Digital Filters Design for Signal and Image Processing

10.3. Appendix: resultant of two polynomials

Let:

two non-constant polynomials of variable z and with complex coefficients. We assume, without any loss of general information, that:

We try to discover at what conditions the two polynomials P and Q admit a zero, or at least a non-constant factor, that they share. The results that we can establish are based on the lemma shown below. A demonstration of this lemma is found in [BEN 99].

LEMMA 10.1. – the polynomials P and Q have a zero in common if and only if there exist L and M non-constant polynomials in z that satisfy the following conditions:

LP + MQ = 0 and deg(M) < n and deg(L) < m.

By writing:

and:

the conditions of Lemma 10.1 is written: there are m + n complex numbers λ₀,…,λ_m−1, μ₀, …, μ_n−1 all not identically zero so that:

In other words, the polynomial family:

is linearly dependent in the vectorial space of the complex polynomials ...