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 λ_{0},…,λ_{m−1}, μ_{0}, …, 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 ...

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