Results. Math. 69 (2016), 93–103 c  2015 Springer Basel 1422-6383/16/010093-11 published online September 24, 2015 DOI 10.1007/s00025-015-0490-y Results in Mathematics On the Convergence of Chebyshev’s Method for Multiple Polynomial Zeros Stoil Ivanov Abstract. In this paper we investigate the local convergence of Cheby- shev’s iterative method for the computation of a multiple polynomial zero. We establish two convergence theorems for polynomials over an ar- bitrary normed field. A priori and a posteriori error estimates are also provided. All of the results are new even in the case of simple zero. Mathematics Subject Classification. 65H04, 12Y05. Keywords. Chebyshev’s method, polynomial zeros, multiple zeros, local convergence, error estimates. 1. Introduction A lot of iterative methods for finding multiple zeros of nonlinear equations have been studied by Osada [1], Neta [2], Chun and Neta [3], Homeier [4], Proinov [5], Ren and Argyros [6], Sharma and Sharma [7], Bi et al. [8], Zhou et al. [9] and many others. On the other hand, there are a lot of scientific works devoted to numerical methods for zeros of polynomials (see, e.g. Pan [10], Kalantari et al. [11], McNamee [12], McNamee and Pan [13] and the references therein). Recently, Proinov [5, 14, 15] has presented general convergence theorems for the Picard iteration x k+1 = Tx k , k =0, 1, 2,..., (1) where T : D ⊂ X → X is an iteration function in a metric space X. In particu- lar, in [5, Sections 4–5] Newton’s method for polynomial zeros with multiplicity m (Schr¨ oder [16]),