2011/01/23 by Adam Strzeboński, Adam Strzebonski, Strzebonski, Adam +2
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Cryptography and Residue Arithmetic #F.2 #FOS: Computer and information sciences #FOS: Mathematics #I.1 #Mathematical Software (cs.MS) #Numerical Analysis (math.NA) #Numerical Methods and Algorithms #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #cs.MS #cs.NA #cs.SC #math.AG #math.NA
paper · pdf · doi:10.48550/arxiv.1101.4369
16 pages
openalex publication_date 2011/01/23 · arxiv created 2011/05/09 · arxiv updated 2011/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L=\QQ(α) is a simple algebraic extension of the rational numbers. We consider two approaches for tackling the problem. In the first approach using resultant computations we perform a reduction to a polynomial with integer coefficients. We compute separation bounds for the roots, and using them we deduce that we can isolate the real roots of Bα in \sOB(N10), where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for real roots and we prove that they are optimal, under mild assumptions. For isolating the roots we consider a modified Sturm's algorithm, and a modified version of \funcdescartes' algorithm introduced by Sagraloff. For the former we prove a complexity bound of \sOB(N8) and for the latter a bound of \sOB(N7). We implemented the algorithms in \funcC as part of the core library of \mathematica and we illustrate their efficiency over various data sets. Finally, we present complexity results for the general case of the first approach, where the coefficients belong to multiple extensions.