A Fast Version of the Schur–Cohn Algorithm

Abstract

In this paper we describe a fast algorithm for counting the number of roots of complex polynomials in the open unit disk, classically called the Schur–Cohn problem. For degree d polynomials our bound is of order dlog2d in terms of field operations and comparisons, but our approach nicely allow us to integrate the bit size as a further complexity parameter as well. For bit size σ of input polynomials of Z[X] our running time is of order d2σ·log(dσ)·loglog(dσ)·log(d) in the multi-tape Turing machine model, thus improving all previously proposed approaches.