Algebraic computation of the number of zeros of a complex polynomial in the open unit disk by a polynomial representation

We present a general method for the exact computation of the number of zeros of a complex polynomial inside the unit disk, assuming that the polynomial does not vanish on the unit circle. We prove the existence of a polynomial sequence. This sequence involves a reduced number of arithmetic operations and the growth of intermediate coefficients remains controlled. We study the singular case where the constant term of a polynomial of this sequence vanishes.