Fraction-free computation of the unit-circle resultant with any singularity profile

Abstract

We show that an algorithm established in a recent publication in this journal to provide a fraction-free version for the Bistritz test can also be used to compute the unit-circle resultant. The consequence is an effective new algorithm to find this resultant for any complex polynomial that is also an efficient integer algorithm for polynomial with real or Gaussian integer coefficients. The algorithm ends always with the resultant expressed as the minimal degree multivariate polynomial of the polynomial's (indeterminate) coefficients. A simpler form of the algorithm, that equally reaches this expression in all circumstances, but is not guaranteed to stay over integers during possible intermediate abnormal gaps, is also suggested along with the simplest form that the algorithm takes in the strongly regular case.

Keywords:

• The Schur–Cohn problem
• Sylvester matrix
• T-Bezoutian
• subresultants
• discrete-time linear systems stability
• symmetric polynomials
• integer algorithm
• immittance algorithms

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Israel Science Foundation [1989/16].