A certified iterative method for isolated singular roots

In this paper we provide a new method to certify that nearby polynomial system has singular isolated root and compute its multiplicity structure. More precisely, given f=(f1,…,fN)∈C[x1,…,xn]N, present Newton iteration on an extended deflated locally converges, under regularity conditions, small deformation of f such deformed exact root. The simultaneously converges the coordinates coefficients so-called inverse describes structure at We use α-theory test quadratic convergence, give bounds size approximation error. approach relies analysis punctual Hilbert scheme, for which description. show in particular some strata can be rationally parametrized exploit these parametrizations certification. numerical experimentation how approximate computed as starting point iterations fast convergence with structure, certified by our criteria.