Complete numerical isolation of real roots in zero-dimensional triangular systems

We present a complete numerical algorithm of isolating all the real zeros of a zero-dimensional triangular polynomial system Fn ⊆ Z[x1, . . . , xn]. Our system Fn is general, with no further assumptions. In particular, our algorithm successfully treat multiple zeros directly in such systems. A key idea is to introduce evaluation bounds and sleeve bounds. We also present a much more efficient algorithm for zero-dimensional triangular systems without multiple roots. We implemented our algorithms and promising experimental results are shown.