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.
the conditions of Lemma 10.1 is written: there are m + n complex numbers λ0,…,λm−1, μ0, …, μn−1 all not identically zero so that:
In other words, the polynomial family:
is linearly dependent in the vectorial space of the complex polynomials ...