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 \(B \in L[x]\), where \(L=\mathbb {Q} [ \lg (\alpha )]\) and \(\alpha \) 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 \({\mathtt {C}}\) as part of the core library of mathematica for the case \(B \in \mathbb {Z} [ \lg (q)][x]\) where q is positive rational number and we demonstrate its efficiency over various data sets.