Let $f, g$ be two polynomials in $S[t]$ where the coefficient ring is $S = \mathbb{C}[a_1..a_n]$. The resultant of $R(f,g)$ gives some measure as to whether or not $f$ and $g$ share a common factor.

My question is what happens once we set a factor of the resultant equal to zero.

For example, suppose that for some nonconstant polynomials $c, p, q$ (with $p$ and $q$ relatively prime) $f = cp + a_1F$

$g = cq + a_1G$.

Clearly, if we set $a_1 = 0$, then $f$ and $g$ contain a common factor $c$. So $a_1$ divides the resultant. But now if we set $a_1 = 0$, and then cancel the common factor, it is reasonable to next study the resultant of $p$ and $q.$

Question: What is the relationship between $R(p,q)$ and $R(f,g)$?

We first thought that an irreducible factor of $R(p,q)$ must be [(some factor of $R(f,g)$) modulo $a_1$]. This is not true (see example below), however I hope some version of it will be true. The biggest difficulty, is that once we set $a_1$ to zero, we have to divide our polynomials by a common factor, and it's very difficult to say what happens to either the roots of the polynomials, or to the Sylvester matrix - the main tools we have to study resultants.

In even the simplest case: If $R(f,g)$ is a monomial in $S$, is $R(p,q)$ a monomial?

Thanks for your help!

Example:

$f = t*t + at^3;$

$g = t*(t+b) + a(t^2 + 1);$

Then $R(f,g) = a^2(a^3-ba+a+1)$, but upon setting $a=0$, these factors become $0$ and $1$ respectively, whereas the resultant $R(p,q) = R(t,t+b) = b$.