Computing Roots of Polynomials using Bivariate Quadratic Clipping

The paper presents a new algorithm to compute all real roots of a system of two bivariate polynomial equations over a given domain. Using the Bernstein-Bézier representation, we compute the best linear approximant and the best quadratic approximant of the two polynomials with respect to the L norm. Using these approximations and bounds on the approximation errors, we obtain a linear strip bounding the first polynomial and a quadratic strip bounding the second polynomial. By intersecting these strips, we obtain a new reduced subdomain enclosing the roots. This algorithm is combined with bisection steps, in order to isolate the roots. It is applied iteratively until a certain accuracy is obtained. Experimental results are also presented. Mathematics Subject Classification (2000). Primary 12D10; Secondary 65H20.