Univariate Real Root Isolation over a Single Logarithmic Extension of Real Algebraic Numbers

Abstract

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial B ∈ L[x], where L = Q[lg(α)] and α is a positive real algebraic number. The algorithm approximates the coefficients of B up to a sufficient accuracy and then solves the approximate polynomial. For this we derive worst case (aggregate) separation bounds. We also estimate the expected number of real roots when we draw the coefficients from a specific distribution and illustrate our results experimentally. A generalization to bivariate polynomial systems is also presented. We implemented the algorithm in C as part of the core library of mathematica for the case B ∈ Z[lg(q)][x] where q is positive rational number and we demonstrate its efficiency over various data sets.