Verified Error Bounds for Isolated Singular Solutions of Polynomial Systems

In this paper, we generalize the algorithm described by Rump and Graillat, as well as our previous work on certifying breadth-one singular solutions of polynomial systems, to compute verified and narrow error bounds such that a slightly perturbed system is guaranteed to possess an isolated singular solution within the computed bounds. Our new verification method is based on deflation techniques using smoothing parameters. We demonstrate the performance of the algorithm for systems with singular solutions of multiplicity up to hundreds.