THE SCHUR-COHN ALGORITHM REVISITED

The sim of the Schur-Cohn algorithm is to compute the number of roots of a complex polynomial in the open unit disk, each root counted with its multiplicity. Unfortunately, in its original form, it does not wor k with all polynomials. Using technics similar to those of the sub-res ultants, we construct a new sequence of polynomials, the Schur-Cohn su b-transforms. For this, we propose an algorithm in only O(d(2)) arithm etical operations (d being the degree of the polynomial studied), whic h is well adapted to computer algebra and supports specialization. We then show how to use bezoutians and hermitian forms to compute the num ber of roots in the unit disk with the help of the Schur-Cohn sub-tran sforms we have built. (C) 1998 Academic Press.