Complete Numerical Isolation of Real Zeros in General Triangular Systems∗

We consider the computational problem of isolating all the real zeros of a zero-dimensional triangular polynomial system Fn ⊆ Z[x1, . . . , xn]. We present a complete numerical algorithm for this problem. Our system Fn is general, with no further assumptions. In particular, our algorithm is the first to successfully treat multiple zeros in such systems. A key idea is to introduce evaluation and separation bounds, which are used in conjunction with sleeve bounds to detect zeros of even multiplicity. Our algorithm assumes a computational model of bigfloats with exact ring operations. We have implemented our algorithm and promising experimental results are shown.