We address the problem of computing ideals of polynomials which vanish at a finite set of points. In particular we develop a modular Buchberger-Moeller algorithm, best suited for the computation over QQ, and study its complexity; then we describe a variant for the computation of ideals of projective points, which uses a direct
approach and a new stopping criterion. The described algorithms are implemented in cocoa, and we report some experimental timings
Let X be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal I(X) independent of the data uncertainty. We present a method to compute a polynomial basis B of I(X) which exhibits structural stability, that is, if e X is any set of points differing only slightly from X, there exists a polynomial set e B structurally similar to B, which is a basis of the perturbed ideal I( e X).
We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newton's Iteration. Our method exhibits quadratic convergence when refining isolating intervals of simple roots of polynomials (and other well-behaved functions). We assume the use of arbitrary precision rational arithmetic. Unlike Newton's Iteration our method does not need to evaluate the derivative.
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