new method for polynomial real root isolation

The purpose of this short note is to make known to our community some of the results of my Ph.D. Dissertation [1]. More details can also be found elsewhere [2], [3].