Comparing Real Algebraic Numbers of Small Degree

We study polynomials of degree up to 4 over the rationals or a computable real subfield. Our motivation comes from the need to evaluate predicates in nonlinear computational geometry efficiently and exactly. We show a new method to compare real algebraic numbers by precomputing generalized Sturm sequences, thus avoiding iterative methods; the method, moreover handles all degenerate cases. Our first contribution is the determination of rational isolating points, as functions of the coefficients, between any pair of real roots. Our second contribution is to exploit invariants and Bezoutian subexpressions in writing the sequences, in order to reduce bit complexity. The degree of the tested quantities in the input coefficients is optimal for degree up to 3, and for degree 4 in certain cases. Our methods readily apply to real solving of pairs of quadratic equations, and to sign determination of polynomials over algebraic numbers of degree up to 4. Our third contribution is an implementation in a new module of library synaps v2.1. It improves significantly upon the efficiency of certain publicly available implementations: Rioboo’s approach on axiom, the package of Guibas-Karavelas-Russel [11], and core v1.6, maple v9, and synaps v2.0. Some existing limited tests had shown that it is faster than commercial library leda v4.5 for quadratic algebraic numbers.